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Derivation of various formulas for electromagnetic waves (in the case of absorbing media)

*This chapter is positioned as a continuation of the following chapter (hereafter referred to as the previous chapter). Therefore, the equations (1) to (9) shown in this chapter refer to the equations in the previous chapter.

・Equation for electromagnetic waves in absorbing media

・Application of Snell's law to absorbing media

・Application of Fresnel formula to absorbing media and energy reflectance

・Derivation of the definition formula of ellipsometry

  • Equation for electromagnetic waves in absorbing media

From (2)-①, (2)-③, (1)-④ in the previous chapter,

電磁波に関する様々な公式の導出（吸収体）_129.png

Therefore,

電磁波に関する様々な公式の導出（吸収体）_025.png

Here, the refractive index n is expanded to a complex number, which allows us to express the effect of absorption by the medium mathematically.

In (10)-①, the refractive index n is replaced with the complex refractive index shown below.

電磁波に関する様々な公式の導出（吸収体）_130.png

Here, κ is called the extinction coefficient.

At this time,

電磁波に関する様々な公式の導出（吸収体）_027.png

Therefore, if the light intensity is I,

電磁波に関する様々な公式の導出（吸収体）_026.png
吸収係数

Here, the absorption coefficient α is defined as follows:

吸収係数の定義式

Applying this,

電磁波に関する様々な公式の導出（吸収体）_012.png

This means that the light intensity attenuates exponentially with respect to ek・r , and 1/α represents the depth in the direction of light wave propagation from r = 0 until the light intensity becomes 1/e times. This depth is called the absorption length and is used as a guide to the depth that light can penetrate.

吸収長の図示

Furthermore, (10)-⑥ can be illustrated spatially as follows:

波数ベクトルと光強度の関係について図示

Here, for the sake of simplicity, ekz =0.

  • Application of Snell's law to absorbing media

The refractive index of medium 1 is n1, and the refractive index of medium 2 is n2. Also, with respect to the interface between medium 1 and medium 2, the incident angle from medium 1 is θ, and the outgoing angle to medium 2 is ξ.

電磁波に関する様々な公式の導出（吸収体）_014.png

From Snell's law derived in (4)-⑦-2 and (5)-⑦-2 in the previous chapter,

電磁波に関する様々な公式の導出（吸収体）_016.png

Here, if medium 2 is an absorbing medium, the effect of absorption by medium 2 can be expressed mathematically by replacing the refractive index n2 with the complex refractive index below, as in (10)-②.

電磁波に関する様々な公式の導出（吸収体）_131.png

In this case, since the right-hand side of (4) and (5)-⑦-2 is a real number, ξ must also be a complex number to satisfy the equation. This can be expressed as follows.

電磁波に関する様々な公式の導出（吸収体）_132.png

Here, the range of values for each parameter is determined as follows:

電磁波に関する様々な公式の導出（吸収体）_046.png

Replacing n2 and ξ in (4),(5)-⑦-2 with (11)-①-1,2 respectively and rearranging the equation gives the following:

電磁波に関する様々な公式の導出（吸収体）_133.png

Here, from (11)-①-3 and (11)-②,

[ i ] When θ = 0,

ξr ＝ ξi = 0

[ ii ] When θ ≠ 0,

Since ξr ≠ 0 and ξi ≠ 0,

・Derivation of ξi :

Transforming (11)-② gives us :

電磁波に関する様々な公式の導出（吸収体）_047.png

Here, the following is set:

電磁波に関する様々な公式の導出（吸収体）_048.png

Applying this to the above equation,

電磁波に関する様々な公式の導出（吸収体）_049.png

Since X > 0, A > 0,

電磁波に関する様々な公式の導出（吸収体）_050.png

Here, we put it as follows:

電磁波に関する様々な公式の導出（吸��収体）_051.png

Applying this to the above equation,

電磁波に関する様々な公式の導出（吸収体）_052.png

Here, from (11)-①-3 and (11)-②, the following can be said:

電磁波に関する様々な公式の導出（吸収体）_053.png

Therefore,

電磁波に関する様々な公式の導出（吸収体）_054.png

ξi can also be expressed as follows:

