Derivation of various formulas for electromagnetic waves
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This course systematically derives various formulas for electromagnetic waves. It is recommended that you understand the entire flow.
・Fresnel's formula for p-polarized light (reflection and Snell's law are also derived along the way)
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Parameter definitions and relationships
Each parameter of the electromagnetic wave is defined as follows:
The relationship between each parameter is expressed as follows by definition:
Next, the permittivity and permeability will be described.
The dielectric constant and magnetic permeability in a vacuum are the following constants:
The following relationship exists between the permittivity, permeability and each parameter.
Here, in a non-magnetic medium, the magnetic permeability can be handled as follows.
In this case, (1)-③ becomes as follows:
Here, the angular frequency has the relationship of (3)-⑥, which will be shown later, and finally (3)-⑦ and ⑧ are derived. For more information on this, please refer to the following link.
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Mathematical expression of electromagnetic waves
Electromagnetic waves can be expressed as follows (only the results derived from Maxwell's equations are shown):
Electric field vector:
Magnetic field vector:
The relationship between the electric field and the magnetic field is as follows.
where each vector is a quantity defined as follows:
The relationship between the directions of the vectors shown in (2)-② can be illustrated as follows.
Here, the vector depth method can be expressed as follows:
...From back to front
...From front to back
Using this, the relationship between the vector directions shown in (2)-② can be expressed as follows.
The electromagnetic wave represented by (2)-① can be illustrated as follows. Here, to avoid complicating the diagram, ekz =0, but the same consideration can be given to the case where ekz is not 0.
Moreover, the relationship between the phase lead and the electromagnetic wave lead can be illustrated as follows:
In this way, since the phases of the position advance and the time advance are in opposite directions, the signs of the position component and the time component are opposite.
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Boundary conditions for propagation between media
As shown in the figure below, in the propagation of light between non-magnetic media, the z=0 plane is set as the boundary between the media, and we consider the transmitted and reflected light when an electromagnetic wave is incident on the boundary surface.
Here, the subscripts of each parameter have the following meanings:
i: Value for incident light
r: Value for reflected light
t: Value for transmitted light
The electric field and magnetic field of incident light, reflected light, and transmitted light are expressed as follows from (2)-①.
Now consider the boundary conditions of the electromagnetic field at the boundary surface.
Since the tangential component of the electric field is continuous at z=0,
Since the tangential component of the magnetic field is continuous at z=0,
For any x, y, and t, the conditions for (3)-② and (3)-③ to hold are
In this case, (3)-② and (3)-③ will be as follows.
From (3)-④-3, the ω of incident light, reflected light, and emitted light will be expressed as a common ω without the subscript.
In addition, since ω does not depend on the medium, the following can also be said:
From (3)-⑥, (1)-① can be replaced as follows.
Furthermore, in non-magnetic media, (1)-④ and ⑤ hold, so (3)-⑦ becomes as follows:
Here, there is no loss of generality in determining the y-axis in the direction where 𝑘𝑖𝑦 = 0.
In this case, from (3)-④-1, we get the following.
From this, if the y-axis is determined in the direction where 𝑘𝑖𝑦 = 0, it can be said that the incident light, reflected light, and transmitted light are all within the xz plane .
In addition, since the electric field and magnetic field can each be expressed as the sum of a component that vibrates parallel to the plane of incidence (p-polarized light) and a component that vibrates perpendicular to the plane of incidence (s-polarized light), we will consider these separately in the next chapter.
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Fresnel's formula for p-polarized light
Consider the case where the electric field oscillates parallel to the plane of incidence (p-polarization) when light is incident on the boundary surface of a z=0 plane medium. In this case, the electric and magnetic fields of the incident light, reflected light, and transmitted light are expressed as shown in the figure below.
The subscripts 1 and 2 represent different media.
The directions of the electric and magnetic fields are illustrated according to (2)-②.
It is assumed that propagation occurs between non-magnetic, absorption-free media, and that (3)-⑧ is applicable.
Each vector shown in the figure can be expressed in component form as follows:
Here, from (3)-⑧,
from (4)-④-1,
In addition, from (3)-④-1 and (4)-①,
From (4)-⑤ and (4)-⑥, the following law of reflection and Snell's law can be derived.
Substituting (4)-②③ (electric field and magnetic field component expression for p-polarized light) into (3)-②a, ③a (boundary condition for electric field and magnetic field amplitude), and then applying (4)-④-2 (relationship between electric field and magnetic field amplitude) and (4)-⑦-1 (law of reflection), we obtain the following:
Therefore, if the amplitude transmittance is tp and the amplitude reflectance is rp , the following Fresnel formula (p-polarized light) can be obtained.
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Fresnel's formula for s-polarized light
Consider the case where the electric field oscillates perpendicular to the plane of incidence (s-polarization) when light is incident on the boundary surface of a z=0 plane medium. In this case, the electric and magnetic fields of the incident light, reflected light, and transmitted light are expressed as shown in the figure below.
The subscripts 1 and 2 represent different media.
The directions of the electric and magnetic fields are illustrated according to (2)-②.
It is assumed that propagation occurs between non-magnetic, absorption-free media, and that (3)-⑧ is applicable.
