Plane wave

A plane wave is a wave whose wavefronts, the surfaces of constant phase, are flat planes perpendicular to a single direction of travel, and whose amplitude is the same at every point of each plane. It is the simplest solution of Maxwell's equations in a uniform medium and the reference against which other waves are described: a collimated laser beam behaves locally like one over distances well within its Rayleigh range, and any beam can be written as a sum of them. A plane wave at 1550 nm oscillates at 193.4 THz, with an angular frequency of 1.215 × 10¹⁵ rad/s and a period of 5.17 fs.

Equation and sign convention

In complex notation a monochromatic plane wave is

$$\mathbf{E}(\mathbf{r},t) = \mathbf{E}_0\,e^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)} ,$$

with the physical field given by the real part. This is the $e^{i(kz-\omega t)}$ convention used throughout this site, including the Jones vector and circular polarization entries. Engineering texts often write $e^{j(\omega t - kz)}$ instead; the two describe the same wave but flip the sign of imaginary quantities such as the loss term in a complex index, so the convention must be known before comparing formulas.

The wavevector $\mathbf{k}$ points along the direction of propagation and has magnitude

$$k = \frac{2\pi n}{\lambda_0} = \frac{n\,\omega}{c} ,$$

where $n$ is the refractive index and $\lambda_0$ the vacuum wavelength. In vacuum at 633 nm, $k$ = 9.93 rad/µm; in fused silica ($n$ = 1.444) at 1550 nm, $k$ = 5.85 rad/µm. The wavefronts move at the phase velocity $\omega/k = c/n$, 2.08 × 10⁸ m/s in silica. Light is a transverse wave: in an isotropic medium $\mathbf{E}_0$ is perpendicular to $\mathbf{k}$, the magnetic field is perpendicular to both, and the direction of $\mathbf{E}_0$ sets the wave's polarization.

Intensity and field amplitude

The time-averaged power per unit area carried by a plane wave, the magnitude of the Poynting vector, is

$$I = \tfrac{1}{2}\,n\,\varepsilon_0\,c\,|E_0|^2 .$$

For $I$ = 1 W/cm² = 10⁴ W/m², the field amplitude is

$$E_0 = \sqrt{\frac{2I}{n\,\varepsilon_0\,c}} ,$$

which gives 2.74 kV/m in vacuum and 2.24 kV/m in glass with $n$ = 1.5. The magnetic flux density amplitude is $nE_0/c$, 9.2 nT in the vacuum case. Because intensity scales as $|E_0|^2$, a hundredfold increase in intensity raises the field only tenfold.

Why a plane wave is an idealization

A true plane wave has infinite extent and therefore infinite total power, so it cannot exist. Real beams are bounded, and a bounded beam spreads by diffraction. A Gaussian beam with a 1 mm waist radius at 1550 nm has a Rayleigh range of 2.03 m and a far-field half-angle of 0.49 mrad; within about a meter of its waist its wavefronts are nearly flat and a plane-wave description of reflection, refraction or interference at a surface is accurate. Fresnel coefficients, Snell's law and thin-film formulas are all derived for plane waves and apply to beams to the extent that the beam's angular spread is small.

Plane-wave decomposition

Any field in a plane can be written as a superposition of plane waves traveling in different directions, its angular spectrum. A plane wave tilted by $\theta$ to the axis has a transverse spatial frequency $\sin\theta/\lambda$: at 633 nm in air, a 10° tilt corresponds to 274 cycles/mm. Propagation then reduces to giving each plane-wave component the phase $e^{ik_z z}$, which is the basis of Fourier optics and of most beam-propagation codes. Components with transverse spatial frequency above $n/\lambda_0$ have imaginary $k_z$ and do not propagate; they are evanescent and carry the sub-wavelength detail of a field. Inside a waveguide the same idea appears in reverse: a guided mode can be viewed as a pair of plane waves bouncing between the core boundaries, whose axial wavevector component sets the effective index.

Common questions

What is the difference between a plane wave and a spherical wave?

A spherical wave spreads from a point source, with wavefronts that are concentric spheres and an amplitude that falls as $1/r$. Far from the source a small patch of a spherical wavefront is nearly flat, so it can be treated locally as a plane wave.

Why is the plane wave written with a complex exponential?

The complex form makes derivatives and phase shifts simple products. Only the real part is physical, and quadratic quantities such as intensity are calculated with the time average $\tfrac{1}{2}\mathrm{Re}(\mathbf{E}\times\mathbf{H}^*)$ rather than by squaring the complex field.

Is a laser beam a plane wave?

Approximately, near the waist of a well-collimated beam. Its intensity falls off across the beam, and away from the waist its wavefronts curve and its size grows.

References: Hecht, Optics 5th ed. 2017; Born & Wolf, Principles of Optics 7th ed. 1999; Saleh & Teich, Fundamentals of Photonics 3rd ed. 2019; Goodman, Introduction to Fourier Optics 4th ed. 2017.