In Wave mechanics, one speaks about interferences when two Onde S in the same way standard meet. This phenomenon often appears in Optique with the light waves, but it is also obtained with other types of waves, like the sound waves or the electromagnetic waves.

Definition

A Onde is modelled by a function has ( X , T ), X being the position in space (vector) and T being time.

When one has two distinct sources, two transmitters, creating two waves has ₁ and has ₂, in a point X given, the amplitude of has will be

has ( X , T ) = has ₁ ( X , T ) + has ₂ ( X , T )

In physics, one classically considers two “ideal” phenomena which occur when one mixes two waves sine oïdales:

• the interference when the two waves have the same frequency
• the Battement when the frequencies are slightly different.

This approach is justified by:

• the fact that all function continues can break up into a sum of sinusoidal functions (Transformée of Fourier);
• when the frequencies are very different, the result is much simpler.

Tools of study

One uses Interféromètre S to measure or visualize the interferences.

Let us quote the Fentes of Young, the Interféromètre of Michelson, Interféromètre with double mirror, the interferometer with transparent blade.

Sinusoidal waves

Of the same waves frequency out of phase (formula of the interferences)

One considers two of the same waves pulsation but of phases different (that can be caused by a multiple way of the wave in its propagation) from expressions:

$A\_1\; (T)\; =A\_\; \{01\}\; \backslash \; cos\; \backslash \; left\; (\backslash \; Omega\; T\; -\; \backslash \; varphi\_1\; \backslash \; right)$ and $A\_2\; (T)\; =A\_\; \{02\}\; \backslash \; cos\; \backslash \; left\; (\backslash \; Omega\; T\; -\; \backslash \; varphi\_2\; \backslash \; right)$

One seeks to put the resulting wave in the form:

$A\; (T)\; =A\_1\; (T)\; +A\_2\; (T)\; =A\_0\; \backslash \; cos\; \backslash \; left\; (\backslash \; Omega\; T\; -\; \backslash \; varphi\; \backslash \; right)$

While developing, it comes:

$A\_0\; \backslash \; cos\; \backslash \; Omega\; T\; \backslash \; cos\; \backslash \; varphi\; +\; A\_0\; \backslash \; sin\; \backslash \; Omega\; T\; \backslash \; sin\; \backslash \; varphi\; =\; A\_\; \{01\}\; \backslash \; cos\; \backslash \; Omega\; T\; \backslash \; cos\; \backslash \; varphi\_1+A\_\; \{01\}\; \backslash \; sin\; \backslash \; Omega\; T\; \backslash \; sin\; \backslash \; varphi\_1\; +A\_\; \{02\}\; \backslash \; cos\; \backslash \; Omega\; T\; \backslash \; cos\; \backslash \; varphi\_2+\; A\_\; \{02\}\; \backslash \; sin\; \backslash \; Omega\; T\; \backslash \; sin\; \backslash \; varphi\_2$

One must thus solve simultaneously:

$A\_0\; \backslash \; cos\; \backslash \; Omega\; T\; \backslash \; cos\; \backslash \; varphi\; =\; A\_\; \{01\}\; \backslash \; cos\; \backslash \; Omega\; T\; \backslash \; cos\; \backslash \; varphi\_1+A\_\; \{02\}\; \backslash \; cos\; \backslash \; Omega\; T\; \backslash \; cos\; \backslash \; varphi\_2$
$A\_0\; \backslash \; sin\; \backslash \; Omega\; T\; \backslash \; sin\; \backslash \; varphi\; =\; A\_\; \{01\}\; \backslash \; sin\; \backslash \; Omega\; T\; \backslash \; sin\; \backslash \; varphi\_1+A\_\; \{02\}\; \backslash \; sin\; \backslash \; Omega\; T\; \backslash \; sin\; \backslash \; varphi\_2$

from where:

$A\_0\; \backslash \; cos\; \backslash \; varphi\; =\; A\_\; \{01\}\; \backslash \; cos\; \backslash \; varphi\_1+A\_\; \{02\}\; \backslash \; cos\; \backslash \; varphi\_2$
$A\_0\; \backslash \; sin\; \backslash \; varphi\; =\; A\_\; \{01\}\; \backslash \; sin\; \backslash \; varphi\_1+A\_\; \{02\}\; \backslash \; sin\; \backslash \; varphi\_2$

Calculation of the amplitude of the resulting wave

One obtains the amplitude while raising squared and by summoning member with member the two equations:

