Optical diffraction pattern

Formulas of optics

The principle of the diffraction patterns rests on the same formula being able to be shown either by the geometrical Optique, or by the electromagnetic theory of Maxwell. It is based on the Principe of Huygens-Fresnel.

Calculation on a network is very similar to the calculation made on the Fentes of Young (see this article): the Différence of walk between two features (thus the dephasing of the rays diffused by two close features) is calculated same manner. The difference is that instead of having the sum of two functions of wave, one with the sum of an “infinite” series (the number of features being very large):

$E (X, T) = \ sum_ {I = 0} ^ {\ infty} E_i = E_0 \ cdot \ sum_ {I = 0} ^ {\ infty} \ cos (\ Omega T - I \ cdot \ Delta \ varphi (X))$

by taking again the notations of the article Slits of Young :

X is the X-coordinate of the point on the display screen, on an axis perpendicular to the features of the network;
$E_0 \ cdot sin (\ Omega T)$ is the amplitude of the incidental wave arriving on feature 0, ω being the pulsation;
$\ Delta \ varphi (X) = \ frac {2 \ pi V X} {\ lambda D}$ is dephasing between two close features, with

V the step of the network;
D the distance enters the network and display the screen of the figure of diffraction (screen parallel with the plan of the network).

If one is in condition of diffraction between two features (case of the slits of Young), one is it also between all the features: dephasing is a multiple of 2π everywhere. One thus will have the maximum one of intensity in

$x_k = K \ cdot \ frac {\ lambda D} {V}$

or, if the screen is “ad infinitum” (i.e. with several meter or in the focal plan image of a convergent Lentille), one considers the angle of deviation α giving a maximum of intensity:

\ alpha_k = \ arcsin \ left (K \ cdot \ frac {\ lambda} {V} \ right)

Width of the lines and size of the network

The difference between a network and slits of Young is that the intensity will be cancelled as soon as one deviates from the conditions of diffractions. Instead of having a peak whose form is in cos², there is a very fine peak: if one places oneself in x_k + δ X , then

$\ Delta \ varphi (X) = 2 K \ pi + \ frac {2 \ pi V \ delta X} {\ lambda D}$

a feature I will be in opposition of phase with feature 0 if there exists an entirety J checking

$i \ cdot \ frac {2 \ pi V \ delta X} {\ lambda D} = \ pi + 2j \ pi$

that is to say:

$i = (1 + 2j) \ cdot \ frac {\ lambda D} {2V \ delta X}$

In the case of the slits of Young, there is cancellation only when λ D /2 V δ X is whole; here, it is enough to take J sufficiently large so that the fraction becomes whole. In theory (infinite number of enlightened features), the intensity is thus null except condition of diffraction (the whole of the real is the adherence of the whole of the rational ).

In practice, the network has a finished number of features, and only a portion of the network is enlightened. If one calls NR the number of lit features, then the intensity is cancelled for the first time when

\ delta X = \ frac {\ lambda D} {2NV}

if NR is odd, or in

\ delta X = \ frac {\ lambda D} {2 (N-1) V}

if it is even. The width of the peak is thus divided by NR (or NR -1) compared to the slits of Young.

The case of diffraction ad infinitum can treat in the reciprocal Espace.

Formulate networks

When the light strikes a network, it considered or is transmitted only in certain points, the features of the network. Each feature diffuses the light in all the directions, and these waves interfere.

As the features are laid out in a regular way, there is an alternation constructive interference/destructive interference according to the scattering angle. One can thus calculate, for a wavelength λ given, the angles R for which one will have a constructive interference.

; Network in reflection

Is N ₁ the Indice of the propagation medium of the incidental Onde (of Wavelength λ). That is to say I the Angle of incidence and R the angle of reflection for which one has an interference constructor. Either has the step of the network and m a Integer. There is

n_1 \ sin r=-n_1 \ sin I + m \ frac {\ lambda} {has}

; Network in transmission

Is N ₁ the Indice of the propagation medium of the incidental Onde (of Wavelength λ), and N ₂ the index of the transparent medium in the slit of the network (one can have N ₁ = N ₂ if the slit is simple obviously). That is to say I the Angle of incidence and R the angle of refraction for which one has an interference constructor. Either has the step of the network and m a Integer. There is

n_2 \ sin r=n_1 \ sin I - m \ frac {\ lambda} {has}

In these two formulas, the angles are describe by a algebraic Valeur.

The number m names the “mode”, or “order of diffraction”. In each studied case, the number of modes results from the preceding equations by noting that

-1 ≤ sin R ≤ 1

each wavelength is thus diffracted in several directions. In fact there exist more modes but this remains on the surface of the network.

Vocabulary

; Angular dispersion

One calls angular dispersion the derivative

\ frac {Dr.} {D \ lambda}

; Effectiveness

A_m

the amplitude of the wave thought of the order m .

the effectiveness resembles in all points the coefficient of reflection of a wave. It is defined, with the order m , by:

\ left|A_m \ right|^2 \ frac {\ cos R} {\ cos I}

; Free spectral interval (ISL)

It is defined by the report/ratio

\ frac {\ lambda} {m}

It corresponds to the maximum interval wavelength so that there is not covering of order.

; Resolution

the resolution is limited because the network has a finished dimension (convolution by function carries of a sampled signal, therefore problem of spectral covering). It is given by

\ frac {\ lambda \ cdot m} {V}

Applications

The applications are varied in Spectroscopie because the angle of exit depends on the Wavelength studied. Thus, the networks are used in the Spectroscope S of the type Littrow or in the assembly of Czerny-Turner (see the article dispersive Analyze in wavelength ).

The networks can be used like Monochromateur S: by choosing a direction, one can select only one wavelength. It is thus possible to use them in the reconcilable Laser S.

Moreover, when a network moves a length $x$ , it introduces a dephasing $\ frac {p \ cdot 2 \ pi} {V}$ , therefore thanks to the Interférence S between modes 1 and -1 one can go back to the displacement of the network. One can thus thus produce an incremental position sensor of high Résolution.

The networks are also very useful in teaching because they make it possible to include/understand the properties of the light; they are often used in practical works.

There exist also two-dimensional networks, composed of nonparallel lines or points. At the base, the Holographie consists in creating a two-dimensional network by impressing a photographic film. The restitution of the image is in fact the figure of diffraction on this network. Another example is the light diffraction on a Compact disk, the bit being as many points.

There exist finally three-dimensional networks: the crystals. Each node of the network (atom or molecule) is a site of diffusion. It is the base of the Diffraction of x-rays, of the figure of diffraction in electronic Microscopie in transmission, of the pseudo-lines of Kikuchi used in EBSD (electronic Microscopie with sweeping), and of the Neutron diffraction. See the articles Law of Bragg and Theory of diffraction on a crystal .

We saw above that the less one network with a dimension refers, plus the peaks of diffraction are broad. In the same way, less a Cristallite has atoms (the smaller it is), the more the peaks are broad. That makes it possible to estimate the crystallite size by diffraction of x-rays, to see the article Formule of Scherrer .

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