FRAUNHOFER DIFFRACTION

Let us consider the screen with two slits illuminated by a flat monochromatic wave. Calculations show that the intensity of the light getting pass through the slits will depend upon the angle j between the direction of the light propagation and the perpendicular to screen:

where I0 is the intensity of the light in the center of diffraction pattern when only one slit is opened, b is the width of the slit, d is the distance between the slits, k=2p /l is the wave factor, l is the wavelength, D is the difference of the optical lengths of the interfering rays (in the case, for example, when the wave is incident not perpendicularly to the screen or one slit is covered by glass). The first multiplier of the equation in the square brackets describes the Fraunhofer diffraction on one slit and the second multiplier describes the interference from two point sources. The total energy of the light getting through the slit is proportional to b, while the width of the diffraction pattern is proportional to 1/b. For this reason the intensity of the light I0 in the center of diffraction pattern will be proportional to b2. In the limits of the first diffraction maximum we can see N interference fringes, where N=2d/b.

This figure shows the dependence of the light intensity on the angle in the case of diffraction on one slit (red curve) and for two slits diffraction (blue curve). We can see in this figure that the maximal intensities of the interference fringes follow the curve for diffraction on one slit.

Talking about "Fraunhofer" diffraction we mean the far-field diffraction, i.e. when the point of observation is far enough from the screen with the slits. Quantitatively the criteria of the Fraunhofer diffraction is described by the formula:

z >> d2/l

where z is the distance from the screen with the slits to the point of observation. In the close proximity to the screen with the slits the diffraction pattern will be described by the Fresnel's equations.

Fraunhofer diffraction
| This animation shows the experiment when the width b of the silts is varied, while the distance d between them is constant. We can see in the figure that for the narrower slits the diffraction pattern is wider and the visibility is lower. The frequency of the interferometric fringes is the same.
| This animation shows the experiment when the width b of the silts is constant (1000 nm) and the distance d between them is varied in the range 1000-10000 nm. Wavelength is 600 nm. The frequency of the interferometric fringes is increasing proportionally to the distance d between the slits, while the width of the diffraction pattern is the same and depends only on b.
| This animation shows the Fraunhofer Diffraction on one slit. The width b of the silts is varied in the range 500-1500 nm, wavelength equal to 600 nm.
Click on the picture and then Alt-Enter to see full-screen AVI video-animation