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The principle and description of polarization
~Understanding polarization states derived from electromagnetic wave equation~  ... (PL)

Polarization is a phenomenon determined by how the electric field vector of light vibrates. This article carefully explains, with mathematical formulas, how polarization is generated and described, using the wave equation of electromagnetic waves (EM1) as a guide.

table of contents:

・Description of elliptic polarization due to the p/s polarization component of electromagnetic waves incident on the interface

  • Description of elliptic polarization due to the p/s polarization component of electromagnetic waves incident on the interface

・Description of elliptic polarization using the p/s polarization component

In a scenario where electromagnetic waves are incident on an interface, when the plane containing the normal to the interface and the incident light is defined as the plane of incidence, the component of the electric field that vibrates parallel to the plane of incidence is defined as p-polarized light, and the component of the electric field that vibrates perpendicular to the plane of incidence is defined as s-polarized light. This can be illustrated as follows.

電磁波における様々な公式の導出_286.png

In this case, any electromagnetic wave incident on the interface can be described as the sum of p-polarized and s-polarized waves, as shown below.

偏光の原理とその記述_001.png

Here, we define the unit vector of the amplitude vector as follows.

偏光の原理とその記述_002.png

However, the absolute value of the amplitude vector extends the concept to complex numbers as follows (this is called the complex amplitude). This is to describe the amplitude in a way that includes the relative phase difference δ between p-polarization and s-polarization.

偏光の原理とその記述_003.png

Using this, (1)-②a,b can be rewritten as follows.

偏光の原理とその記述_004.png

Let the scalar components (i.e., the coefficients of the unit vectors) of (1)-② a',b' be as follows.

偏光の原理とその記述_005.png

Furthermore, the real part of (1)-⑤a,b is set as follows.

偏光の原理とその記述_006.png

Transforming (1)-⑥a results in the following:

偏光の原理とその記述_007.png

Substituting (1)-⑥b into this, the position vector and time are eliminated, resulting in the following:

偏光の原理とその記述_008.png

Squaring both sides, we finally obtain the following equation.

偏光の原理とその記述_009.png

This represents an elliptical shape and shows the trajectory of the tip of the electric field vector over time as it changes at an arbitrary position vector.

Especially,

[i] When δ = 2m × π (where m is an integer),

偏光の原理とその記述_010.png

This represents a straight line, so it is called linear polarization.

[ii] When δ = (2m + 1) × π (where m is an integer),

偏光の原理とその記述_011.png

This also represents a straight line, so it is called linear polarization.

[iii] When δ = (m + (1/2)) × π (where m is an integer) and Es = Ep = E,

偏光の原理とその記述_012.png

This represents a circle with radius E, and is therefore called circularly polarized light.

・Derivation of the rotation angle and ellipticity of an ellipse

Here, the elliptical shape of (1)-⑦' is assumed to be the same as the elliptical shape shown below, rotated by an angle γ with respect to the origin (counterclockwise direction is considered positive).

偏光の原理とその記述_013.png

However, 𝑎 and 𝑏 denote the semi‑major (or semi‑minor) axis and the semi‑minor (or semi‑major) axis of the ellipse, respectively, and both take positive values.

Furthermore, the ellipticity χ is defined by the following equation.

偏光の原理とその記述_014.png

The relationships shown so far can be illustrated as follows:

偏光の原理とその記述_015.png

Here, γ is an angle with a direction where counterclockwise is positive, and χ is a positive angle with no direction.

The relationship between (Esrt_Re , Eprt_Re ) and (E'srt_Re , E'prt_Re ) can be expressed as follows:

偏光の原理とその記述_016.png

Substituting (1)-⑩ into (1)-⑦' and rearranging, we get the following:

偏光の原理とその記述_017.png

(1)-⑧ and (1)-⑪ are the same,

偏光の原理とその記述_018.png

From (1)-⑪b and (1)-⑫b,

偏光の原理とその記述_019.png

From (1)-⑨, (1)-⑪a,c, and (1)-⑫a,c, the following is obtained.

偏光の原理とその記述_020.png

Here, we will use the following relationship.

偏光の原理とその記述_021.png

Substituting these into (1)-⑭ and rearranging, we get the following:

偏光の原理とその記述_022.png

Here, from (1)-⑬, cos2γ can be expressed as follows.

偏光の原理とその記述_023.png

Substituting this into (1)-⑭' and rearranging, we obtain the following:

偏光の原理とその記述_024.png

Also, since a and b are positive values, tanχ is a positive value from (1)-⑨, and tanχ can be found from (1)-⑭'' as follows.

偏光の原理とその記述_025.png

Generally, ellipticity is expressed in the form sin2χ, and from (1)-⑭'' and (1)-⑭''', the following equation is finally obtained.

偏光の原��理とその記述_026.png

Here, the compound terms that appeared along the way are reduced to the same equation through subsequent calculations.

  • Related Topics

For a detailed explanation of the electromagnetic wave formula, please refer to the following page.

  • Related literature

[A] Inoue, K. & Nakajima, S., Optics, Kyoritsu Publishing

[B] Tanaka, T., Introduction to Optics, Shokabo Publishing

[C]Max Born & Emil Wolf, Principles of Optics, Cambridge University Press

  • Update History

  • 2026-06  Newly released.

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