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A Hecht Eugene, Optique, 2002, Addison Wesley, Jenkins FA & White HE, Fundamentals of Optics, 1957, 3e édition, McGraw Hill, New York, Longhurst RS, Geometrical and Physical Optics, 1967, 2e édition, Longmans, Londres, Cette page a été modifiée pour la dernière fois le 14 septembre 2020 à 06:48, This page is based on the copyrighted Wikipedia article.

La différence de phase entre les deux ondes est déterminée par la différence de distance parcourue par les deux ondes. The equation was named in honor of Joseph von Fraunhofer although he was not actually involved in the development of the theory.[3].

- = The angles of the first order beams are about 20°; if we assume the wavelength of the laser beam is 600 nm, we can infer that the grating spacing is about 1.8 Î¼m.

If W < λ, the intensity of the diffracted light does not fall to zero, and if D << λ, the diffracted wave is cylindrical. Diffraction par une fente de profondeur infinie. L = t ψ

In the far field, propagation paths for wavelets from every point on an aperture to a point of observation are approximately parallel, and a positive lens (focusing lens) focuses parallel rays toward the lens to a point on the focal plane (the focus point position on the focal plane depends on the angle of the parallel rays with respect to the optical axis). j

= ′ [1][2] In contrast, the diffraction pattern created near the object, in the near field region, is given by the Fresnel diffraction equation. Therefore, the amplitude of the total wave travelling in the direction θ is zero.

r / For example, when a slit of width 0.5 mm is illuminated by light of wavelength 0.6 Î¼m, and viewed at a distance of 1000 mm, the width of the central band in the diffraction pattern is 2.4 mm. If the light travelling at an angle θ from each slit has a path difference of one wavelength with respect to the adjacent slit, all these waves will add together, so that the maximum intensity of the diffracted light is obtained when: This is the same relationship that is given above. The width of the slit is W. The Fraunhofer diffraction pattern is shown in the image together with a plot of the intensity vs. angle θ. β 1 = -

r π {\ displaystyle L}. Généralement, une intégrale bidimensionnelle sur des variables complexes doit être résolue et dans de nombreux cas, une solution analytique n'est pas disponible.

where the array length in rad

′ . The output profile of a single mode laser beam may have a Gaussian intensity profile and the diffraction equation can be used to show that it maintains that profile however far away it propagates from the source.[15]. {\ displaystyle -a / 2} The same applies to the points just below A and B, and so on. sin A [ The form of the function is plotted on the right (above, for a tablet), and it can be seen that, unlike the diffraction patterns produced by rectangular or circular apertures, it has no secondary rings. Des franges d'interférence à double fente peuvent être observées en coupant deux fentes dans un morceau de carte, en éclairant avec un pointeur laser et en observant la lumière diffractée à une distance de 1 m. Si la séparation des fentes est de 0,5 mm et que la longueur d'onde du laser est de 600 nm, alors l'espacement des franges vues à une distance de 1 m serait de 1,2 mm. 1



a = e The finer the grating spacing, the greater the angular separation of the diffracted beams. 1

/ r j Par conséquent, l'amplitude de l'onde totale se déplaçant dans la direction θ est nulle. Lorsque deux ondes sont additionnées, le déplacement total dépend à la fois de l' amplitude et de la phase des ondes individuelles: deux ondes d' amplitude égale qui sont en phase donnent un déplacement dont l'amplitude est le double des amplitudes d'onde individuelles, tandis que deux ondes qui sont en les phases opposées donnent un déplacement nul. θ e

une

c β ⁡ = péché ′ For example, if a 0.5 mm diameter circular hole is illuminated by a laser with 0.6 μm wavelength, the Fraunhofer diffraction equation can be employed if the viewing distance is greater than 1000 mm. Cependant et en intégrant de à , ∫ ≪ 1 If the light is incident at an angle θ0, the grating equation is: The detailed structure of the repeating pattern determines the form of the individual diffracted beams, as well as their relative intensity while the grating spacing always determines the angles of the diffracted beams. ( y

θ

The difference in phase between the two waves is determined by the difference in the distance travelled by the two waves. ′ L'image de droite montre un faisceau laser diffracté par un réseau en n = 0, et ± 1 faisceaux. r r In the double-slit experiment, the two slits are illuminated by a single light beam. ′ {\displaystyle \beta =\omega /c=2\pi /\lambda }
Le graphique et l'image de diffraction simple fente. In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens. ( e (

