For example, incoming and outgoing light can be considered as reversals of each other,[1] without affecting the bidirectional reflectance distribution function (BRDF)[2] outcome.
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).
The Stokes–Helmholtz reversion–reciprocity principle[3][4][5][6][7][8][9][10][11][12][13][1][14][15][16][17][18][19][20][21][22][excessive citations] was stated in part by Stokes (1849)[3] and with reference to polarization on page 169 [4] of Hermann Helmholtz's Handbuch der physiologischen Optik of 1856 as cited by Gustav Kirchhoff[8] and by Max Planck.
[8]Simply put, in suitable conditions, the principle states that the source and observation point may be switched without changing the measured intensity.
[1][12] In his magisterial proof[23] of the validity of Kirchhoff's law of equality of radiative emissivity and absorptivity,[24] Planck makes repeated and essential use of the Stokes–Helmholtz reciprocity principle.
[9][10][11][12] The Helmholtz reciprocity theorem has been rigorously proven in a number of ways,[25][26][27] generally making use of quantum mechanical time-reversal symmetry.
As these more mathematically complicated proofs may detract from the simplicity of the theorem, A.P Pogany and P. S. Turner have proven it in only a few steps using a Born series.
In doing so, the series may be iterated through in the usual way to generate the following integral solution: Noting again the form of the Green's function, it is apparent that switching
For that reason, this principle has important applications in the field of transmission electron microscopy (TEM).
The notion that conjugate optical processes produce equivalent results allows the microscope user to grasp a deeper understanding of, and have considerable flexibility in, techniques involving electron diffraction, Kikuchi patterns,[29] dark-field images,[28] and others.
An important caveat to note is that in a situation where electrons lose energy after interacting with the scattering medium of the sample, there is not time-reversal symmetry.
In the case of inelastic scattering with small energy loss, it can be shown that reciprocity may be used to approximate intensity (rather than wave amplitude).
We can therefore conclude that the distortions to reciprocity due to magnetic fields of the electromagnetic lenses in TEM may be ignored under typical operating conditions.
Generally, polepieces for TEM are designed using finite element analysis of generated magnetic fields to ensure symmetry.
Meanwhile, their excitation polarities are exactly opposite, generating magnetic fields that cancel almost perfectly at the plane of the sample.