The Helmholtz reciprocity principle describes how a ray of light and its reverse ray encounter matched optical adventures, such as reflections, refractions, and absorptions in a passive medium, or at an interface. It does not apply to moving, non-linear, or magnetic media.
For example, incoming and outgoing light can be considered as reversals of each other, without affecting the bidirectional reflectance distribution function (BRDF) outcome. If light was measured with a sensor and that light reflected on a material with a BRDF that obeys the Helmholtz reciprocity principle one would be able to swap the sensor and light source and the measurement of flux would remain equal.
In the computer graphics scheme of global illumination, the Helmholtz reciprocity principle is important if the global illumination algorithm reverses light paths (for example Raytracing versus classic light path tracing).
Physics
The Stokes-Helmholtz reversion-reciprocity principle was stated in part by Stokes (1849) and with reference to polarization on page 169 of Helmholtz's Handbuch der physiologischen Optik of 1856 as cited by Kirchhoff and by Planck.
As cited by Kirchhoff in 1860, the principle is translated as follows:
A ray of light proceeding from point 1 arrives at point 2 after suffering any number of refractions, reflections, &c. At point 1 let any two perpendicular planes a1, b1 be taken in the direction of the ray; and let the vibrations of the ray be divided into two parts, one in each of these planes. Take similar planes a2, b2 in the ray at point 2; then the following proposition may be demonstrated. If when the quantity of light i polarized in the plane a1 proceeds from 1 in the direction of the given ray, that part k thereof of light polarized in a2 arrives at 2, then, conversely, if the quantity of light i polarized in a2 proceeds from 2, the same quantity of light k polarized in a1 [Kirchhoff's published text here corrected by Wikipedia editor to agree with Helmholtz's 1867 text] will arrive at 1.
The most extremely simple statement of the principle is 'if I can see you, then you can see me'. Rayleigh stated the basic idea of reciprocity as a consequence of the linearity of propagation of small vibrations, light consisting of sinusoidal vibrations in a linear medium.
Like the principles of thermodynamics, this principle is reliable enough to use as a check on the correct performance of experiments, in contrast with the usual situation in which the experiments are tests of a proposed law.
In his magisterial proof of the validity of Kirchhoff's law of equality of radiative emissivity and absorptivity, Planck makes repeated and essential use of the Stokes-Helmholtz reciprocity principle.
When there are magnetic fields in the path of the ray, the principle does not apply. Departure of the optical medium from linearity causes departure from Helmholtz reciprocity. When there are moving objects in the path of the ray, the principle may be entirely inapplicable.
Helmholtz reciprocity referred originally to light. This is a particular form of electromagnetism that may be called far-field radiation. For this, the electric and magnetic fields do not need distinct descriptions, because they propagate feeding each other evenly. So the Helmholtz principle is a more simply described special case of electromagnetic reciprocity in general, which is described by distinct accounts of the interacting electric and magnetic fields. The Helmholtz principle rests mainly on the linearity and superposability of the light field, and it has close analogues in non-electromagnetic linear propagating fields, such as sound. It was discovered before the electromagnetic nature of light became known.