[14] This technique can be used in a process called apodization—the aperture is covered by a Gaussian filter, giving a diffraction pattern with no secondary rings. 1 / e The Airy disk can be an important parameter in limiting the ability of an imaging system to resolve closely located objects. The phase of the contributions of the individual wavelets in the aperture varies linearly with position in the aperture, making the calculation of the sum of the contributions relatively straightforward in many cases. / A


/ t sin r The spacing of the fringes is also inversely proportional to the slit dimension. A detailed mathematical treatment of Fraunhofer diffraction is given in Fraunhofer diffraction equation. 2 ]

Un traitement mathématique détaillé de la diffraction de Fraunhofer est donné dans l'équation de diffraction de Fraunhofer .



Soit le tableau de longueur a parallèle à l'axe y avec son centre à l'origine comme indiqué sur la figure de droite. 1 Then the differential field is:[11], d = e Dans le champ lointain, les chemins de propagation des ondelettes de chaque point d'une ouverture à un point d'observation sont approximativement parallèles, et une lentille positive (lentille de focalisation) focalise des rayons parallèles vers la lentille vers un point sur le plan focal (la position du point focal sur le plan focal dépend de l'angle des rayons parallèles par rapport à l'axe optique). {\ displaystyle r_ {1} = ry \ sin \ theta} This is often known as the far field and is defined as being located at a distance which is significantly greater than W2/λ, where λ is the wavelength and W is the largest dimension in the aperture. When the distance between the slits and the viewing plane is z, the spacing of the fringes is equal to zθ and is the same as above: A grating is defined in Born and Wolf as "any arrangement which imposes on an incident wave a periodic variation of amplitude or phase, or both". π β If the width of the slits is small enough (less than the wavelength of the light), the slits diffract the light into cylindrical waves. a

ω Example: 9 What is the width of a parallel beam of 546 and width of b=0.5mm after propagation of 10 meter? {\displaystyle dE={\frac {A}{r_{1))}e^{j\omega [t-(r_{1}/c)]}dy={\frac {A}{r_{1))}e^{j(\omega t-\beta r_{1})}dy}.

Des exemples de diffraction Fraunhofer. The path difference between two waves travelling at an angle θ is given by. Les maxima des franges d'interférence se produisent à des angles, où λ est la longueur d' onde de la lumière.

{\ displaystyle A ^ {'} = {\ frac {Ae ^ {j (\ omega t- \ beta r)}} {r_ {1}}}}, E

β The image on the right shows a laser beam diffracted by a grating into n = 0, and ±1 beams. 2 Les angles des poutres du premier ordre sont d'environ 20 °; si l'on suppose que la longueur d'onde du faisceau laser est de 600 nm, on peut en déduire que l'espacement du réseau est d'environ 1,8 µm. Si la lumière est incidente à un angle θ 0 , l'équation de réseau est: La structure détaillée du motif répétitif détermine la forme des faisceaux diffractés individuels, ainsi que leur intensité relative tandis que l'espacement du réseau détermine toujours les angles des faisceaux diffractés.

L'équation a été nommée en l'honneur de Joseph von Fraunhofer bien qu'il n'ait pas été réellement impliqué dans le développement de la théorie.



β L'espacement angulaire des franges est donné par. For example, if a 0.5 mm diameter circular hole is illuminated by a laser with 0.6 μm wavelength, the Fraunhofer diffraction equation can be employed if the viewing distance is greater than 1000 mm. r ′



{\ displaystyle {\ frac {W ^ {2}} {L \ lambda}} \ ll 1}, λ r

The width of the slit is the distance AC.

Lorsque les deux ondes sont en phase, c'est-à-dire que la différence de trajet est égale à un nombre entier de longueurs d'onde, l'amplitude sommée, et donc l'intensité sommée est maximale, et lorsqu'elles sont en anti-phase, c'est-à-dire que la différence de trajet est égale à la moitié une longueur d'onde, une longueur d'onde et demie, etc., alors les deux ondes s'annulent et l'intensité additionnée est nulle. )

j β where Close examination of the double-slit diffraction pattern below shows that there are very fine horizontal diffraction fringes above and below the main spot, as well as the more obvious horizontal fringes. The diffraction pattern given by a circular aperture is shown in the figure on the right. =


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