Petula Clark

The theory of diffraction , in its elementary form, rests on the principle of Huygens-Fresnel. According to this principle, each point reached by a Onde behaves like a secondary source. The figure of Diffraction observed results from the interference of the waves emitted by the whole of the secondary sources. Although this theory does not utilize the nature of the wave (sound, luminous…), the vocabulary and the illustrations of this article will be borrowed from the field of optics.

The principle of Huygens-Fresnel is an approximation of the rigorous solution to the problem of diffraction given by the resolution of the equation of wave. It is valid within the framework of the paraxial approximation: i.e. when the distance between the object and the figure of diffraction is large in front of at the same time the size of the object and the size figure of diffraction.

Principle of Huygens-Fresnel

Statement

That is to say an incidental monochromatic wave on an opening. According to the principle of Huygens-Fresnel, any element of surface of the opening can be regarded as a secondary source, being gradually propagated (Huygens) and the amplitude of the wave emitted by this secondary source is proportional to the sum of each element of surface of the incidental wave (Fresnel). The waves emitted by these various sources interfere between them to give the diffracted wave.

One finds sometimes an expression corrected which holds account owing to the fact that in any rigor the point source is not isotropic. As it emits in a privileged direction, one adds sometimes a “factor of obliqueness”.

Mathematical expression

One considers an opening (Σ) contained in the $z=0$ plan. That is to say $E (P) =E (X, Y)$ the amplitude of the incidental wave in an unspecified point P of the opening, coordinates $(X, Y)$ . The amplitude of the wave emitted by the secondary source of surface dΣ around P is form $KE (P) D \ Sigma$ where K is a constant which it is not useful to seek to determine here.

When it arrives at the point of observation M, of coordinates $(X, there)$ in the $z=r$ plan, this wave has as an amplitude, in complex notation:

dE (M) = \ frac {1} {PM} KE (P) D \ Sigma e^ {J \ varphi_ {PM}} = \ frac {1} {PM} KE (P) D \ Sigma e^ {J \ frac {2 \ pi PM} {\ lambda}}

The factor 1/PM gives an account of the attenuation of the spherical wave emitted out of P and $\ varphi_ {PM}$ represents the dephasing of the wave between P and Mr.

The total amplitude in M is obtained by summoning the contributions of all the points of the opening, that is to say:

E (M) =K \ iint_ {\ Sigma} \ frac {E (P)}{PM} e^ {J \ frac {2 \ pi PM} {\ lambda}} D \ Sigma

Factor of transmission

The diffracting objects are not inevitably openings letting spend 100% of the wave on the level of the opening and anything to side. They can be objects attenuating the wave in a way different according to the point considered (slide for example, for the light) and/or from objects introducing a dephasing depending there too on the point considered.

To take into account these various possibilities, one introduces the factor of transmission, or transmission, T (P) =t (X, Y), of an object which is the relationship between the amplitude of the wave right after the object with that of the wave right before the object.

By noting E₀ (P) the amplitude of the wave right before the diffracting object, the amplitude of the diffracted wave is written then:

E (M) =K \ iint \ frac {E_0 (P)}{PM} T (P) e^ {J \ frac {2 \ pi PM} {\ lambda}} D \ Sigma

The transmission being defined for any point P pertaining to the plan of the diffracting object, the integrals are calculated of - ∞ with +∞

This formalism will be used for the diffraction of Fraunhofer.

Examples:

square Ouverture on side has letting pass 90% of the wave.

$T (X, Y) =0,9 \; \ mathrm {if} \; |X|$

T (X, Y) =0 \; \ mathrm {if not}

square Prisme on side has, index N and point angle α (small).

an incidental luminous ray at a distance from the top of the prism undergoes a dephasing φ=2πα there there (n-1)/λ where λ is the length of onde.

By supposing the perfectly transparent prism, the transmission is written:

t (X, Y) = \ exp (J \ phi) \; \ mathrm {if} \; |X|

t (X, Y) =0 \; \ mathrm {if not}

Diffraction of Fresnel

Under the usual conditions of observation, the sizes of the opening and phenomenon of diffraction observed are small in front of the distance R of observation. One a:

PM= \ sqrt {r^2+ (x-X) ^2+ (there there) ^2}

One can thus use a development limited to write:

PM \ simeq R \ left
\ frac {1} {2} \ left (\ frac {there there} {R} \ right) ^2 \ right]

By replacing PM by this expression in the exponential one complexes and 1/PM by 1/r (this approximation is sufficient here because 1/PM is not a periodic function), one obtains then:

E (M) =K' \ iint_ {\ Sigma} E (P) \ exp \ left \ frac {J \ pi} {\ lambda R} \ left [(x-X) ^2+ (there there) ^2 \ right \ right] dXdY

where $K'= \ frac {K} {R} e^ {J \ frac {2 \ pi R} {\ lambda}}$ . This integral is called '' transformation of Fresnel '' and makes it possible to determine the figure of diffraction observed remotely finished diffracting opening. This kind of diffraction can for example be observed on the edges of the geometrical shade of a screen, like below illustrated.

Diffraction of Fraunhofer

The diffraction of Fraunhofer or diffraction ad infinitum is a very important particular case where the plan of observation is located far from the diffracting object, this one being lit by a plane wave (point source ad infinitum) and being defined by its factor of transmission T (X, Y).

It is this phenomenon which fixes the ultimate limit of resolution that one can hope for of an optical instrument.