$A\_0^2\; (\backslash \; cos^2\; \backslash \; varphi+\; \backslash \; sin^2\; \backslash \; varphi)\; =A\_\; \{01\}\; ^2\; (\backslash \; cos^2\; \backslash \; varphi\_1+\; \backslash \; sin^2\; \backslash \; varphi\_1)\; +A\_\; \{02\}\; ^2\; (\backslash \; cos^2\; \backslash \; varphi\_2+\; \backslash \; sin^2\; \backslash \; varphi\_2)\; +2A\_\; \{01\}\; A\_\; \{02\}\; (\backslash \; cos\; \backslash \; varphi\_1\; \backslash \; cos\; \backslash \; varphi\_2+\; \backslash \; sin\; \backslash \; varphi\_1\; \backslash \; sin\; \backslash \; varphi\_2)$

And as follows:

$A\_0^2=A\_\; \{01\}\; ^2+A\_\; \{02\}\; ^2+2A\_\; \{01\}\; A\_\; \{02\}\; \backslash \; cos\; \backslash \; left\; (\backslash \; varphi\_1-\; \backslash \; varphi\_2\; \backslash \; right)$

If the waves are superimposed in phase ($\backslash \; varphi\_1=\; \backslash \; varphi\_2$), one obtains:

$A\_0^2=\; \backslash \; left\; (A\_\; \{01\}\; +A\_\; \{02\}\; \backslash \; right)\; ^2$

One speaks about constructive interference , because the resulting power increases.

If the waves are superimposed into antiphase ($\backslash \; varphi\_1=\; \backslash \; varphi\_2+\; \backslash \; pi$), one obtains:

$A\_0^2=\; \backslash \; left\; (A\_\; \{01\}\; -\; A\_\; \{02\}\; \backslash \; right)\; ^2$

One speaks about destructive interference , because the resulting power decreases.

Between these two extremes, the total power varies according to the cosine of the difference of the phases.

Calculation of the phase of the resulting wave

One obtains the phase by reporting member to member the two preceding equations:

$\backslash \; tan\; \backslash \; varphi=\; \backslash \; frac\; \{A\_\; \{01\}\; \backslash \; sin\; \backslash \; varphi\_1+A\_\; \{02\}\; \backslash \; sin\; \backslash \; varphi\_2\}\; \{A\_\; \{01\}\; \backslash \; cos\; \backslash \; varphi\_1+A\_\; \{02\}\; \backslash \; cos\; \backslash \; varphi\_2\}$

Finally, the expression of the resulting wave is summarized with:

$A\; (T)\; =\; \backslash \; sqrt\; \{A\_\; \{01\}\; ^2+A\_\; \{02\}\; ^2+2A\_\; \{01\}\; A\_\; \{02\}\; \backslash \; cos\; \backslash \; left\; (\backslash \; varphi\_1-\; \backslash \; varphi\_2\; \backslash \; right)\}\backslash \; cos\; \backslash \; left\; (\backslash \; Omega\; T\; -\; \backslash \; arctan\; \backslash \; frac\; \{A\_\; \{01\}\; \backslash \; sin\; \backslash \; varphi\_1+A\_\; \{02\}\; \backslash \; sin\; \backslash \; varphi\_2\}\; \{A\_\; \{01\}\; \backslash \; cos\; \backslash \; varphi\_1+A\_\; \{02\}\; \backslash \; cos\; \backslash \; varphi\_2\}\; \backslash \; right)$

Waves of close frequency (beats)

See the detailed article Beat.

Current applications

Out of radio

Out of radio, an interference is the superposition of two or several waves. It is frequent, for the frequencies higher than a few hundred kilocycles, than a reception antenna receives simultaneously the direct wave coming from the transmitter and one (or several) wave reflected by an obstacle. The two signals will be superimposed and, according to the difference in phase between them, will see their amplitudes being added or will be withdrawn. This kind of interference is responsible for fading, term Anglo-Saxon indicating a more or less fast variation of the amplitude of the received signal. But the phenomenon is not limited to the only radio waves.

In the common direction, for the radio, that took the direction of “parasite” (it is acted in fact of the interference between the radio wave and a parasitic wave).

Sound and visual interferences

On a stereo chain, one can also reverse the connection of one of the two loudspeakers; then, while walking in the part, there will be places where the sound is cancelled, disappears. It is this phenomenon which is used in the helmets anti-sound.

