2.1. Complex amplitude of light wave and radiation intensity

The electromagnetic nature of the light waves was theoretically substantiated by British physicist Maxwell in his paper [15]. Ruling out the currents from his equation system, he obtained an equation that describes the propagation of electromagnetic disturbances:

$$
\overrightarrow{\Delta f} - \frac{\frac{1}{c^2} d^2 \overrightarrow{f}}{dt^2} = 0,\tag{1}
$$

where f ! is the electric E ! or magnetic H ! field vector tension, and с is light velocity.

Note that this equation exactly coincides with the equation of disturbance motion in elastic no compressive medium.

Further, let us target on plane monochrome waves that are a special case of a solution to Eq. (1). These are the waves wherein the electric and magnetic field vary according to the cosine law and vector f ! is a function of a single coordinate and time. Let us restrict ourselves to discussing the tension of the electric-filed vector. In a form independent from the origin selection for a plane monochrome wave, we have [16]:

$$\overrightarrow{E} = \text{Re}\{\overrightarrow{E}\_0 \mathbf{e}^{+i(\overrightarrow{k}\cdot\overrightarrow{r}-\omega t)}\},\tag{2}$$

where E ! <sup>0</sup> is some constant complex vector, k ! is a wave vector equaling ðω=cÞ n ! ¼ ð2π=λ<sup>Þ</sup> <sup>n</sup> !, r ! is a radius vector of a point of space, ω is the wave frequency, λ is the wavelength, and n ! is a single vector coinciding with the direction of the light wave propagation.

Let us further accept that all the waves have the same direction of the electric field vectors, then, when the waves impose their amplitudes, they can be added as scalar values. Besides, further in Eq. (2), we shall omit symbol Re and operate exponents instead of cosines. It is possible because in the problems that we are discussing below the final result will differ by an insufficient factor. Instead of Eq. (2), we have:

$$E(\mathbf{x}, y, z, t) = A(\mathbf{x}, y, z) \exp\left(-i\omega t\right),\tag{3}$$

where expression Aðx, y, zÞ¼jAðx, y, zÞjexp½iθðx, y, zÞ� was named complex amplitude in optics, jAðx, y, zÞj is the module of the complex amplitude, and θðx, y, zÞ is the wave phase at the observation point.

Now let us introduce the notion of light intensity as a value proportionate to volumetric density of radiation energy averaged by the time interval substantially exceeding the wave oscillation period:

$$I(\mathbf{x}, y, z) = \lim\_{T \to \infty} \frac{1}{T} \int\_{-T/2}^{T/2} |E(\mathbf{x}, y, z, t)|^2 dt = |A(\mathbf{x}, y, z)|^2. \tag{4}$$

Thus, the radiation intensity at a point of space equals the squared complex amplitude module.

#### 2.2. Interference of two waves

Let us discuss the light intensity distribution in superposition of two monochrome waves. Suppose that two waves of the same length λ were emitted by one point source in various directions, then two plane waves 1 and 2 crossing at angle θ were shaped by the optical systems. Let us take some point in the area of beam superposition. For certainty, let the wave amplitudes be the same equaling A0, but their initial phases ϕ differ. In compliance with Eq. (4), we have:

$$I = A \times A^\* = \left[ A\_0 e^{i(\vec{k\_1}\vec{r} + \varphi\_1)} + A\_0 e^{i(\vec{k\_2}\vec{r} + \varphi\_2)} \right] \times \left[ A\_0 e^{-i(\vec{k\_1}\vec{r} + \varphi\_1)} + A\_0 e^{-i(\vec{k\_2}\vec{r} + \varphi\_2)} \right]$$

$$= 2I\_0 + 2I\_0 \cos\left[ (\vec{k\_1} - \vec{k\_2})\ \vec{r} + \varphi\_1 + \varphi\_2 \right],\tag{5}$$

where <sup>I</sup><sup>0</sup> <sup>¼</sup> <sup>A</sup><sup>2</sup> <sup>0</sup>. Eq. (5) describes periodic light intensity distribution in the neighborhood of point r !, which was called light interference by T. Jung. Elementary calculations can demonstrate [17, 18] that minimum distance Δ between neighboring intensity maximums or minimums called bandwidth or period of the interference fringes is determined by formula (Eq. (6)):

$$
\Delta = \frac{\lambda}{2\sin\frac{\theta}{2}}.\tag{6}
$$

It follows from Eq. (6) that if θ tends to zero, Δ tends to infinity, which corresponds to tuning of the interferometer to "endless" band. If angle θ between vectors k ! <sup>1</sup> and k ! <sup>2</sup> equals 180°, then Δ ¼ λ=2, which corresponds to the wave interference in colliding beams. For θ = 60° value, Δ ¼ λ.

Now let us discuss the contrast of the interference fringes γ introduced by Michelson and determined by formula γ ¼ ðImax−IminÞ=ðImax þ IminÞ, where Imin and I max are the minimum and the maximum intensity values, respectively. From Eq. (5), it follows that in the case of a point light source discussed here and constant wavelength λ, contrast γ ¼ 1. Experience shows that if the light source is not point and (or) it emits light in some wavelength interval, the fringes contrast is less than 1.

It is commonly believed that case γ ¼ 0 corresponds to completely incoherent light; if 0 < γ < 1, the light is partially coherent, and coherence is the ability of waves to interfere. Interference of partially coherent light can be studied in Ref. [19]. In the text below, we will suppose that the light waves discussed here are completely coherent, i.e., two waves of the same amplitude generate an interference pattern with the contrast equal to 1. We can suppose with practical precision that similar waves are generated by laser light sources.
