JP2011022889A - Method for simulating electromagnetic wave propagation - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a new method for simulating electromagnetic wave propagation which calculates behaviors of electromagnetic waves by using a computer when electromagnetic waves are incident on an aggregation including media that include a plurality of particles and by which simulation is performed even when particles are non-spherical, the particles are condensed, or distributed unevenly. <P>SOLUTION: The method for simulating electromagnetic wave propagation is for calculating behaviors of electromagnetic waves when the electromagnetic waves are incident on the aggregation including media that include a plurality of particles by using a computer, and calculates the behaviors of the electromagnetic waves when the electromagnetic waves are incident on the aggregation in which the particles are distributed at random or according to a certain rule by using an FDTD (Finite Difference Time Domain) method and radiative transfer equation. <P>COPYRIGHT: (C)2011,JPO&INPIT

Description

  The present invention relates to an electromagnetic wave propagation simulation method for calculating the behavior when an electromagnetic wave is incident on a medium containing a large number of particles using a computer.

  The electromagnetic wave propagation simulation method that uses a computer to calculate the behavior of light when electromagnetic waves are incident on a medium containing many particles is designed for diffuser plates such as displays and color filters, as well as ink, paint, plastic, dyeing, etc. It is useful for the design of various measuring devices in the coloring industry, remote sensing, meteorological science and medical fields.

  In particular, the liquid crystal display includes a diffusion plate and a color filter. Since the color filter is formed by dispersing pigment particles in a medium made of a transparent resin, light is irradiated on the resin medium containing many pigment particles. In this case, light behavior such as light scattering, diffraction, and absorption by pigment particles is calculated using a computer, and it is required to obtain data for designing an optimal diffuser plate and color filter for displays. Yes.

  Conventionally, simulation by Mie scattering has been performed as an electromagnetic wave propagation simulation method for calculating the behavior when an electromagnetic wave is incident on a medium containing a large number of particles using a computer. However, when the concentration of particles in the medium is large or the thickness of the medium is large and multiple scattering cannot be ignored, calculation is impossible.

  Therefore, a simulation method based on the MOM (Method of Moments) method for obtaining a statistical average of the positions and attributes of particles dispersed statistically in advance has been proposed (see, for example, Non-Patent Document 1). When applied to particles having a large absolute value of complex dielectric constant, such as metal particles, the accuracy was insufficient.

  In addition, there is a radial transfer method (RT method) as a method for calculating multiple scattering. This is a method for calculating the electromagnetic wave propagation characteristics of the target space by expressing the scattering characteristics of individual scattering particles or a minute area of the space with a phase function. When the scattering particles are spherical, the phase function can be obtained using Mie's formula, etc., but when there are non-spherical particles, aggregates, or uneven distribution, the error becomes large and the application becomes difficult. There was a problem of becoming.

`` Monte Carlo Simulation of Electromagnetic Wave Propagation in Dense Random Media with Dielectric Spheroids '', IEICE Trans. Electron., Vol. E83-C, No. 12, December 2000, p1797-1801

  Accordingly, an object of the present invention is an electromagnetic wave propagation simulation method for calculating, using a computer, the behavior of an electromagnetic wave when the electromagnetic wave is incident on an aggregate composed of a medium including a plurality of particles, where the particles are non-spherical. Another object of the present invention is to provide a new electromagnetic wave propagation simulation method capable of performing simulation even when particles are aggregated or unevenly distributed.

In order to solve the above-mentioned problems, the present inventor has conducted intensive studies on an electromagnetic wave propagation simulation method for calculating the behavior when an electromagnetic wave is incident on a medium containing a large number of particles using a computer. By incorporating this calculation method into the simulation, it was found that when the particles are non-spherical, the simulation is possible even when the particles are aggregated or unevenly distributed. The invention has been completed.
That is, the present invention provides the following <1> to <5> inventions.