電磁波に関する様々な公式の導出（吸収体）_055.png

Here, from (11)-①-3 and (11)-②, the following can be said:

電磁波に関する様々な公式の導出（吸収体）_056.png

Therefore,

電磁波に関する様々な公式の導出（吸収体）_057.png

However, ξi < 0 (∵ (11)-③-5)

・Derivation of ξr :

Once again, transforming (11)-② gives us :

電磁波に関する様々な公式の導出（吸収体）_090.png

Here, we put it as follows:

電磁波に関する様々な公式の導出（吸収体）_059.png

Applying this together with (11)-③-1b shown previously to the above equation, we get

電磁波に関する様々な公式の導出（吸収体）_060.png

Since Y > 0, A > 0,

電磁波に関する様々な公式の導出（吸収体）_061.png

Again, applying (11)-③-2a,b,

電磁波に関する様々な公式の導出（吸収体）_062.png

Here, from (11)-①-3, the following can be said:

電磁波に関する様々な公式の導出（吸収体）_063.png

Therefore,

電磁波に関する様々な公式の導出（吸収体）_064.png

However, 0 < ξ r < π /2 (∵ (11)-①-3)

ξr can also be expressed as follows:

電磁波に関する様々な公式の導出（吸収体）_065.png

Here, from (11)-①-3, the following can be said:

電磁波に関する様々な公式の導出（吸収体）_070.png

Therefore,

電磁波に関する様々な公式の導出（吸収体）_069.png

However, 0 < ξ r < π /2 (∵ (11)-①-3)

·Summary

To summarize the above results,

電磁波に関する様々な公式の導出（吸収体）_073.png

… (11)-③-5,7, (11)-④-5,7

電磁波に関する様々な公式の導出（吸収体）_072.png

… (11)-③-2a,b

  • Application of Fresnel formula to absorbing media and energy reflectance.

[ i ] When θ=0,

At normal incidence, p-polarized light and s-polarized light are in the same state, so they are shown together.

The Fresnel formula for normal incidence (s,p polarization) derived in the previous chapter is shown below.

電磁波に関する様々な公式の導出（吸収体）_082.png

Here, by replacing the refractive index n2 with the complex numbers shown in (11)-①-1, the amplitude reflectance (s, p polarization) when incident on an absorbing medium can be expressed as follows:

電磁波に関する様々な公式の導出（吸収体）_134.png

Therefore, from (6)-④,⑦, the energy reflectance (s, p polarization) when incident on an absorbing medium is as follows:

電磁波に関する様々な公式の導出（吸収体）_136.png

[ii] When θ≠0,

​・In the case of p-polarized light

The Fresnel formula (p-polarized light) derived in the previous chapter is shown below.

電磁波に関する様々な公式の導出（吸収体）_137.png

Here, by replacing the refractive index n2 and the exit angle ξ with the complex numbers shown in (11)-①-1,2, the amplitude reflectance (p-polarized light) when incident on an absorbing medium can be expressed as follows:

電磁波に関する様々な公式の導出（吸収体）_138.png

Therefore, from (6)-④, the energy reflectance (p-polarized light) when incident on an absorbing medium is as follows:

電磁波に関する様々な公式の導出（吸収体）_139.png

Here, from (11)-③-5,7 and (11)-④-5,7, ξr and ξi can be expressed as functions of θ.

​・In the case of s-polarized light

The Fresnel formula (s-polarized light) derived in the previous chapter is shown below.

電磁波に関する様々な公式の導出（吸収体）_080.png

Here, by replacing the refractive index n2 and the exit angle ξ with the complex numbers shown in (11)-①-1,2, the amplitude reflectance (s-polarized light) when incident on an absorbing medium can be expressed as follows:

電磁波に関する様々な公式の導出（吸収体）_140.png

Therefore, from (6)-⑦, the energy reflectance (s-polarized light) when incident on an absorbing medium is as follows:

電磁波に関する様々な公式の導出（吸収体）_141.png

Here, from (11)-③-5,7 and (11)-④-5,7, ξr and ξi can be expressed as functions of θ.

