Huygens' principle
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Image:Wellen-Brechung.png Huygens' principle (named for Dutch physicist Christiaan Huygens) is a method of analysis applied to problems of wave propagation (both in the far field limit and in near field diffraction). It recognizes that each point of an advancing wave front is in fact the center of a fresh disturbance and the source of a new train of waves; and that the advancing wave as a whole may be regarded as the sum of all the secondary waves arising from points in the medium already traversed. This view of wave propagation helps better understand a variety of wave phenomena, such as diffraction.
For example, if two rooms are connected by an open doorway and a sound is produced in a remote corner of one of them, a person in the other room will hear the sound as if it originated at the doorway. As far as the second room is concerned, the vibrating air in the doorway is the source of the sound. The same is true of light passing the edge of an obstacle, but this is not as easily observed because of the short wavelength of visible light.
In essense, Huygens principle is the same statement as the main postulate of quantum field theory stating that any particle (real or virtual) propagates along any and all allowed paths - and thus the final amplitude (of probability to find it at certain point) is simply a sum of all amplitudes of its wave function as integrated along all allowed paths (i.e. is proportional to sum of all path integrals).
Diffraction
The most common application of Huygens' principle is for the case of a Plane wave (usually light) incident on an aperture of arbitrary shape. In this case, Huygens' principle simply states that a large hole can be approximated by a collection of many small holes so each is practically a point source (which contribution is easy to calculate). A point source generates waves that travel spherically in all directions (similar to circular waves caused by dropping small stone in a pond).
Consider the case of single slit diffraction, where we have one slit through which we shine light onto a distant screen. Suppose, we want to calculate at which point on the screen interference minima (dark stripes)occur. We then replace this relatively wide slit by an increasing number of narrow ones (subslits), and add waves produced by each. Obviousely, two small slits interfere destructively when their path lengths differ by <math>\lambda/2</math> (are in 180 phase difference). We can calculate (using phasors or a similar wave addition math) that for three waves from three slits to cancel each other the phases of slits must differ 120 degrees, thus path difference from the screen point to slits must be <math>\lambda/3</math>, and so forth. In the limit of approximating the single wide slit with an infinite number of subslits the path length difference between edges of slit must be exactly <math>\lambda</math> to get complete destructive interference (and so a dark stripe on the screen).
Fourier transforms
The qualitative argument we used to glean some understanding of single slit diffraction using only Huygens' principle is difficult to apply in general to apertures of truly arbitrary shape. The wave that emerges from a point source has amplitude <math>\psi</math> at location r that is given by the solution of the wave equation for a point source. That's exactly what we mean by the Green's function for the wave equation, which is in spherical coordinates
- <math>\psi(r)\propto \frac{e^{ikr}}{r}</math>
Therefore, if we approximate the amplitude from an aperture as coming from many point sources, we should sum together an infinite number of point sources. But that just describes a surface integral. Thus,
- <math>\Psi(r)\propto \int_\mathrm{aperture}\frac{e^{ikr}}{r}~da,</math>
which is simply the spatial Fourier transform of the aperture. Huygens' principle when applied to an aperture simply says that the far-field diffraction pattern is the Fourier transform of the aperture.
See also
de:Huygenssches Prinzip es:Principio de Huygens fr:Principe de Huygens-Fresnel ko:하위헌스의 원리 he:עקרון הויגנס ja:ホイヘンスの原理 fi:Huygensin periaate sv:Huygens princip th:หลักการของไฮเกนส์ zh:惠更斯原理