Interference (wave propagation)
Interference (Lat. inter, between '. Ferire and over altfrz s'entreferir, hitting each other ' ) describes the superposition of two or more waves according to the superposition principle - that is, the addition of their displacements ( not the intensities) during their penetration. Interference occurs in all types of waves, ie in terms of sound, light, matter waves, etc.
The waves delete it from each other, it is called (complete ) destructive interference. Increase the amplitudes, then one speaks of constructive interference. The pattern of locations of constructive and destructive interference is referred to as the interference pattern. In the experimental setup appear alternately characteristic interference maxima and interference minima. A well-known example is the stripe pattern behind a double-slit arrangement. The occurrence of interference in the physical experiment is your proof of the wave nature of the investigated radiation. That in a double slit experiment also an interference pattern can be observed when the light (photons) are replaced by electrons caused at the beginning of the 20th century a new understanding of matter. The destructive interference is physically impossible for gravitational waves.
- 3.1 Interference of two waves of the same frequency and amplitude, but different phase
- 3.2 Interference of two waves of the same frequency but different amplitude and phase
- 3.3 superimposition of circular waves
- 4.1 Intuitive Explanation
- 4.2 Mathematical version
Based on interference physical effects
Beat and standing wave
Superimposed are two waves with unequal but close frequencies and so is given by the beat a pattern as shown in the first figure to the right ( lower graph). It forms a rapid oscillation (in dark red ), whose amplitude varies with a slow frequency (blue). Considering the intensities with a detector, a time averaging is additionally carried out on the sampling, wherein the sampling frequency of the detector.
For normal lighting and frequencies as far apart that beat practically irrelevant the ( time-averaged ) interference pattern is the sum of the interference patterns of the individual frequencies. This is because the interference between the waves of different frequencies - is eliminated in the time averaging - due to the lack of a fixed phase relationship. For dichromatic light is obtained in this case:
Where the Poynting vector.
To tune musical instruments you can change the appropriate settings until one perceives no longer beat together with a reference tone ( for example from a tuning fork ). The measurement of the beat signals can be used also for measuring otherwise ( for the instrument ) at high frequencies. This, however, a signal source is necessary, provides the signals with very stable and precise frequency.
The interference of two waves of the same wavelength but opposite propagation direction leads to a standing wave.
Coherence
The wave field, the waves resulting from the interference of two ( or more ), may only be stable over time when these shafts with each other have a (temporary ) fixed phase relationship. This is called coherent waves. If the waves are not monochromatic, ie consisting of a number of frequency components, so we define a coherence time, which describes how the waves can be up shifted against each other to still produce a stable interference pattern. This coherence time (or derived from coherence length ) is an important measure of the physical light source.
Double -slit experiment
A well-known experiment which illustrates the effect of the interference, the double -slit experiment ( Thomas Young, 1802). Here, in an electron or light beam an aperture of two narrow ( order of a wavelength ) is set up columns. Behind it is a detector or a screen on which the electrons or photons are detected. If only one slit open, the typical diffraction pattern of a single slit forms. Are you both column -free, so do not arise only two light strips, but several. The strip spacings are inversely proportional to the distance of the column. Interference and diffraction cause overlap as described above, the waves of light, thus forming an array of bright and dark stripes. In the double -slit experiment will be discussed again below in the section on interference in quantum mechanics.
Interferometer
In metrology, interferometers are used. These use interference phenomena to measure lengths or phase shifts with very high resolution. For this purpose a ( light ) beam is split into two coherent parts that are later superimposed again. The two beams lay there different routes and back. These differ by an integer multiple of the wavelength, is obtained at the output of the interferometer constructive interference. Do they differ by half a wavelength ( phase shift), we get destructive interference. If, now the interferometer initially to constructive interference, and then performs an additional phase shift in one of the two arms of a, one can determine this by the intensity at the output of the interferometer.
There are different implementations of this principle: Mach- Zehnder interferometers, Michelson interferometers, Sagnac interferometers, Fabry -Perot interferometer, etc.
Diffraction and resolution of optical devices
In the diffraction of waves at a gap, a wavefront encounters a gap. According to Huygens' principle now, from all points along the gap new hemisphere waves that interfere behind it. Thus, the typical diffraction pattern of a single slit is formed (see figure at right ). In this case, the diffraction pattern is the narrower, the wider the gap, and the smaller the wavelength.
