Harmony

The models and formulas that describe the world are so compact and beautiful.
And they convert into each other.

Preamble

In the mid of 19th century, James Clerk Maxwell wrote down his great equations for electric and magnetic fields. This was done based on the analysis of known experiments (sic!) Maxwell's equations form a complete system: they describe all (any) phenomena in electrodynamics (!). They played an important role in the emergence of the special theory of relativity. They turn out to be applicable in quantum mechanics when considering the motion, for example, of charged particles in external electromagnetic fields. Maxwell's equations are also in demand in astrophysics and cosmology ...

A large class of problems is associated with the propagation of electromagnetic waves (light). In the absence of charges and currents, the electric and magnetic fields are described by the wave equation (this directly follows from Maxwell's equations). For a harmonic signal, it all comes down to the Helmholtz equation:

(1)

Here

Any plane wave

(2)

is a solution to equation (1). Thus, any combination (sum) of plane waves is also solution of this equation.

Now let's analyze one particular problem of the propagation of a beam of waves (light) in one direction.
For example, toward the right from the lens of the "illuminator" shown in the figure

Based on the specifics of the task, let's seek the electric field in the form of a wave propagating along the 0Z axis

(3)

Here the amplitude A depends on the coordinates (is not a constant, as in (2)). That is, this is not a plane wave, but something more complex, propagating along 0Z. Let's try to find this amplitude. Write the derivatives:

If the amplitude A does not change too "fast/abruptly" along direction of propagation 0Z (and why would it under normal conditions?), then:

And now from (1) we have :

(4)

Pay attention... Wow! This is the diffusion equation!
Just the diffusion coefficient is complex, and instead of time here is Z-coordinate.

So, the complex amplitude A changes (“diffuses” from “high density” to “low one“) in the X0Y plane (perpendicular to the propagation) as the beam propagates - it follows directly from (4)!
And this effect.... In optics, this is called diffraction.

Indeed, equation (4) describes near-field diffraction (Fresnel diffraction)!

A very efficient numerical solution of (4) is implemented in the Beam Propagation Method (BPM). The method is used to solve a variety of problems. For example, it allows to estimate diffraction distortions at holes in screens when the screen thickness cannot be neglected.

This is how general Maxwell equations are "converted" to the Helmholtz equation, for specific conditions (the absence of charges and currents). Even more specificity allows to go over to the equation of "complex diffusion", which describes ... Fresnel diffraction.

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