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Transmission and Reflection Intensity in Two‑Interface Thin‑Film Interference: Derivation from the Electromagnetic Wave Equations

The intensity of interference can be quantitatively described as the superposition of electromagnetic waves. In this chapter, starting from the fundamental electromagnetic wave equation (EM1), we derive the general expression for the light intensity observed when multiple waves overlap. This general formula provides a unified basis for explaining a wide range of interference phenomena—from interference fringes formed at curved interfaces, such as Newton’s rings, to the transmission and reflection intensity formulas in thin‑film interference dominated by multiple reflections at two interfaces. It enables us to understand interference behavior that cannot be captured by the apparent bright and dark patterns alone, using a description grounded in electromagnetic wave theory.

  • Two-interface thin-film interference in the normal incidence approximation

・ Two-interface thin-film interference in the normal incidence approximation derived from the electromagnetic wave equation

Here, we derive the two-interface interference under the normal incidence approximation based on the electromagnetic wave equation (EM1) . First, the overview and parameters of the system model are shown below.

  • With respect to the z-axis, the region from -∞ to 0 is defined as medium 1, the region from 0 to d as medium 2, and the region from d to +∞ as medium 3.

  • Assume there is no absorption in each medium.

  • The incident light travels from z=-∞ in the +z direction and enters medium 2.

  • Let Ei be the electric field vector of the incident light, Et be the electric field vector of the transmitted light, and Er be the electric field vector of the reflected light.

  • The electric field vector of transmitted light is given as the sum of the electric field components corresponding to each transmission order in multiple reflections, and each component is represented by Etm (where m is a natural number).

Furthermore, the electric field vector of the reflected light is given as the sum of the electric field components corresponding to each reflection order in multiple reflections, and each component is represented by Erm (where m is a non-negative integer).

This can be expressed as follows:

電磁波の式から導く干渉縞の強度分布の定量表現_060.png

  • Transmission and reflection occur at the interface between medium 1 and 2, and at the interface between medium 2 and 3, respectively. The amplitude transmittance and amplitude reflectance at each surface are expressed as follows.

・When incident from medium 1 to medium 2,

        Amplitude transmission: t12

        Amplitude reflectance: r12

・When incident from medium 2 to medium 1,

        Amplitude transmission: t'12

        Amplitude reflectance: r'12

・When incident from medium 2 to medium 3,

        Amplitude transmission: t23

        Amplitude reflectance: r23

  • The wavenumbers of light in media 1, 2, and 3 are defined as k1 , k2 , and k3 (positive in the +z direction), and the angular frequency is defined as ω.

Based on these factors, the transmission and reflection of each surface can be illustrated as follows.

電磁波の式から導く干渉縞の強度分布の定量表現_122.png

The electric field vectors of the incident and reflected light are expressed as follows:

電磁波の式から導く干渉縞の強度分布の定量表現_092.png

Here, let EAi denote the electric‑field amplitude vector of the incident light.
Let EAtm (with 𝑚 a natural number) denote the electric‑field amplitude vector of the transmitted component corresponding to the 𝑚-th transmission order in the multiple‑reflection process.
Likewise, let EArm (with 𝑚 a non‑negative integer) denote the electric‑field amplitude vector of the reflected component corresponding to the 𝑚-th reflection order.

Here, the amplitude vector of the electric field is expressed as the product of the amplitude (scalar quantity) and the unit direction vector eEA , as follows.

電磁波の式から導く干渉縞の強度分布の定量表現_062.png

Substituting (1)-③ into (1)-② yields the following:

電磁波の式から導く干渉縞の強度分布の定量表現_093.png

Substituting (1)-②' into (1)-① yields the following:

電磁波の式から導く干渉縞の強度分布の定量表現_094.png

Here, we set it as follows:

電磁波の式から導く干渉縞の強度分布の定量表現_065.png

Substituting this into (1)-①', we get the following:

電磁波の式から導く干渉縞の強度分布の定量表現_095.png

Here, all changes in amplitude and phase that occur from just before the electric field is incident on the interface between medium 1 and medium 2 until just after it exits in the +z and -z directions after multiple reflections at the interface are included in the amplitude vectors of the transmitted and reflected light components corresponding to each transmission and reflection order. In this case, each amplitude is expressed as follows.

