Plane Waves

Author: the photonics expert Dr. Rüdiger Paschotta (RP)

What are Plane Waves?

Plane waves are very often considered in wave optics as well as in other areas where waves play a role. They are the kind of waves with the simplest geometric form and mathematical description. By definition, they have plane wavefronts: at any moment of time, the locations of constant phase are planes. Also, they must have a uniform optical intensity, since otherwise the wavefronts could not remain plane during further propagation.

Plane waves satisfy wave equations for homogeneous media or in free space; therefore, one can say that plane waves are free-space modes.

A monochromatic plane wave is most easily characterized by a wave vector, with which the wave field (here treated as a scalar wave) can be described as the complex amplitude

with the wave vector, the magnitude of which is the wavenumber ($k$), and the angular frequency ($\omega$). In optics, the oscillating quantity is often the electric field strength ($E$), which can be taken to be the real part of the complex amplitude. The wave vector indicates in which direction the wave travels, and its magnitude tells the phase change per unit length (for a fixed time). A plane wave has a well-defined direction of propagation with no divergence.

Note that there are different sign conventions in wave optics; the above equation is based on physicist's convention. According to a common convention in engineering, phasors involve the phase factor ($\exp \left(i \omega t - i \vec k \vec r \right)$), i.e., are the complex conjugate of the phasors used here.

Plane waves need to be extended infinitely because otherwise any usual wave equation would not be fulfilled. Therefore, plane waves actually never occur in reality. However, a real wave may at least approximate a plane wave over some volume.

Electromagnetic waves, being transverse waves, can also exhibit polarization. As plane waves have flat wavefronts with uniform amplitude and phase, they can have only linear, circular or elliptical polarization, and not e.g. a spatially variable polarization direction.

The spacing of the wavefronts is the wavelength. That quantity is specifically defined for plane waves, or for waves which at least approximate them. Note that in a laser beam, for example, which converges to a focus and then diverges, the wavefront spacing necessarily needs to undergo some changes. This is true even for that spacing on the beam axis; for Gaussian beams, there is the Gouy phase shift which somewhat affects that spacing in the region of the focus.

Note that other properties, which are frequently attributed to light in general, apply only to plane waves. For example, the phase velocity of light is defined for plane waves. Also, one often calculates chromatic dispersion based on the assumption of plane waves, and the results are not valid for light propagating in a waveguide, for example (guided waves).

Due to the rather short wavelengths in optics, a reasonable approximation of plane waves is possible without covering large volumes of space. For example, some volume within a laser beam with a beam radius of only a few millimeters can be taken as such an approximation.

Fourier optics is an area of optics where light beams and other light waves are essentially decomposed into plane waves, using spatial Fourier transforms.

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