Each optical device includes an inlet opening (eg: opening or first lens of a lens or primary mirror of a telescope ). This can be modeled as a gap and its diffraction pattern limits the resolution of the device. This limitation is due to the fact that each optical apparatus, an image of this entrance slit generated in any way, which is given by the width of the diffraction pattern.
Interference colors
White light, which on thin layers of optically transparent materials ( such as a film of oil on water, a thin oxide layer on metals, or simply bubbles ) is reflected, often appears in color. Thereby interfere the light that is reflected at the upper and lower boundary surface of the thin layer. With the direction of the light of a particular wavelength is then extinguished, and all that remains is the complementary color of the extinguished light left.
In the schematic drawing right the principle is shown in thin layers. Beam 2 is reflected at the surface, beam 1 after passing by the light blue shown thin layer. He travels a longer path around the track. If the path difference is an odd multiple of half a wavelength that is:
Then the beams 1 and 2 delete from. They interfere destructively. For a certain angle, this condition is satisfied only for certain wavelengths. These wavelengths are therefore not included in the reflected light. Light of other wavelength is reflected, however. The reflection of white light so that gets the complementary color of the light, the reflection interferes destructively. If blue light is not included in the reflection, the reflected light will thus yellow.
Interference colors in broken ice
Interference colors in nail polish layer
Interference colors in the plumage of the purple -throated nymph
Black Opal with a full, opalescent colors game
A known example of the appearance of interference colors of two closely adjacent surfaces of the Newton rings. Here is a converging lens with a long focal length on a flat glass plate. To the point of contact is formed between the glass surfaces around a gap slowly outwardly increasing thickness. If this arrangement is illuminated with monochromatic light from above, occur both in reflection and in transmission of light and dark concentric rings around the point of contact of lens and glass plate. If the test set is illuminated with white light, then produced colored concentric rings. The width of the rings and the intensity of their colors decreases with increasing radius.
The iridescent colors of the opalescence are also a result of interference. The light is scattered at small structures in the interior of the material. The colors of many butterflies, some particularly gorgeous iridescent birds or the gemstone opal based on this effect.
White light interference
The overlay continuously varying amplitude and wavelength (spectrum) generates an interference pattern within the coherence length. In white light interferometry, this behavior is exploited in order to obtain a unique length measurement. Another example can be found in optical coherence tomography, which can thereby capture three-dimensional structures.
Laser speckle
The light of an expanded laser beam has an almost perfect coherence perpendicular to the beam. This leads to laser light after the reflection at the uneven surface is not capable of generating interference. Then each point of the surface serves as a scattering center / point source of secondary spherical wave. A visual illustration of these point sources superimposed on the image, the light reaching the image point in different ways. This superposition leads to interference at the image point. Their result depends on the exact run-lengths of light between point source and pixel. A path length difference in the size of half a wavelength of the light determines destructive or constructive interference. Overall, a randomly distributed point pattern on the image of the figure.
Anti-noise
In the acoustic interference is utilized for reduction of disturbing noises, see anti-sound. This principle is used, for example in headphones for airplane pilots used to dampen engine noise locally.
Applications in wireless technology
By shifting the phase between the antenna elements of a phased array antenna, the viewing direction can be changed very quickly.
Through detailed analysis of the phase shifts between the individual antennas of radio telescopes toward the distant radiation sources can be very accurately determined.
An antenna pattern shows the radiation pattern of the individual antennas or antenna groups, the shape of which is determined by the interference. In the Yagi -Uda antenna is focused in this way, the radiation energy in a narrow forward lobe, resulting in the desired directivity.
In Balanced duplexer a gas discharge tube is ignited at high transmission power, which almost acts like a short circuit on the waves. By clever power distribution to two separate branches of a waveguide having a different phase shift, and then merging the two units it is achieved that the transmission energy to the antenna flows ( constructive interference ) and not to the receiver ( destructive interference ).
A diplexer possible by destructive or constructive interference in separate branches of an array of waveguides that two way radios of different wavelengths can be used with an antenna. Similarly, the sum or difference of two gleichfrequenter signals is formed in a ring coupler.