電磁波の式から導く干渉縞の強度分布の定量表現_096.png

Here, from Stokes' relation ( (EM1)-(9)-② ), the following can be said.

電磁波の式から導く干渉縞の強度分布の定量表現_068.png

Substituting (1)-⑥ into (1)-⑤, we get the following:

電磁波の式から導く干渉縞の強度分布の定量表現_109.png

Substituting this into (1)-④ and rearranging, we get the following:

電磁波の式から導く干渉縞の強度分布の定量表現_098.png

Therefore, the amplitude reflectance r and amplitude transmittance t of the entire system are as follows.

電磁波の式から導く干渉縞の強度分布の定量表現_099.png

From (EM1)-(6)-④ , the energy reflectance and energy transmittance can be calculated as follows.

電磁波の式から導く干渉縞の強度分布の定量表現_100.png

Therefore, the sum of the energy reflectance and energy transmittance is as follows.

電磁波の式から導く干渉縞の強度分布の定量表現_102.png

Here, if we denote the refractive indices of media 1, 2, and 3 as n1 , n2 , and n3 , then according to Fresnel's law ( (EM1)-(4)-⑨'c, (EM1)-(5)-⑨'c ), the amplitude reflectance and amplitude transmittance at each interface can be expressed as follows using the refractive index of each medium.

電磁波の式から導く干渉縞の強度分布の定量表現_103.png

Applying this to the relevant part in (1)-⑨, we get the following:

電磁波の式から導く干渉縞の強度分布の定量表現_075.png

Therefore, the numerator and denominator of (1)-⑨ are equal, and it can be said that the sum of the energy transmittance and energy reflectance is 1, as follows.

電磁波の式から導く干渉縞の強度分布の定量表現_104.png

・Complex plane representation of film thickness change and intensity change in two-interface thin film interference

From here on, we will focus solely on reflected light and consider the light intensity distribution resulting from interference.

If the amount of reflected light is Ir , then Ir is proportional to the square of the absolute value of the electric field vector of the reflected light, and can be calculated as follows.

電磁波の式から導く干渉縞の強度分布の定量表現_105.png

If we let the wavelength in a vacuum be λ0, then k2 = 2πn2 / λ0,

電磁波の式から導く干渉縞の強度分布の定量表現_106.png

The conditions for maximum and minimum interference between two interfaces can be reversed depending on the refractive index relationship of the medium. This reversal is difficult to grasp intuitively from the equations alone unless one understands how the phase change that occurs with each reflection affects the direction of the amplitude.
Therefore, by representing the reflection amplitude as a vector on the complex plane and geometrically tracking the change in its direction, we can intuitively follow the reversal of the interference conditions.

Therefore, from here on, we will find the trajectory on the complex plane for the reflection component of (1)-④', with d as the variable.

First, we find the standard form of the reflection component of (1)-④' as follows.

電磁波の式から導く干渉縞の強度分布の定量表現_107.png

Therefore, if we let h be the center of the locus of a circle on the complex plane and R be the radius of curvature, then the following holds.

電磁波の式から導く干渉縞の強度分布の定量表現_079.png

Here, from the reflection component of (1)-④' and (1)-⑬, the following relationship is obtained.

電磁波の式から導く干渉縞の強度分布の定量表現_108.png

From the above, the circular locus on the complex plane can be obtained as follows. This circular locus represents the phase rotation of the reflection amplitude with respect to the change in film thickness d, and the conditions for the maximum and minimum of the reflection intensity can be understood geometrically.