Each vector shown in the figure can be expressed in component form as follows:
Here, from (3)-⑧,
(5)-④-1,
In addition, from (3)-④-1 and (5)-①,
From (5)-⑤ and (5)-⑥, the following law of reflection and Snell's law can be derived.
Substituting (5)-②③ (electric field and magnetic field component expression for s-polarized light) into (3)-②a, ③a (boundary condition for electric field and magnetic field amplitude), and then applying (5)-④-2 (relationship between electric field and magnetic field amplitude) and (5)-⑦-1 (law of reflection), we obtain the following:
Therefore, if the amplitude transmittance is ts and the amplitude reflectance is rs, the following Fresnel formula (s-polarized light) can be obtained.
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Transmittance and Reflectance
The volumes of the incident light, reflected light, and transmitted light passing through the infinitesimal region dxdy on the z=0 plane per unit time are dV i , dV r , and dV t , respectively . These are illustrated as follows.
From the figure, dV i , dV r , and dV t can be expressed as follows.
The energy densities of the incident light, reflected light, and transmitted light are expressed as follows:
Therefore, from (6)-①, ② and (1)-⑤, the amount of energy of the incident light, reflected light, and transmitted light passing through the infinitesimal region dxdy on the z=0 plane per unit time can be expressed as follows:
Therefore, the energy transmittance Tp and energy reflectance Rp of p-polarized light are as follows:
From (6)-④, the following calculation results confirm that the law of conservation of energy is satisfied.
At normal incidence:
Moreover, the energy transmittance Ts and energy reflectance Rs of s-polarized light are respectively as follows:
From (6)-⑦, the following calculation results confirm that the law of conservation of energy is satisfied.
For normal incidence, it is as follows. Since normal incidence of p-polarized light and normal incidence of s-polarized light are in the same state, it goes without saying that (6)-⑥ and (6)-⑨ have the same equation.
For the case of n1=1 and n2=2, if we graph (4)-⑨, (5)-⑨, (6)-④, and (6)-⑦, we get the results shown below.
When n 1 > n 2 , total reflection occurs as θ is increased, as will be explained later.
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Brewster's Angle
As can be seen from the graph of the incidence angle dependence of energy reflectance shown in the previous chapter, p-polarized light is not reflected at a certain incidence angle. This angle is called the Brewster angle θB , and we will derive it.
In (4)-⑨, when r p =0,
(4)-⑦-2 (Snell's law)
(7)-① and ②,
Here, the following angle conditions are considered:
The solution obtained at this time is as follows:
Again, from (4)-⑦-2 (Snell's Law),
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Total reflection and evanescent waves
When n1 > n2 , as the angle of incident light is increased, the exit angle ξ of the transmitted light becomes π/2 at a certain angle of incidence θ. The angle of incidence at this time is called the critical angle, and according to (4)-⑦-2 and (5)-⑦-2 (Snell's Law), it is as follows for both p-polarized light and s-polarized light.
At angles of incidence greater than this, Snell's law does not have a real solution ξ. In this case, all of the incident light is reflected. This is called total reflection. When n1 > n2 , the incidence angle dependence of the energy transmittance and reflectance is as follows:
Even under the condition of total reflection, if ξ is expanded to a complex number, Snell's law can have a solution. In other words, under the condition of total reflection, sinξ is a real value greater than or equal to 1, but ξ is a complex number.
The complex solution under the total reflection condition can be found as follows.
For p-polarized light, the electric field and magnetic field of the transmitted light under the total reflection condition are as follows, substituting (4)-①, ②, and ③ into (3)-① and applying (8)-②, (4)-④-2, (4)-⑤-2, (4)-⑥, and (4)-⑦-2.
For s-polarized light, the electric field and magnetic field of the transmitted light under the total reflection condition can be obtained by substituting (5)-①, ②, and ③ into (3)-① and applying (8)-②, (5)-④-2, (5)-⑤-2, (5)-⑥, and (5)-⑦-2.
The result obtained is merely a mathematically obtained solution, but among the following z-dependent components that are common to both p-polarized and s-polarized light, the negative complex signs represent evanescent waves that actually exist physically (the positive complex signs do not exist physically).
This means that the amplitude decays exponentially with respect to z, and the distance from the interface (z=0) until the amplitude decays to 1/e (penetration depth) is expressed as follows (applying ③-⑧ by transforming the equation):
Regarding the effect of the penetration of evanescent waves on the amount of reflected light (energy reflectance), substituting (8)-② into (6)-④ and ⑦ gives Rp =1, Rs =1, which shows that there is no loss of light due to total reflection. However, when substituting (6)-④ and ⑦, the value of Tp does not become zero, and the denominator of Ts becomes zero, making it impossible to calculate. This is because evanescent light is not considered in the process of deriving the formulas for Tp and Ts .
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Extinction and Absorption Coefficients
Once again, (2)-① is shown below.
If the z-axis is taken in the same direction as the direction of light travel, then kx = ky = 0, and therefore k = kz , which can be expressed as follows.
Here, from (1)-①,
Now, replace the refractive index n by the complex refractive index shown below.
Here, κ is called the extinction coefficient.
At this time,
Therefore, if the electric field strength is I,
Here, the absorption coefficient α is defined as follows.
At this time,
This means that the electric field strength decays exponentially with z, where α is the depth from z=0 at which the strength falls off by a factor of 1/e.