In fact also interferences are at the origin of the phenomena of Diffraction (for example irisation of a thin oil film, decomposition of the light by a Compact disk).

When one grants a Musical instrument, one takes a sound of reference (Diapason, the of the Hautbois) and one adjusts the instrument so that the two sounds agree. When the difference in frequency is weak, sound beats are perceived, and these beats slow down when the frequencies approach.

Quantum interferences

The interferences by waves (vague, its, light) can be obtained by a mathematical reasoning. Elementary particles, as the electron S should not then interfere only these waves in the same way. However according to the quantum Mechanical , one cannot be restricted with the name of particle . Indeed, those are also waves. One should rather say that it is neither of the particles nor of the waves: they are objects presenting sometimes undulatory and sometimes corpuscular aspects. This concept of Dualité wave-corpuscle makes it possible to include/understand that what one often calls of the particles can also exhiber an undulatory behavior and thus to present interferences.

Thus, in 1961, C. Jönsson with Tübingen carried out an experiment where a wire of spider metallized separating an electron beam into two produced an interference of electrons. In practice, the electrons form impacts on a photographic plate, and the distribution of these impacts present of the fringes, same manner as for the light.

When it was possible to detect the photons and the electrons individually, one as could show as there does not need an assembly of particles to make interferences: when they arrive one by one, there are also interferences. That makes it possible to confirm celebrates it assertion of Dirac “each photon interferes only with him even” and the experiment of thought described by Feynman in its famous courses, where it raised the question to know if the figure of interference appears even if the electrons arrived the ones after the others in front of two slits.

It is in 1989 qu ' a team of researcher of Hitachi (manufacturing electron microscopes) succeeds in controlling the production of electrons and detection one by one and observing the appearance in time, electron after electron of the figure of interferences. The team of Hitachi can affirm that, in their experiment, the electrons passed one by one as indicated in the experiment of thought of Feynman. What one observes is that the successive impacts form the interference rings gradually.

Let us recall that Davisson and Thomson divided the Nobel Prize of physics of 1937 for “the discovery of phenomena of interferences which occur when one exposes crystals to an electron beam” what confirmed the theoretical thesis of Louis de Broglie which accepted the Nobel Prize in 1929 for its discovery of the undulatory aspect of the electron.

Since, interferences were observed with Neutron S, Atome S and even of the Molécule S like the Carbone 60. Indeed with a Condensate of Bump-Einstein, it is possible to make interferences while crossing in both and observing the two halves mix. What is remarkable in these results, it is that the assertion of Dirac seems to apply to any particle, whether it is a Boson or a Fermion.

The same result exists for the interferences produced by photons: to see???.

Holography

An article HTTP: /Natural www-ssrl.slac.stanford.edu/research/highlights_archive/lensless.html also appeared in , 432,885 (2004) makes it possible to see how the interferences make it possible to make Holographie in x-rays.

It is imagery without lenses. (But in opposite direction, if one uses lenses to make diffraction or interferences that also becomes of holography on line.)

The Cristallographie X-ray is a technique of interference which by transformations of Fourier makes it possible to make reconstitutions or “images” 3D of molecules on a nanometric scale, but that is not done by recording the phase.

Stefan Eisebitt (BESSY, the Berlin Electron Storage Boxing ring Company for Synchrotron Radiation, Berlin, Germany) and Jan Luning (Stanford Synchrotron Labor Radiation, the USA), succeeded in making holography with X-ray in illuminant an object positioned on a hole of 1.5 micrometer with at side a hole of reference of 100 nanometer, these two holes being enlightened with a beam of x-rays coherent.

That produced a figure of interference or hologram which is recorded with a camera CCC.

The Transformée of fast Fourier makes it possible to obtain a “image”.

They thus observed the magnetic fields of Multicouche S cobalt-platinum.

Nanometric structures on the scale should be able to be observed: nanocristaux and components of micro-electronics, cells and proteins complex.

Another imminent technological advancement should make progress this field of holography x-rays:

The lasers with electron which will give a pulsated source of some femtosecondes very intense and coherent of x-rays will make it possible to observe the movement of nanoparticules or even of atoms

" Lensless Imaging off Magnetic Nanostructures by X-ray Spectro-Holography" - S. Eisebitt, J. Luning, W.F. Schlotter, Mr. Lorgen, W. Eberhardt and J. Stohr

See too

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