(i)

電磁波の式から導く干渉縞の強度分布の定量表現_032.png

At that time,

電磁波の式から導く干渉縞の強度分布の定量表現_110.png

(ii)

電磁波の式から導く干渉縞の強度分布の定量表現_034.png

At that time,

電磁波の式から導く干渉縞の強度分布の定量表現_111.png

(iii)

電磁波の式から導く干渉縞の強度分布の定量表現_036.png

At that time,

電磁波の式から導く干渉縞の強度分布の定量表現_112.png

(iv)

電磁波の式から導く干渉縞の強度分布の定量表現_039.png

At that time,

電磁波の式から導く干渉縞の強度分布の定量表現_113.png

(v)

電磁波の式から導く干渉縞の強度分布の定量表現_041.png

At that time,

電磁波の式から導く干渉縞の強度分布の定量表現_114.png

(vi)

電磁波の式から導く干渉縞の強度分布の定量表現_043.png

At that time,

電磁波の式から導く干渉縞の強度分布の定量表現_115.png

(vii)

電磁波の式から導く干渉縞の強度分布の定量表現_045.png

At that time,

電磁波の式から導く干渉縞の強度分布の定量表現_116.png

(viii)

電磁波の式から導く干渉縞の強度分布の定量表現_047.png

At that time,

電磁波の式から導く干渉縞の強度分布の定量表現_117.png

(ix)

電磁波の式から導く干渉縞の強度分布の定量表現_049.png

At that time,

電磁波の式から導く干渉縞の強度分布の定量表現_118.png

(x)

電磁波の式から導く干渉縞の強度分布の定量表現_051.png

At that time,

電磁波の式から導く干渉縞の強度分布の定量表現_119.png

(xi)

電磁波の式から導く干渉縞の強度分布の定量表現_053.png

At that time,

電磁波の式から導く干渉縞の強度分布の定量表現_120.png

(xii)

電磁波の式から導く干渉縞の強度分布の定量表現_055.png

At that time,

電磁波の式から導く干渉縞の強度分布の定量表現_121.png

From the above, we can conclude the following:

  • When the refractive index of medium 2 is higher than that of mediums 1 and 3, or lower than that of mediums 1 and 3,

・When d = mλ0/2n2, the intensity is minimal (especially when n1 = n3 , the intensity is zero).

・When d = (2m + 1)λ0/4n2, the intensity is at its maximum.

  • When the refractive index of medium 2 is higher than that of medium 1 and lower than that of medium 3, or higher than that of medium 3 and lower than that of medium 1,

・When d = mλ0/2n2, the intensity is at its maximum.

・When d = (2m + 1)λ0/4n2, the intensity is minimal.

Understanding this point allows us to determine, for example, the conditions under which the center of Newton's rings [1] becomes brighter and the conditions under which it becomes darker. Normally, in Newton's rings, medium 2 is air, so its refractive index is lower than that of mediums 1 and 3, and if they are in contact at the center, the center will be darker. This behavior is illustrated below.

・Example: Interference intensity distribution in Newton's rings

Consider the case where medium 1 has a gentle curvature with radius R and is in contact with medium 3 at x=0, as shown in the figure below.

電磁波の式から導く干渉縞の強度分布の定量表現_086.png

In this case, d can be expressed as a function of x as follows:

電磁波の式から導く干渉縞の強度分布の定量表現_087.png

Furthermore, under the condition that interference occurs (i.e., when the thickness of medium 2 is on the order of the wavelength), the relation 𝑥≪𝑅 holds, and therefore the following approximation is generally applied.

電磁波の式から導く干渉縞の強度分布の定量表現_090.png

Here, we set the parameters as follows:  

   n1=1.5

   n2=1

   n3 = 1.5

   λ0 = 0.00055 mm

   R=10000mm

In this case, from (1)-⑩, (1)-⑪', and (1)-⑮, the light intensity distribution with respect to x is represented as shown in the graph below, and since this corresponds to case (ix) in the complex plane, the values of d at the local maximum and minimum can also be indicated in the graph as follows.

電磁波の式から導く干渉縞の強度分布の定量表現_088.png

Up to this point, we have focused on the case of normal incidence for simplicity. However, in actual optical phenomena, oblique incidence is more common. In future articles, we plan to cover the cases of oblique incidence as well, including both s‑polarization and p‑polarization.

  • Related Topics

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

  • References

[1] Newton's rings - Wikipedia

  • 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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