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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.
The equation was named in honor of Joseph von Fraunhofer although he was not actually involved in the development of the theory. This article explains where the Fraunhofer equation can be applied, and shows the form of the Fraunhofer diffraction pattern for various apertures. A detailed mathematical treatment of Fraunhofer diffraction is given in Fraunhofer diffraction equation.
When a beam of light is partly blocked by an obstacle, some of the light is scattered around the object, light and dark bands are often seen at the edge of the shadow — this effect is known as diffraction. Huygens postulated that every point on a primary wavefront acts as a source of spherical secondary wavelets and the sum of these secondary wavelets determines the form of the proceeding wave at any subsequent time.
Fresnel developed an equation using the Huygens wavelets together with the principle of superposition of waves, which models these diffraction effects quite well.
It is not a straightforward matter to calculate the displacement amplitude given by the sum of the secondary wavelets, each of which has its own amplitude and phase, since this involves addition of many waves of varying phase and amplitude. When two waves are added together, the total displacement depends on both the amplitude and the phase of the individual waves: two waves of equal amplitude which are in phase give a displacement whose amplitude is double the individual wave amplitudes, while two waves which are in opposite phases give a zero displacement.
Generally, a two-dimensional integral over complex variables has to be solved and in many cases, an analytic solution is not available. The Fraunhofer diffraction equation is a simplified version of the Kirchhoff's diffraction formula and it can be used to model the light diffracted when both a light source and a viewing plane the plane of observation are effectively at infinity with respect to a diffracting aperture.
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. With a distant light source from the aperture, the Fraunhofer approximation can be used to model the diffracted pattern on a distant plane of observation from the aperture far field. Practically it can be applied to the focal plane of a positive lens.
When the distance between the aperture and the plane of observation on which the diffracted pattern is observed is large enough so that the optical path lengths from edges of the aperture to a point of observation differ much less than the wavelength of the light, then propagation paths for individual wavelets from every point on the aperture to the point of observation can be treated as parallel.
The Fraunhofer equation can be used to model the diffraction in this case. For example, if a 0. 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. So, if a positive lens with a sufficiently long focal length so that differences between electric field orientations for wavelets can be ignored at the focus is placed after an aperture, then the lens practically makes the Fraunhofer diffraction pattern of the aperture on its focal plane as the parallel rays meet each other at the focus.
In each of these examples, the aperture is illuminated by a monochromatic plane wave at normal incidence. The width of the slit is W. The Fraunhofer diffraction pattern is shown in the image together with a plot of the intensity vs. Most of the diffracted light falls between the first minima. The size of the central band at a distance z is given by.
For example, when a slit of width 0. The fringes extend to infinity in the y direction since the slit and illumination also extend to infinity.
We can find the angle at which a first minimum is obtained in the diffracted light by the following reasoning. The width of the slit is the distance AC. The same applies to the points just below A and B , and so on. We have:. We can develop an expression for the far field of a continuous array of point sources of uniform amplitude and of the same phase.
Let the array of length a be parallel to the y axis with its center at the origin as indicated in the figure to the right. Then the differential field is: . The form of the diffraction pattern given by a rectangular aperture is shown in the figure on the right or above, in tablet format. The dimensions of the central band are related to the dimensions of the slit by the same relationship as for a single slit so that the larger dimension in the diffracted image corresponds to the smaller dimension in the slit.
The spacing of the fringes is also inversely proportional to the slit dimension. If the illuminating beam does not illuminate the whole vertical length of the slit, the spacing of the vertical fringes is determined by the dimensions of the illuminating beam. 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. It can be seen that most of the light is in the central disk. The angle subtended by this disk, known as the Airy disk, is. The Airy disk can be an important parameter in limiting the ability of an imaging system to resolve closely located objects.
The diffraction pattern obtained given by an aperture with a Gaussian profile, for example, a photographic slide whose transmissivity has a Gaussian variation is also a Gaussian function. 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. 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.
In the double-slit experiment , the two slits are illuminated by a single light beam. If the width of the slits is small enough less than the wavelength of the light , the slits diffract the light into cylindrical waves. These two cylindrical wavefronts are superimposed, and the amplitude, and therefore the intensity, at any point in the combined wavefronts depends on both the magnitude and the phase of the two wavefronts.
The spacing of the fringes at a distance z from the slits is given by . If the slit separation is 0. The difference in phase between the two waves is determined by the difference in the distance travelled by the two waves.
If the viewing distance is large compared with the separation of the slits the far field , the phase difference can be found using the geometry shown in the figure. When the two waves are in phase, i.
This effect is known as interference. The angular spacing of the fringes is given by. 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". This is known as the grating equation.
The finer the grating spacing, the greater the angular separation of the diffracted beams. 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.
In optics , the Fresnel diffraction equation for near-field diffraction is an approximation of the Kirchhoff—Fresnel diffraction that can be applied to the propagation of waves in the near field. In contrast the diffraction pattern in the far field region is given by the Fraunhofer diffraction equation. The near field can be specified by the Fresnel number , F of the optical arrangement. However, the validity of the Fresnel diffraction integral is deduced by the approximations derived below. Specifically, the phase terms of third order and higher must be negligible, a condition that may be written as.