<1>
An electromagnetic wave propagation simulation method for calculating the behavior of an electromagnetic wave when an electromagnetic wave is incident on an aggregate composed of a medium including a plurality of particles, using a computer, and the electromagnetic wave is generated in an aggregate in which particles are randomly distributed according to a certain rule. An electromagnetic wave propagation simulation method characterized in that the behavior of an electromagnetic wave when an incident light enters is calculated using an FDTD method and a radial transfer acquisition.
<2>
<1> The electromagnetic wave propagation simulation method according to <1>, wherein the radial transfer equation is a radial transfer equation with four or more light beams.
<3>
The electromagnetic wave propagation simulation method according to <1> or <2>, wherein the calculation target space is a flat plate having a constant thickness and an infinite width.
<4>
The electromagnetic wave propagation simulation method according to any one of <1> to <3>, wherein the incident wave is an electromagnetic wave having low coherence.
<5>
A computer-readable recording medium, an electromagnetic wave propagation simulation method that uses a computer to calculate the behavior of an electromagnetic wave when the electromagnetic wave is incident on an aggregate composed of a medium containing a plurality of particles. Alternatively, the scattering characteristics of the aggregates distributed according to a certain rule are obtained by the FDTD method, and the electromagnetic wave propagation characteristics in a space where a large number of aggregates exist are calculated by the radial transfer acquisition using the scattering characteristics of the aggregates. The recording medium which stored the program for performing the electromagnetic wave propagation simulation characterized by the above-mentioned.

  According to the present invention, when performing electromagnetic wave propagation simulation for calculating the behavior when an electromagnetic wave is incident on an aggregate composed of a medium containing a large number of particles using a computer, the particles are aggregated when the particles are non-spherical. Even if the particles are distributed or unevenly distributed, simulation is possible, and even if the concentration of particles in the medium is high, simulation can be performed within a realistic time range. In particular, the behavior of reflection, transmission, absorption, scattering, and the like of electromagnetic waves when the size and shape of particles are arbitrarily set can be simulated whether the electromagnetic waves are coherent or non-coherent. Therefore, it can be used for designing optical members such as color filters and diffusers for displays, notch filters having absorption or transmission performance in a narrow wavelength range, and new optical members having special optical performance, and remote sensing. The present invention is extremely useful industrially because it can also be applied to measurement and inspection related to medical treatment.

FIG. 6 is a diagram showing a model in which silver spherical particles having a radius of 10 nm are randomly distributed using a Monte Carlo method so as to form a cluster having a radius of 100 nm in a medium having a refractive index of 1.5. The figure which shows the result of the change by the wavelength of an absorption cross section and a scattering cross section in Example 1 which performed the simulation by the method of this invention targeting the model in which the silver spherical particle | grains of a radius of 10 nm formed the cluster. In Example 2 in which simulation was performed by the method of the present invention for a model in which cylindrical silver particles having a radius of 10 nm and a height of 30 nm formed clusters, the results of changes in the absorption cross section and the scattering cross section depending on the wavelength are shown. FIG. In Example 3 in which simulation was performed by the method of the present invention for a model in which silver cylindrical particles having a radius of 10 nm and a height of 30 nm formed clusters, the results of changes in the absorption cross section and the scattering cross section depending on the wavelength are shown. FIG. The figure which shows the result of the change by the wavelength of an attenuation cross section and a scattering cross section in the comparative example 1 which performed the simulation about the spherical particle of silver and silver cylindrical particle | grains using the conventional Mie theory. A simulation of the amount of scattered light and the amount of absorbed light when a reflecting plate having a reflectance of 0.8 exists on the surface opposite to the incident surface of a flat plate having a thickness of 100 μm and the incident light is perpendicularly incident on the incident surface of the flat plate. In Example 4 which performed this by the method of this invention, the figure which shows the result of the change by the wavelength of scattered light amount and absorbed light amount. For a model in which spherical silver particles with a radius of 100 nm are contained in a 3 μm thick flat plate made of a resin with a refractive index of 1.5, coherent light and non-coherent light are mixed equally. The figure which shows the result of Example 5 which performed the simulation of this invention about the case where the performed light was perpendicularly incident on the flat plate. For a model in which spherical silver particles with a radius of 10 nm form clusters in a 3 μm thick flat plate made of resin with a refractive index of 1.5, coherent light and non-coherent light are equally divided. The figure which shows the result of Example 6 which performed the simulation of this invention about the case where the light mixed in (1) injects perpendicularly | vertically on the flat plate. Coherent light and incoherence are targeted for a model in which cylindrical silver particles with a radius of 10 nm and a length of 30 nm form clusters in a 3 μm thick flat plate made of a resin with a refractive index of 1.5. The figure which shows the result of Example 7 which performed the simulation of this invention about the case where the light of which the light of 1 part was equally divided and injected into the flat plate perpendicularly | vertically. A model in which spherical silver particles with a radius of 100 nm are randomly dispersed on a 3 μm thick flat plate made of a resin with a refractive index of 1.5 so that the concentration of silver particles is 1% by volume is possible. Example 8 in which the simulation of the present invention was performed on two light fluxes having angles of 30.556 degrees and 70.124 degrees with respect to the normal of the flat plate of light in which coherent light and incoherent light were equally divided FIG. A model in which spherical silver particles with a radius of 10 nm are randomly distributed using a Monte Carlo method so as to form a cluster with a radius of 100 nm on a 3 μm thick flat plate made of a resin with a refractive index of 1.5 is created. Implementation of simulation of the present invention for two luminous fluxes having angles of 30.556 degrees and 70.124 degrees with respect to the normal of the flat plate of light in which coherent light and incoherent light are equally mixed The figure which shows the result of Example 9. On a 3 μm thick flat plate made of a resin having a refractive index of 1.5, cylindrical silver particles having a radius of 10 nm and a length of 30 nm were randomly distributed using a Monte Carlo method so as to form a cluster having a radius of 100 nm. A model is created, and the simulation of the present invention is performed with respect to two light fluxes having angles of 30.556 degrees and 70.124 degrees with respect to the normal of the flat plate of light in which coherent light and incoherent light are mixed equally. The figure which shows the result of Example 10 which performed.