Mathematical representation
A wave is usually written by a function of place and time This expresses that a wave propagates both in space and in time. Now superimposing several waves propagate in one place so the wave field can be there as a superposition (sum ) of the individual waves represent:
Interference of two waves of the same frequency and amplitude, but different phase
The superposition of two waves of equal frequency and amplitude can be calculated using the trigonometric addition theorems. The two shafts and with the common frequency, the amplitude and the phase, and by
Described, we obtain for the resulting superposition of waves
That is, it results in a wave of the same frequency, the amplitude of which depends on the difference of the phases of the two original waves, and whose phase is the average of the phases of the original waves. For the same phases of the waves (), the cosine is one. The result is an amplitude, that is, the amplitude is doubled compared to the initial amplitude, which corresponds to constructive interference. A phase difference of 180 degrees (), the cosine is zero, i.e., the resulting wave disappears. This corresponds to destructive interference.
Interference of two waves of the same frequency but different amplitude and phase
For the same frequency of the waves, but different amplitudes and phases can be calculated by using the resulting wave pointer arithmetic. The two shafts, and have the common frequency, and the amplitudes and the phases and
The resulting superimposition of the waves in the form of:
With the amplitude of:
And phase
Superimposition of circular waves
The left figure shows the interference of two circular wave groups of the same wavelength and amplitude. The crosses mark the location of the sources that circles the maxima of the respective partial wave. On white spots constructive interference occurs, in the positive direction of black constructive interference in the negative direction on. On the gray points there destructive interference. It will be appreciated that the minima are located on a hyperbola, whose foci are the same as the source locations of the shafts. We therefore speak of two point sources of a hyperbolic interference. The hyperbola is the curve of all points that have the travel time difference to the two source locations. The vertex distance corresponds to the time difference, if and represent the time reference of the two dining time functions and represents the media speed of propagation.
In the right figure, the change in the interference pattern as a function of wavelength ( increases from top to bottom ) and in dependence on the distance of the sources ( increases from left to right ) demonstrated. In the dark areas (around the interference minima ) there is destructive and bright ( maxima ) constructive interference.
Interference of two circular waves in dependence on the wavelength and the source distance
Interference in quantum mechanics
Intuitive Explanation
In quantum mechanics, interference phenomena play a crucial role. Particles (and more generally any states of a system ) are described by wave functions. These are the solutions of Schroedinger equation, similar to a wave equation may take a form. Thus, particles, that matter, in quantum mechanics behave like waves and also interfere (see wave -particle duality, matter waves). A well-known example is the interference of electrons in a double slit experiment (see also: the images on the right! ) Or the interference of two Bose -Einstein condensates.
The research group of Anton Zeilinger has succeeded in 1999 to observe an interference pattern of fullerenes ( molecules composed of 60 or 70 carbon atoms). These are by no means the heaviest particles could be observed for the quantum interference. The research group around Markus Arndt continued the experiments initiated by Zeilinger at the University of Vienna and, in 2010 Quantum interference with molecules of up to 430 atoms and masses up to nearly 7000 show Atomic mass units.
The remarkable thing about this form of interference, however, is that the measurement of which path has chosen a quantum object ( "Which Way" information), results that only this "used" is - so no interference occurs. In a double-slit arrangement, the interference pattern thus depends on whether you can figure out which way ( through slit 1 or slit 2) took the quantum object. This is true even if the path of the quantum object is not already noticed when passing the column but only later ( delayed measurement process ). Only when a recovery of the "Which Path " information never was or has been repaid by a quantum eraser again, arises behind the double slit, an interference image.
Mathematical version
In the Bra- Ket notation to an arbitrary quantum state in an orthonormal basis () can be represented. Here, the complex coefficients:
The probability that a system in state in the state is then measured as follows:
Important here is that not the probabilities of the particles are superimposed, but the (complex) wave functions themselves the probabilities Would superimposed, one would in the above formula lose the rear interference component and the interference pattern disappears.
De Broglie postulated already in the early 20th century that all massive particles have a wavelength can be attributed, in which the momentum of the particle and is the Planck constant. At this wavelength can directly construct the wave function of a particle, and so calculate the interference pattern with the above-described methods for the light.