Waves

Narrow gap

slit diffraction

Above, we have considered diffraction of a plane wave at one screen. Let's complicate the task 😉

Now, two opaque half-planes with straight edges form a gap (slit).

Near the slit (in the geometrical optics approximation), we get a simple stepwise intensity distribution. But if viewed from a far, or if the slit width is comparable to the wavelength... Again, as with round hole diffraction, the result will depend on how many Fresnel zones fit in the slit width. "Oscillations" of the intensity can be observed along the perpendicular to the slit. Of course, the intensity will not change along the slit. At points under the center of the slit, the intensity can again be either greater or less than "normal".

If we replace the slit with an opaque bar of the same width (and remove the half-planes) the diffraction pattern will change. In this case, the distributions of the E-M field (waves) from the slit and from the bar complement each other so that their sum will be the same as in the absence of all obstacles (in a completely open area). But not the sum of the intensities. We repeat again: the waves of the E-M field propagate and interfere with each other, while we observe the intensity (the square of the amplitude of the E-M field).

Diffraction at slit of 1 mm wide. The distance from the slit to the observing plane is 0.3 m. The wavelength is 0.6 micrometer.
https://www.edp-open.org/images/stories/books/fulldl/eas_59/eas59_pp037-058.pdf

Diffraction at opaque strip of 1 mm wide. The distance from the strip to the observing plane is 0.3 m. The wavelength is 0.6 micrometer.
https://www.edp-open.org/images/stories/books/fulldl/eas_59/eas59_pp037-058.pdf

As we have already seen, the diffraction pattern changes dramatically with a change in the distance from the illuminated/observed object (obstacles, slits, etc.) to the observing plane. This is due to the change in the number of "visible" Fresnel zones. As the distance increases more and more, the wave front near the observation point is formed by a significantly smaller area than the first Fresnel zone. In this case, secondary (spherical) waves within the observation area can be considered plane (as coming from an infinitely distant object). Now, if one put a lens in the path of these waves, then each direction will correspond to a point in the focal plane of the lens where these waves converge. Specifics of such a diffraction pattern is that its appearance (shape) does not change with distance (the pattern is formed by less than one Fresnel zone). So, it will be possible to observe the diffraction pattern from an “infinitely” distant object.

This kind of diffraction is called Fraunhofer diffraction (after Joseph Fraunhofer).

Fraunhofer diffraction. The horizontal scale is determined by the ratio of the wavelength to the slit width.

Recall that for the radius of the first Fresnel zone we have

Then the Fraunhofer diffraction applicability condition — the size of the object from the observation point must be much smaller than the first Fresnel zone — can be expressed as (now r is the size of the observed object):

Fraunhofer diffraction

For the scope of Fresnel diffraction (when an object “covers” several zones) we have:

Fresnel diffraction

Finally, for the area of application of geometric optics (when an object "covers" a lot of Fresnel zones):

geometric optics

Joseph Ritter von Fraunhofer (1787-1826) - was a German physicist, optician, inventor. The eleventh child of a glazier, his parents died when he was only eleven years old. Shortly on achivements: he described absorption lines in the solar spectrum (Fraunhofer lines). In 1821, he first used a diffraction grating to study spectra. He proposed a method for observing the diffraction of light in parallel rays. Since 1823, the keeper of the physics cabinet of the University of Munich and a member of the Bavarian Academy of Sciences.

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