  The electromagnetic wave propagation simulation method of the present invention is an electromagnetic wave propagation simulation method for calculating the behavior of an electromagnetic wave when the electromagnetic wave is incident on an aggregate composed of a medium including a plurality of particles using a computer, and the particles are random or are present. The behavior of the electromagnetic wave when the electromagnetic wave is incident on the aggregate distributed according to the rule is calculated using the FDTD method and the radial transfer acquisition method.

Hereinafter, the present invention will be described in detail.
The present invention is an electromagnetic wave propagation simulation method for calculation using a computer, a recording medium for storing a program for executing the calculation, a device for reading the program from the recording medium, and a storage device for temporarily storing the program and calculation results And a computer having a CPU and an output device.

  In the present invention, an aggregate model to be calculated is first created. In order to distribute the particles randomly in the medium, it is convenient to use the Monte Carlo method. The Monte-Carlo method is a method for generating random numbers having a predetermined statistical property. In the present invention, the Monte-Carlo method is used to generate particles dispersed with a predetermined statistical property in a medium. . What is necessary is just to set the coordinate of particle | grains in the space of calculation object using the function which generate | occur | produces the random number used normally. Predetermined statistical properties correspond to conditions such as complete homogeneity and a certain number of aggregates of particles at a certain rate.

  In the present invention, not only the case where the particles are randomly distributed in the medium, but also the case where the particles are distributed according to a certain rule can be carried out.

The FDTD (Finite Difference Time Domain) method (finite difference time domain method) used in the present invention is a known method described in, for example, “Electromagnetic field and antenna analysis by FDTD method”, 1998, Corona, This is a simulation method in which Maxwell's equations are divided spatially and temporally, space and time derivatives are approximated by finite differences, and time changes of electromagnetic fields are tracked and calculated.
Note that the FDTD method can be applied to both a structure having an irregular boundary surface where a finite size assumption is effective and a structure capable of infinite assumption.

Usually, incident light (electromagnetic wave) in an arbitrary polarization state can be divided into two orthogonal polarization states, that is, a TM mode and a TE mode.
In the FDTD method used in the present invention, the calculation can be efficiently performed by using the inductive convolution method. For the TE mode in electromagnetic waves, the first electromagnetic field analysis equation obtained by applying the inductive convolution method to the wave equation derived from the Maxwell equation is solved by a computer using the inductive relational equation, For the TM mode, the second electromagnetic field analysis formula obtained by applying the recursive convolution method to the Maxwell equation is solved by a computer using the recursive relational expression, and for the TM mode and the TE mode, It is preferable to calculate the electromagnetic field in the spatial and temporal divisions of the FDTD method based on the electromagnetic field obtained in this way (see Japanese Patent Application No. 2008-68139).

（１）

（２） In the FDTD method, the following Maxwell equations can be used.

(1)

(2)

は微分演算子であり、

及び

及び

をｘ軸方向及びｙ軸方向及びｚ軸方向の単位ベクトルとしたとき、

（３）
で定義される微分演算子である。 Here, μ is the magnetic permeability, E is the strength of the electric field, and H is the strength of the magnetic field.

Is a differential operator,

as well as

as well as

Is a unit vector in the x-axis direction, the y-axis direction, and the z-axis direction,

(3)
Is a differential operator defined by.

（４）

式（４）中のεは計算対象の物体（粒子）の誘電率である。 D in Formula (2) represents the electric flux density and is given by the following formula.

(4)

In equation (4), ε is the dielectric constant of the object (particle) to be calculated.