​・Case studies

If the refractive index of aluminum at a wavelength of 589.3 nm is 1.44 and the extinction coefficient is 5.23, the relationship between the incident angle θ and the energy reflectance when light is incident on aluminum from the air can be calculated using (12)-②,③-2, and the following graph is obtained.

電磁波に関する様々な公式の導出（吸収体）_089.png

  • Derivation of the definition formula of ellipsometry

First, we examine the following range of values:

電磁波に関する様々な公式の導出（吸収体）_171.png

From (11)-②,

電磁波に関する様々な公式の導出（吸収体）_092.png

If n2 > n1 , then

電磁波に関する様々な公式の導出（吸収体）_103.png

Therefore, from (11)-①-3,

電磁波に関する様々な公式の導出（吸収体）_104.png

Also, from (11)-①-3 and (13)-②-1,

電磁波に関する様々な公式の導出（吸収体）_172.png

Based on (13)-②-1,2,3 and ξi < 0 (∵(11)-③-5), when n2 > n1 , (12)-②-1 and (12)-③-1 can be expressed as follows using the amplitude and argument :

電磁波に関する様々な公式の導出（吸収体）_142.png
電磁波に関する様々な公式の導出（吸収体）_144.png

Here, the blue part in the formula indicates that the case distinction has been incorporated into the formula, where the value of θ+ξr is zero when it is less than π/2, and is -π when it is greater than π/2.

And, from (11)-③-5,7 and (11)-④-5,7, ξr and ξi can be expressed as functions of θ.

Using this result, ψ and Δ defined in ellipsometry can be expressed as follows:

電磁波に関する様々な公式の導出（吸収体）_145.png

As an example, the calculation results for ψ and Δ for θ when n=1.44 and κ=5.23 are shown below.

電磁波に関する様々な公式の導出（吸収体）_112.png
電磁波に関する様々な公式の導出（吸収体）_113.png

If the p- and s-polarized components of the electric field of the incident light are Eip and Eis , respectively, and the p- and s-polarized components of the electric field of the reflected light are Erp and Ers , respectively, the p- and s-polarized components of the amplitude reflectivity can be expressed as follows:

電磁波に関する様々な公式の導出（吸収体）_146.png

Here, since the left-hand sides of each equation are complex numbers, the electric field on the right-hand side is also a complex number.

Now, let us set the electric field of the incident light as follows:

電磁波に関する様々な公式の導出（吸収体）_148.png

In this case, from (13)-⑤ and ⑥, we get the following.

電磁波に関する様々な公式の導出（吸収体）_149.png

Therefore, from (13)-④-1, the following relational expression is obtained.

電磁波に関する様々な公式の導出（吸収体）_150.png

Here, if the phases of the p- and s-polarized reflected light, including the wave vector and time, are PHp (r, t) and PHs (r, t), respectively, then

電磁波に関する様々な公式の導出（吸収体）_173.png

Therefore, when viewed on the time axis, s-polarized light is ahead of p-polarized light by Δ/ω.

Based on the above, ψ and Δ can be illustrated on the time axis as follows:

電磁波に関する様々な公式の導出（吸収体）_170.png

Here, the electric field of the incident light is normalized to 1 in absolute value in addition to the condition (13)-⑥.

Furthermore, the phases of the p- and s-polarized reflected light, including the wave vector and time, can be expressed as follows:

電磁波に関する様々な公式の導出（吸収体）_174.png

Therefore, when viewed on the spatial axis, s-polarized light is delayed by Δ/kr relative to p-polarized light.

Therefore, ψ and Δ can be illustrated on the spatial axis as follows:

電磁波に関する様々な公式の導出（吸収体）_169.png

Here again, the electric field of the incident light is normalized to 1 in absolute value in addition to the condition (13)-⑥.

Here, we will mention the sign of phase. In the fields of physics and optics, the sign of the phase on the space axis and the time axis is reversed according to their respective conventions. Therefore, the sign of the value of Δ is also reversed in the fields of physics and optics. Please note that this website defines the sign based on the physics convention, so the sign of Δ will be reversed when defined in the field of optics .

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