Expression of the diffracted wave

It will be supposed here that the source is located on the axis of the system and thus that E (P) is constant in the diffracting plan. One can then write:

\ exp \ left J \ frac {\ pi} {\ lambda R} (x-X) ^2 \ right \ simeq \ exp \ left J \ frac {\ pi} {\ lambda R} x^2 \ right exp \ left - J \ frac {2 \ pi} {\ lambda R} xX \ right

if $\ frac {X^2} {\ lambda R} \ L 1$ whatever the X corresponding to a point of the opening.

Therefore, if D is a dimension characteristic of the opening (ex: diameter for a circular opening), one is under the conditions of the diffraction of Fraunhofer if

\ frac {d^2} {\ lambda R} \ L 1

d²/λr is sometimes called number of Fresnel .

The same reasoning being of course valid also for the term (there there) ², the amplitude of the wave diffracted under the conditions of Fraunhofer is thus written:

E (M) =K

\ iint T (X, Y) \ exp \ left - \ frac {2j \ pi} {\ lambda R} (xX+yY) \ rightdXdY
where the constant term of phase during integration, $\ exp \ left J \ frac {\ pi} {\ lambda R} (x^2+y^2) \ right$ , and the constant amplitude E₀ (P) are contained in the new K" constant;.
It is noticed that the diffracted amplitude is proportional to the Transformée of Fourier of the transmission T (X, Y). More precisely, if one notes t the transform of Fourier of T, E (M) is proportional to t (x*k, y*k) with K the number of wave equal to $2 \ Pi/r$ .
Physically, it is noted that the amplitude of the diffracted wave depends only on the direction of observation (defined by the angles x/r and y/r), which justifies the name of diffraction ad infinitum.

In practice, to observe ad infinitum means enough being far from the object diffracting so that the number of Fresnel is much lower than 1 or, in the case of optics, to place itself at the hearth image of a lens. In this last case, the distance R must be replaced by the focal distance from the lens, F, in the preceding formulas.

Examples of figures of diffraction

Diffraction by a slit

Diffraction by a rectangular opening

A rectangular opening on sides has and B corresponds to a transmission T (X, Y) defined by:

T (X, Y) = 1 if |X| T (X, Y) = 0 if not

The calculation of the intensity diffracted by such an opening, i.e. square of the module of the amplitude E (M), gives:

I (X, there) =|E (X, there)|^2=I_0 sinc^2 \ left (\ frac {\ pi xa} {\ lambda R} \ right) sinc^2 \ left (\ frac {\ pi yb} {\ lambda R} \ right)

In this expression, I₀ corresponds to the maximum intensity on the screen (in the center) and “sinc” is the function cardinal Sinus defined by sinc (X) =sin (X) /x.

The figure opposite is a simulation of the figure of diffraction of Fraunhofer obtained with a rectangular opening on sides a=0,1 mm and b=0,3 Misters One took λ=0,5 μ m and one placed oneself at the hearth image of a lens of focal distance f=1 Mr.

The intensity of the maximum secondaries was artificially raised in order to make them visible.

In the absence of diffraction, the figure obtained would have simply been a point shining in the center of the screen corresponding to focusing by the lens of the incidental rays parallel with the axis.

It is noticed that the smallest side corresponds to the greatest spreading out of the light. Indeed, dimensions of the central spot are:

Δx=2λf/a=10 mm

Δy=2λf/b=3,3 mm

Diffraction by a curtain

It is an application of the preceding example. When the light is reflected on an object for example, while looking through a curtain, one can observe figures of diffraction such this one:

It results from the light diffraction by the curtain, whose fabric constitutes a whole whole of square openings. The measurement of the angle between the central spot and its neighbor makes it possible to obtain the step of the curtain.

The irisations of the spots come owing to the fact that each wavelength builds its own figure of diffraction, slightly different from that a close wavelength. The places where the figures coincide are white (in particular the central spot), the others are coloured. It is noted that the distribution of the colors is logical because maximum cardinal sine is regularly obtained (all the $\ Pi/2$ and X, distance from a center point of the task, is proportional to $\ lambda$

Diffraction by a circular opening

It is here more convenient to use the polar Coordonnées $(R, \ theta)$ rather than the Cartesian coordinates $(X, Y)$ . A circular opening of diamètre D corresponds then to a transmission $t (R, \ theta)$ definite par :

$t (R, \ theta) = 1$ if $|R| < d/2$ ,

t (R, \ theta) = 0

if not.

The calculation of the intensity diffracted by such an opening gives:

$I (\ rho) =|E (\ rho)|^2=4I_0 \ left (\ frac {J_1 (\ eta)}{\ eta} \ right) ^2$
where $\ eta = \ frac {\ pi \ rho D} {\ lambda R}$ and $\ rho = \ sqrt {x^2 + y^2}$

The distribution of intensity has the same symmetry of revolution as the pupil. The figure obtained is called figure of Airy . In this expression, $I_0$ corresponds to the maximum intensity on the screen (in the center) and $J_1 (\ eta)$ indicates the Fonction of Bessel of order 1.

The figure opposite is a simulation of the figure of diffraction of Fraunhofer obtained with a circular opening of diameter d = 0,2 Misters λ was taken; = 0,5 µm and one placed themselves at the hearth image of a lens of focal distance f = 1 Mr.

The intensity of the maximum secondaries was artificially raised in order to make them visible.

The ray of the central spot est
$\ Delta \ rho = 1.22 \ lambda F/d$ = 3 Misters.

By noting the ray of the opening $r = D/2$ , one obtient
$\ Delta \ rho = 0.61 \ lambda F/r$ = 3 Misters.

Résolution of an instrument of optique

Some general properties

See too

External bonds

Diffraction Fraunhofer
Diffraction Fraunhofer
a video explanatory on the principle of diffraction
a video explanatory on the diffraction of Fraunhofer
a video explanatory on the assembly of Fourier

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