The FDTD method used in the present invention is a three-dimensional extension of the FDTD method relating to the TM mode described in Japanese Patent Application No. 2008-68139.
The value of the electromagnetic field at a point far from the particle is calculated using the far-field transformation method using the electromagnetic field calculated by the FDTD method.

（５） The value of the electromagnetic field at the far point is calculated using equation (5).

(5)

はＳの上でそれぞれの点での法線の外部向きの単位vectorである。 [Psi] in Equation (5) indicates the strength of any of the x, y, and z components of the electromagnetic field. The subscript F indicates the infinity point (far field). The integration region is an isolated surface S surrounding the scatterer.

Is a unit vector pointing outwardly of the normal at each point on S.

  The radial transfer theory used in the present invention is the amount of light that is absorbed (attenuated) when the incident light interacts with a medium or particle, and the total amount of reflected, scattered, absorbed, and transmitted light. This is a calculation method using a relational expression that agrees with. The Radial Transfer Equation (RTE) resulting from the theory is a formula for the propagation of electromagnetic energy in a medium in which particles are distributed, and is a coherent light, It is a calculation formula that can handle partially coherent light and diffused light (incoherent light). The calculation method using RTE is the RT method. The energy of electromagnetic waves is calculated as a luminous flux. The luminous flux is related to light intensity or illuminance. If the incident light is completely diffused light, it can be calculated using a two-beam RTE as a luminous flux obtained by integrating the output. When the incident light is partially coherent light, RTE of 4 light fluxes and N light fluxes (4 ≦ N) is used. N of the N light flux RTE corresponds to N channels set in the propagation direction. Each channel is usually defined with respect to a polar angle θ and an azimuth angle φ. (See P.S.Mufgett and L.W.Richards, "Multiple Scattering Calculations for Technology", Applied Optics, vol. 10, No. 7, 1971, pp1485.)

（６） The basic RTE equation shows the conservation law of energy at any point in the medium.
The distance gradient of the coherent beam is given by equation (6).

(6)

（７） K in equation (6) is the absorption rate. K is the absorption rate relative to the sum of the media and particle absorption. Equation (6) shows that light is lost from the propagation channel due to absorption and scattering of the coherent beam. The distance gradient of the diffused light beam is expressed by equation (7).

(7)

The absorptance _{Kd in} equation (7) is related to the absorption loss of the diffused light flux. S _d is related to the scattering loss of the diffused light flux. S _{d +} f _d indicates that diffused light propagating to other channels is scattered and enters the target propagation channel. S _c f _c is a portion scattered from the coherent light channel and entering the diffusion channel. That is, the second term after Expression (7) represents the gain of the diffused light beam. All the coefficients of K and S described in equation (7) can be calculated from the absorption and scattering cross sections of the particles and the phase function. The phase function is the amplitude of the scattered wave field. For spherical particles, the phase function can be calculated from Mie theory, but for particles of any shape other than spherical, it cannot be derived from Mie theory. In the present invention, the absorption cross section, scattering cross section, and phase function of any shape of particles (including aggregate particles) can be calculated using the FDTD method (far-field transformation method is applied). .

（８）

（９） Absorption rate K and scattering rate S used in equations (6) and (7) relate to the absorption and scattering cross section of one particle. When incident light is diffuse light, K and S are expressed as follows (P. Kubelka, "New contributions to the optics of intensely light-scattering materials. Part I", Journal of Optical Society of America, vol. 38, no. 5, 1948, p. 448.).

(8)

(9)

Here, C _abs is an absorption cross-section, and C _scat (scattering cross-section) is a scattering cross-section. In this case, the light beam is not classified according to the viewing angle, and the calculation method corresponds to the RT method or the Kubelka-Munk method of two light beams in a one-dimensional reciprocating direction.

（１０）

（１１） In the two-beam method, the reflectance R and the transmittance T are given by the following equations (10) and (11).

(10)

(11)

（１２）

（１３）
ＫとＳはそれぞれ、式（８）と（９）から計算できる。 Here, d is the thickness of the diffusion plate, ρ _g is the reflectance of the back plane of the diffusion plane, and a and b are given as follows.

(12)

(13)
K and S can be calculated from equations (8) and (9), respectively.

In the case of a perfectly spherical particle, C _abs and C _scat can be calculated by Mie theory. However, in the case of arbitrary shaped particles, or in the case of an aggregate of particles, C _abs and C _scat cannot be calculated analytically. In the present invention, numerical values of C _abs and C _scat are calculated using the FDTD method. That is, C _abs , C _scat, and extinction cross-section C _ext are obtained from the following formula (CF Bohren, DR Huffman, “Absorption and scattering of light by small particles”, Wiley-VCH Verlag GmbH & Co. KGaA, 2004, Weinheim, Chap. 3-4.).

