Impulse optics

The impulse optical is the branch of the Optique which treats interactions between the impulses of Lumière and the Matière.

The Laser development of the S.A. allowed to produce luminous impulses increasingly shorter which go down today below one duration a Femtoseconde. These impulses have the characteristic to have a very broad frequency spectrum, so that their propagatives properties different from those of the longer impulses are used today industrially.

Generation of luminous impulses

The continuous development of the laser techniques during the past decades led to the production of luminous impulses increasingly shorter in duration and always more intenses.

Laser

In this part the elements of physics of the cavities Laser are developed which allow the generation of the luminous impulses of short duration. A laser is composed of two basic elements: an amplifying medium and a pair of mirror forming an interferometer of Fabry-Perot.

Stimulated emission

In 1915, Albert Einstein to make compatible between them the theory of the radiation of the black Body and the concept of Photon, found ten years earlier, invents the stimulated emission of light. When an electronic system is in an excited state, if one makes it cross by light whose photons have an energy resounding with the possible electronic transitions, the electrons in the excited levels fall down by emitting twin photons of the incidental photons. By twins one understands photons which have same energy, the same direction of propagation, the same direction of polarization and with which the waves associated are in phase.
A material system in which the electrons have an excess of potential energy can thus be used as amplifying medium for the light.

Fabry-Perot cavity

It is enough to put a feedback on an amplifying medium to produce an oscillator. The best way of including/understanding this point is to consider the Effet Larsen which occurs when a microphone approaches a loudspeaker, in a theater for example. Its product by the loudspeakers is the result of the amplification of an electrical signal, if part of its product is collected by the microphone, this one creates an electrical signal which is in its amplified turn and addition with the signal of origin. The system enters in oscillation in the form of a very intense acute sound. Feedback on the amplifying medium is played by the signal of the microphone and one can say that the Larsen effect is the equivalent of the laser effect in acoustique.
How to make a feedback in optics? It is very simple: it is enough to return part of the light emitted by an amplifying medium using a mirror. Best is still to define a line of propagation by opposing two parallel mirrors between which one places an amplifying medium. The light can then make many return tickets between these two mirrors and be amplified with each passage in the amplifying medium. But by making this type of assembly one reaches the interferometer of Fabry-Perot which has optical properties completely surprising.

Axial modes of a laser

Thus let us consider a pair of distant mirrors the length L and parallels between them. To simplify it will be supposed that they have an infinite reflection any light which fall above is considered. If one sends a photon to the surface of the first mirror, outside, it is considered with a probability equal to 1! With what can thus serve the second mirror of the interferometer so already the first stops all the light? It is the reasoning in photon which disturbs experimental reality here. To include/understand the experimental behavior of the interferometer of Fabry-Perot, it is necessary to consider the light under its undulatory aspect. Indeed, the wave associated with the photon has an infinite space extension in its direction of propagation and is thus on both sides two mirrors. One shows without difficulty which if the distance L between the two mirrors is a multiple entirety of half the wavelength of the light which one considers, then the multiple reflections of the wave which occur between the two mirrors constructivement interfere in the direction of the propagation and for this value the pair of mirror is completely transparent! Each time one increases or decreases the length L of the cavity of half the wavelength of the light the interferometer is transparent, if not it reflects the light completely. One calls axial modes of a laser cavity, the whole of the frequencies amplified in the amplifying medium which can there be established. These frequencies are separated from a quantity equalizes with

c/2L

, C being speed of light. The reverse of this quantity

2L/C

being simply the time of flight of light during a return ticket enters the two separate mirrors of the distance L.

orders of magnitude

The electric field of a red light having a wavelength 620 Nm oscillates at the frequency of

\ nu_0=4,84 X 10^ {14} Hz=484THz

. A 1,5 m length cavity with for Eigen frequency

f=c/2L=1x10^8Hz=100 MHz

which corresponds to the time of flight of the light in the laser cavity

T=2L/c=1x10^ {- 8} s=10 ns

Synchronization of mode

The generation of luminous impulses rests on the phenomenon running of Battement between close frequencies. Two slightly désaccordées notes of music produce a single sound whose intensity varies periodically at a frequency which is worth the difference of the starting frequencies. The same applies with the light: two close frequencies beat with the difference in the frequencies. This comes from what two arches of the waves are added while being superimposed, then deviate to arrive at the opposition of phase and finally to reconsider in superposition an time interval which is the reverse of their difference in frequency.

Duality “time-frequency”

One can describe a luminous impulse by his electric field:

\ epsilon (T) = \ epsilon_0 (T) e^ {I \ omega_0 T}

where

\ epsilon_0 (T)

is a function of time in the shape of bell which describes the form of the impulse and

\ omega_0

is the pulsation of the carrier wave described by the sinusoidal function complexes

e^ {I \ omega_0 T}

.
By taking the transform of Fourier of this function

E (\ Omega) = \ int_ {- \ infty} ^ {+ \ infty} \ epsilon (T) e^ {I \ Omega T} dt

one obtains a new function

E (\ Omega)

called spectrum of the impulse. One returns to the starting function by the mathematical transformation:

\ epsilon (T) = {1 \ over 2 \ pi} \ int_ {- \ infty} ^ {+ \ infty} E (\ Omega) e^ {- I \ Omega T} D \ Omega

The two functions $\ epsilon (T)$ and $E (\ Omega)$ are not independent, they are Transformée of Fourier one of the other; one them known as dual of Fourier. He results from this duality that the quantities which are used to measure these two functions are not independent one of the other.

Thus the duration of the impulse $\ Delta {T}$ and the width of its spectrum $\ Delta \ omega$ are dual one of the other and one shows that there is between them the universal relation:

\ Delta \ Omega \ cdot \ Delta {T} \ Ge {1 \ over2}

An important consequence of this relation is quite simply that more one will want durations of impulse short plus the spectral width of these impulses will have to be broad. There one will recognize a property described by the relations of uncertainty of Heisenberg in quantum Mécanique: better is the knowledge of the position of a particle, less good is the precision of the knowledge its speed, for example. Quantum mechanics generalized this property of the traditional waves to the waves associated with the matière.

For a given spectral width, there exists an infinity of possible durations for an impulse which correspond each one to a law of phase different between the components from frequency from the spectrum. There exists, in particular, a particular impulse whose components of frequency are not out of phase between them, it is the shortest impulse which one can build with contents in frequency given. One speaks about transformed impulse limit, because it is in extreme cases low in duration; it corresponds to the equality in the expression above which connects lasted and spectral width.

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