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  • Equation for electromagnetic waves in absorbing media

From (EM1)-(2)-①, (EM1)-(2)-③, and (EM1)-(1)-④, the equation for electromagnetic waves can be expressed as follows:

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

*The definition of the sign of phase differs between the fields of physics and optics, according to their respective conventions. See (SG) for details.

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,

吸収係数
電磁波に関する様々な公式の導出（吸収体）_026.png

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

Snell's Law, derived in (EM1) -(4)-⑦-2 and ( EM1 ) -(5)-⑦-2, is shown again below.

電磁波に関する様々な公式の導出（吸収体）_215.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 sides of (EM1)-(4)-⑦-2 and (EM1)-(5)-⑦-2 are real numbers, 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 (EM1)-(4)-⑦-2 and (EM1)-(5)-⑦-2 with (11)-①-1 and (11)-①-2 respectively and rearranging the equations gives the following:

電磁波に関する様々な公式の導出（吸収体）_213.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）_216.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 (EM1)-(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 (EM1) is shown below.

電磁波に関する様々な公式の導出（吸収体）_217.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

When n2 >n1 , (12)-②-1 can be expressed as follows, separated into amplitude and phase (however, the rule of (CM) is applied).

電磁波に関する様々な公式の導出（吸収体）_200.png

*(12)-②-2b-r will be derived later.

Here, the blue parts in the equation represent cases where the value of θ+ ξr is zero when it is less than π /2, and -π when it is greater than π /2 (this is based on the definition of (CM) ).

Here, the amplitude reflectance (p-polarization) rp is defined as the ratio of the electric field amplitude Eip of the incident light to the electric field amplitude Erp of the reflected light, but the fact that the left side is a complex number means that at least one of the amplitudes of the electric field on the right side is a complex number (complex amplitude).

電磁波に関する�様々な公式の導出（吸収体）_201.png

Also, from (EM1)-(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）_203.png

However, since ξr and ξi are given as functions of θ in (11)-③-5,7 and (11)-④-5,7, (12)-②-1,2,4 are also functions of θ.

​・In the case of s-polarized light

The Fresnel formula (s-polarized light) derived in (EM1) 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

When n2 >n1 , (12)-③-1 can be expressed as follows, separated into amplitude and phase (however, the rule of (CM) is applied) :

電磁波に関する様々な公式の導出（吸収体）_206.png

*(12)-③-2b-r will be derived later.

Here, the amplitude reflectance (s-polarization) rs is defined as the ratio of the electric field amplitude Eis of the incident light to the electric field amplitude Ers of the reflected light, but the fact that the left side is a complex number means that at least one of the amplitudes of the electric field on the right side is a complex number (complex amplitude).

電磁波に関する様々な公式の導出（吸収体）_176.png

Also, from (EM1)-(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）_205.png

However, since ξr and ξi are given as functions of θ in (11)-③-5,7 and (11)-④-5,7, (12)-③-1, 2, 4 are also functions of θ.

​・Derivation of angle range

We will derive (12)-②-2b-r and (12)-③-2b-r.

First, from (11)-①-3 and θ≠0 and n2 >n1 ,

電磁波に関する様々な公式の導出（吸収体）_208.png

Also, from (11)-②,

電磁波に関する様々な公式の導出（吸収体）_210.png

From (12)-④-1,2, the following can be said.

電磁波に関する様々な公式の導出（吸収体）_211.png

From (12)-④-1,3, the following can also be said.

電磁波に関する様々な公式の導出（吸収体）_212.png

Therefore, (12)-②-2b-r and (12)-③-2b-r are shown.

​・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

It is important to note that in a non-absorbing medium, there is a Brewster angle, which is the angle at which the reflectance of p-polarized light becomes zero, but in an absorbing medium, there is no angle at which the reflectance becomes zero, so the Brewster angle cannot be defined in the sense of the angle at which the reflectance becomes zero.

  • Related Topics

Ellipsometry is an application of the principles described above to the non-destructive measurement of thin film thickness and complex refractive index. See below for the principles of ellipsometry.

Conventions for the signs of phase differ between physics and optics. Here we use the conventions of physics; see below for differences between this and optics.

For the method of calculating the absorption coefficient in (10)-⑤ from the actual transmittance measurement of the material, please refer to the following.

  • Related literature

[A] Shigenobu Sunagawa, Denzikigaku (Electromagnetism). Iwanami Shoten.

  • Update History

  • 2026-03 "Derivation of the Definition of Ellipsometry" deleted (moved to a separate page)

  • 2025-12  New release

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