（１４）

（１５）

(14)

(15)

（１６） Here, θ and φ are the polar angle and the azimuth angle, respectively. Ψ ₀ used in equations (14) and (15) results from

(16)

（１７） Ψ _{F in} Expression (16) is calculated using Expression (5). Ψ _i is the electromagnetic field strength of the incident light. The absorption cross-sectional area C _abs is obtained by the equation (17).

(17)

（１８） RTE can be performed for more than two fluxes by setting multiple channels. The scattering rates S and S _{d ±} are determined by the scattering cross section C _scat and the phase function. The phase function is the amplitude of the scattered electromagnetic field in a certain direction. When the Phase function is expanded using the Legendre polynomial, the coefficient of each term can be calculated as follows.

(18)

は１次の係数であり、ｐ(θ)はPhase関数で、Ｐ_iは１次のLegendre多項式を示す。ｐ(θ)はある状態に従ってＭｉｅ理論またはＦＤＴＤ方法を用いて計算できる。 here,

Is a first order coefficient, p (θ) is a Phase function, and P _i is a first order Legendre polynomial. p (θ) can be calculated using Mie theory or FDTD method according to a certain state.

In the present invention, it is preferable to use a combination of the three calculation methods of the FDTD method, the Monte Carlo method, and the RTE method.
Further, in the RTE method, it is preferable to use the RTE method of four or more light beams because it is suitable for simulation when coherent light is included. The number of light beams is usually 4 or more and 12 or less, and the calculation speed is fast, so the cases of 4 and 6 are particularly preferable.

  In the present invention, the shape of particles to be simulated can be arbitrarily set. The shape of the particles can be set to any shape, for example, a spherical shape, a cylindrical shape, a prism shape, a polyhedron other than a prism shape, a scale shape, a donut shape, or a hollow cylindrical shape. In addition, when some of the two particles are associated with each other, the case where the particles are aggregated can also be set. Furthermore, when two or more kinds of particle shapes and substances are mixed, a case where there is a distribution in size, and a case where there is a bias in the distribution of particles in the medium can be set.

  It can also be set when the concentration of particles in the medium is a high concentration of 5% by volume or more. The present invention can be suitably applied when the particle concentration in the medium is 5 to 90% by volume, more preferably 5 to 50% by volume, and even more preferably 6 to 20% by volume. Can do.

  In the present invention, if the medium to be simulated is a uniform medium that transmits light, by setting the light transmittance and refractive index as constants, vacuum, air, water, organic solvent, solution, glass, Any of the resins can be set. The present invention can be suitably applied particularly when glass or resin that does not cause movement of particles is used as a medium.

  In the present invention, the aggregate to be calculated in the simulation is usually a case where the space occupied by the medium and the particles is a flat plate, which has a certain thickness and is an infinitely wide flat plate. . The space occupied by the medium and the particles is a flat plate, that is, the medium and the particles exist in a space existing between two parallel planes with a constant interval.

  In the present invention, the light source to be simulated is any one of a laser light source (a light source that emits light having high coherence), a normal light source (a light source that emits light having low coherence), a continuous light source, and a pulse light source. However, for display design, the present invention can be preferably applied in the case of a normal light source and a continuous light source, and the calculation is simplified in the case of a parallel light source. preferable.

Next, specific embodiments of the present invention will be described.
[1] Setting of particle dispersion medium As a medium to be calculated, an appropriate light transmittance (or light absorption rate) and refractive index are set in order to represent a uniform and light-transmitting substance (including vacuum). A flat plate is set as the shape of the medium, two parallel planes with a predetermined interval corresponding to the thickness of the flat plate are set, and an infinitely large space existing between them is set as an approximate shape for the flat plate.
Determine the shape of the particles to be dispersed. Appropriate light transmittance (or absorptance) and refractive index are set to represent the substance constituting the particle.
As a light source, a parallel beam of normal light is set for use in display design.
The particles are randomly distributed in the medium so as to have a predetermined concentration by the Monte Carlo method.

[2] Execution of simulation The scattering and the absorption cross section of an aggregate in which a plurality of particles are randomly distributed or distributed according to a certain rule are calculated using the FDTD method.

  [3] The electromagnetic field propagation in the medium randomly distributed in the aggregate or in a certain rule is calculated using a radial transfer excitation (or a radial transfer method).

EXAMPLES Next, although an Example demonstrates this invention further in detail, this invention is not limited to these Examples.
In the examples, a program described in the PG Fortran 90 language was created, and calculation was performed using Mathcad (trade name) manufactured by Mathsoft Engineering & Education inc.

Example 1
A model in which silver spherical particles having a radius of 10 nm were randomly distributed using a Monte Carlo method so that one cluster having a radius of 100 nm was formed in a medium having a refractive index of 1.5. The content of particles in the cluster was 50% by volume. An image of this model is shown in FIG. The three views of FIG. 1 show cross sections along the XY plane, the YZ plane, and the XZ plane in order from the left.

For this aggregate model, the incident light is completely scattered light (fully incoherent light), and the amount of scattered light and the amount of absorbed light are calculated when the wavelength of the incident light is changed from 400 nm to 700 nm at intervals of 20 nm. did. Light scattering and absorption were calculated by the Kubelka-Munk method, which is a two-beam RT method. The scattering and absorption cross sections for that were calculated by a combination of the three-dimensional FDTD method and the far-field transformation method. The far-field transformation method is a calculation method for knowing the electromagnetic field at a point far from the particle. The calculation results are shown in FIG.
Spherical silver particles with a radius of 10 nm, which is much smaller than the wavelength of light, can be assumed that if they do not form clusters, the interaction with light is small and the amount of scattering and absorption is small. When an aggregate is prepared by dispersing silver particles in a medium at a high concentration, the silver particles are aggregated to form clusters. This model is a model of this actual aggregate. As a result of the simulation by the method of the present invention, when the concentration is high and it is close to the actual formation of the cluster, a considerable amount of scattering and absorption occurs. Therefore, it has been found that the simulation method of the present invention can correctly simulate the actual situation.

(Example 2)
A model in which cylindrical silver particles having a radius of 10 nm and a height of 30 nm are randomly distributed using a Monte Carlo method so that one cluster having a radius of 100 nm is formed in a medium having a refractive index of 1.5. Created. The orientation of the cylindrical particles was such that the axis of the cylinder was parallel to a plane perpendicular to the traveling direction of incident light, and was aligned in a direction perpendicular to the polarization direction. The content of particles in the cluster was 50% by volume.

Using this aggregate model, the amount of scattered light and the amount of absorbed light were calculated when the wavelength of incident light was changed from 400 nm to 700 nm at 20 nm intervals in the same manner as in Example 1. The results are shown in FIG.
Presumably, cylindrical silver particles with a radius of 10 nm and a height of 30 nm, which is much smaller than the wavelength of light, have little interaction with light if they are not clustered and have little scattering and absorption. However, when silver particles are actually dispersed at a high concentration in a medium to produce an aggregate, the silver particles are aggregated to form clusters. This model is a model of this actual aggregate. As a result of the simulation by the method of the present invention, when the concentration is high and it is close to the actual formation of clusters, a considerable amount of scattering and absorption occurs, and the amount of scattered light and the amount of absorbed light increase as the wavelength of light is shorter. Therefore, it has been found that the simulation method of the present invention can correctly simulate the actual situation.

(Example 3)
Similar to Example 2, except that the axis of the cylinder was perpendicular to the plane perpendicular to the traveling direction of the incident light and was aligned in a direction parallel to the polarization direction, and an aggregate model was created. Using this model, the amount of scattered light and the amount of absorbed light when the wavelength of incident light was changed from 400 nm to 700 nm at intervals of 20 nm were calculated in the same manner as in Example 1. The results are shown in FIG.
As in Example 2, a considerable amount of scattering and absorption occurred, and the shorter the wavelength of light, the larger the scattered light amount and absorbed light amount. Therefore, as in Example 2, it was found that the simulation method of the present invention can simulate the actual situation correctly.

(Comparative Example 1)
Applying the conventional Mie theory to the same aggregate model as in Example 1, the incident light is completely scattered (when there is no coherence), and the wavelength of the incident light is changed from 400 nm to 700 nm at intervals of 20 nm. The amount of scattered light and the amount of absorbed light were calculated. In the Mie calculation, since the scattering characteristics of the cluster composed of the aggregate of particles cannot be calculated, the cluster having a radius of 100 nm is replaced with a homogeneous equivalent particle. The radius of the equivalent particle is 79 nm. The results are shown in FIG.
A large peak occurs in the scattered light amount and the absorbed light amount in the vicinity of 420 nm, which indicates that the actual situation cannot be simulated correctly.

Example 4
Similar to the model of Example 1, a silver spherical particle having a radius of 10 nm is formed on a 1000 μm-thick flat plate so that one cluster having a radius of 100 nm is formed in a medium having a refractive index of 1.5. A model randomly distributed using the method, and cylindrical silver particles having a radius of 10 nm and a height of 30 nm as in Example 2, and one cluster having a radius of 100 nm in a medium having a refractive index of 1.5. The model was randomly distributed using the Monte Carlo method to form a spherical silver particle having a radius of 10 nm and the Monte Carlo method without forming a cluster in a medium having a refractive index of 1.5. Three models were created: a randomly distributed model. Assuming that a reflection plate having a reflectance of 0.8 exists on the surface opposite to the incident surface of the flat plate, the amount of scattered light and the amount of absorbed light were calculated when incident light was perpendicularly incident on the incident surface of the flat plate.

The incident light is completely scattered (when there is no complete coherence), and the amount of scattered light and the amount of absorbed light when the wavelength of the incident light is changed from 400 nm to 700 nm is the Kubelka-Munk method, which is a two-beam RT method. Calculated with The scattering and absorption cross sections for that were calculated by a combination of the three-dimensional FDTD method and the far-field transformation method. The simulation result is shown in FIG.
Spherical silver particles having a radius of 10 nm, which is much smaller than the wavelength of light, and silver cylindrical particles having a radius of 10 nm and a height of 30 nm have a small interaction with light when there is no cluster, and scattering and absorption. Is small (r in FIG. 6), but in the case where the concentration is close to the reality of forming clusters, the case of cylindrical particles (r_cyl in FIG. 6) rather than the case of spherical particles (r_sph in FIG. 6) It was found that the amount of scattered light and the amount of absorbed light were larger.

(Example 5)
The cross-sectional area and phase function of spherical silver particles with a radius of 100 nm are calculated by Mie theory on a 3 μm thick flat plate made of a resin with a refractive index of 1.5, and coherent light and incoherent light are calculated. The reflectance and transmittance were calculated when light mixed in equal parts was incident on the flat plate perpendicularly. The concentration of silver particles was as low as 1% by volume. For the calculation, RTE of four luminous fluxes that handle electromagnetic waves (light) propagating in one direction and in the opposite direction with respect to coherent light and non-coherent light was used.

Calculates the coherent light transmittance, coherent light reflectance, incoherent light transmittance, and non-coherent light reflectance change with wavelength in the wavelength range of 400 to 700 nm. The results are shown in FIG. In FIG. 7, the transmittance of coherent light is indicated by “x”, the reflectance of coherent light is indicated by a dotted line “..., And the transmittance of non-coherent light is indicated by a one-dot chain line“ − ”. "-" And the reflectivity of incoherent light is indicated by a solid line "___".
For both coherent light and non-coherent light, the transmittance shows a low value with little change depending on the wavelength, and it can be seen that much light is reflected or absorbed. The reflectivity of coherent light was around 4%, and the reflectivity of non-coherent light increased with wavelength. The reflectivity of 4% for coherent light is mostly caused by Fresnel reflection at the interface.

(Example 6)
A Monte Carlo method in which a spherical silver particle having a radius of 10 nm is formed in a flat plate made of a resin having a refractive index of 1.5 and having a radius of 10 nm to form a cluster having a radius of 100 nm and the particle concentration in the cluster is 50% by volume A model was created in which a plurality of clusters were distributed in the flat plate so that the particle concentration in the flat plate was 1% by volume. The reflectance and transmissivity were calculated when light in which coherent light and non-coherent light were equally divided and entered perpendicularly to the flat plate. For the calculation, as in Example 5, RTE of four light beams was used together with the FDTD method.

Calculates the coherent light transmittance, coherent light reflectance, incoherent light transmittance, and non-coherent light reflectance change with wavelength in the wavelength range of 400 to 700 nm. The results are shown in FIG. In FIG. 8, the transmittance of coherent light is indicated by “x”, the reflectance of coherent light is indicated by a dotted line “..., And the transmittance of non-coherent light is indicated by a dashed line“ − ”. "-" And the reflectivity of incoherent light is indicated by a solid line "___".
Coherent light, non-coherent light, transmittance and reflectance change little with wavelength, but coherent light transmittance is high, and then incoherent light is reflected. The rate of transmission of incoherent light and the reflectivity of coherent light were low. The propagation behavior of electromagnetic waves was completely different from the flat plate in which silver particles of 100 nm in Example 5 were dispersed without forming clusters. Even in an actual flat plate, it has been found that the light propagation behavior is completely different between the case where the pigment forms a cluster and the case where the pigment is uniformly dispersed.

(Example 7)
A model similar to that in Example 6 was created except that the silver particles had a cylindrical shape with a radius of 10 nm and a length of 30 nm. The same calculation as in Example 6 was performed, and the results are shown in FIG. The line in FIG. 9 shows the same thing as FIG.
Although the result similar to Example 6 was obtained, the reflectance of non-coherent light increased as the wavelength increased.

(Example 8)
Similarly to Example 5, on a 3 μm thick flat plate made of a resin with a refractive index of 1.5, spherical silver particles with a radius of 100 nm were placed on a Monte Carlo so that the concentration of silver particles was 1% by volume. A randomly distributed model was created by the method. Similar to Example 5, coherent light and non-coherent light were equally mixed, but unlike Example 5, it was 30.556 degrees with respect to the normal to the surface of the flat plate. The reflectance and transmittance were calculated when light propagated at an angle of. The calculation was performed for two light fluxes of 30.556 degrees and 70.124 degrees with one light flux in a range of ± 20 degrees. For the calculation, an RTE of 6 light fluxes handling electromagnetic waves (light) propagating in one direction and the opposite direction of each of the two light fluxes for coherent light and incoherent light was used together with the FDTD method.

Since the light propagating through the medium is incident on the light emitting surface at an angle of 70.124 degrees with respect to the normal line and totally reflected, the calculation result is shown in FIG. 10 only for channel 1 of 30.556 degrees. .
It was found that the transmittance was low and most of the light was reflected or absorbed. The wavelength dependence of the reflectance was small.

Example 9
The flat plate model is the same as in Example 6, and the Monte Carlo method is used so that a silver spherical particle with a radius of 10 nm is formed into a cluster with a radius of 100 nm in a 3 μm thick flat plate made of a resin with a refractive index of 1.5. A randomly distributed model was used. The light propagation channel was the same as in Example 8, and the reflectance and transmittance were calculated. For the calculation, RTE of 6 luminous fluxes was used together with the FDTD method as in the case of Example 8.

Since the light propagating through the medium is incident on the light emitting surface at an angle of 70.124 degrees with respect to the normal line and totally reflected, the calculation result is shown in FIG. 11 only for channel 1 of 30.556 degrees. .
It was found that the transmittance was low and most of the light was reflected or absorbed. The wavelength dependence of the reflectivity was small, and was lower than when 100 nm silver particles were uniformly dispersed.

(Example 10)
The flat plate model was the same as in Example 7, and a model similar to that in Example 6 was created except that the plate was a cylinder with a radius of 10 nm and a length of 30 nm. The light propagation channel was the same as in Example 8, and the reflectance and transmittance were calculated. For the calculation, RTE of 6 luminous fluxes was used together with the FDTD method as in the case of Example 8.

Since light incident at an angle of 70.124 degrees with respect to the normal to the surface of the flat plate is totally reflected, the calculation result is shown in FIG. 12 only for channel 1 of 30.556 degrees.
It was found that the transmittance was low and most of the light was reflected or absorbed. The wavelength dependence of the reflectance is small, lower than when 100 nm silver particles are uniformly dispersed, and is not significantly different from the case where silver particles are spherical.

Claims (5)

  An electromagnetic wave propagation simulation method for calculating the behavior of an electromagnetic wave when an electromagnetic wave is incident on an aggregate composed of a medium including a plurality of particles, using a computer, and the electromagnetic wave is generated in an aggregate in which particles are randomly distributed according to a certain rule. An electromagnetic wave propagation simulation method characterized in that the behavior of an electromagnetic wave when an incident light enters is calculated using an FDTD method and a radial transfer acquisition.   The electromagnetic wave propagation simulation method according to claim 1, wherein the radial transfer equation is a radial transfer equation with four or more light beams.   The electromagnetic wave propagation simulation method according to claim 1, wherein the space to be calculated is a flat plate having a constant thickness and an infinite width.   The electromagnetic wave propagation simulation method according to claim 1, wherein the incident wave is an electromagnetic wave having low coherence.   A computer-readable recording medium, an electromagnetic wave propagation simulation method that uses a computer to calculate the behavior of an electromagnetic wave when the electromagnetic wave is incident on an aggregate composed of a medium containing a plurality of particles. Alternatively, the scattering characteristics of the aggregates distributed according to a certain rule are obtained by the FDTD method, and the electromagnetic wave propagation characteristics in a space where a large number of aggregates exist are calculated by the radial transfer acquisition using the scattering characteristics of the aggregates. The recording medium which stored the program for performing the electromagnetic wave propagation simulation characterized by the above-mentioned.

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