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Overview


Diffraction slits, and diffraction gratings in general, are optical components on which coherent light can diffract and interfere with itself to produce a wide variety of interference patterns which are then projected onto a screen.


Figure 1: A single diffraction slit. (Left) Front. (Right) Back.
Figure 1: A single diffraction slit. (Left) Front. (Right) Back.

Tutorial


The K-Optics Starter’s Kit and both Advanced Kits (2x1 and 2x2) come equipped with 9 different diffraction gratings. Each of which finds use in the Slit Diffraction Experiment. Coherent light is shined upon the slit from a LASER to produce the interference pattern characteristic of that slit which can then be measured.

Figure 2: The variety of slits in the kit. (Top row) Single slits of varying widths. (Middle row) Double slits of varying separation. (Bottom row) From left to right: Lines, Grid, and a single barrier (negative slit).
Figure 2: The variety of slits in the kit. (Top row) Single slits of varying widths. (Middle row) Double slits of varying separation. (Bottom row) From left to right: Lines, Grid, and a single barrier (negative slit).

Technical Specifications


a.    Single Slits

Table 1: Technical specifications of single slits. All measurements in mm.
Table 1: Technical specifications of single slits. All measurements in mm.

b.  Double Slits


Table 2: Technical specifications of double slits. All measurements in mm.
Table 2: Technical specifications of double slits. All measurements in mm.Table 2: Technical specifications of double slits. All measurements in mm.

c. Special


Table 3: Technical specifications of special gratings.
Table 3: Technical specifications of special gratings.


Theoretical Background


Diffraction and interference are byproducts of light’s superposition principle. That is, when one light wave meets another light wave, their amplitudes add up. If these waves are coherent then an interference pattern can be readily observed. When coherent waves are identical, nothing very interesting happens apart from pure constructive interference. It is only when the optical path each wave takes is distinct that interference patterns become more interesting.

Figure 3: Interference of two coherent waves. The resultant field is shown in blue.
Figure 3: Interference of two coherent waves. The resultant field is shown in blue.

For two coherent waves – that is, they have the same polarization and wavelength, as well as a constant phase difference between them – we can observe, mathematically, the effect of interference. Suppose two coherent waves of identical amplitudes but different starting locations are given:


Figure 4: Coherent waves with optical path difference attributed to different starting positions.
Figure 4: Coherent waves with optical path difference attributed to different starting positions.

We can mark each starting point as the starting phase:

Then, by the superposition principle

and the intensity can be shown to be:

Therefore, for distinct phase differences:

there are distinct effects on the total intensity, in particular:

which is how we get light and dark spots – or, in the case of slits, fringes.

Diffraction from slits is slightly more involved than the one-dimensional demonstration shown above. In general, the pattern projected onto a screen at a distance z due to aperture opening A(x,y) can be calculated by using the Fraunhofer integral:

Where (x',y' ) are coordinates on the aperture, z is the distance to the screen, (x,y) are coordinates on that screen, and λ is the wavelength.


Figure 5: An illustration of the Fraunhofer integral setup for a double slit.
Figure 5: An illustration of the Fraunhofer integral setup for a double slit.

In essence, this integral suggests that the pattern projected on the screen is a Fourier transform of the aperture. This is easily calculable for many different shapes of apertures. Namely, for the single slit case, the result is the famous sinc function which is the Fourier transform of a slit opening (of width b):

E(x)∝sin⁡(πbx/λz)/(πbx/λz)=sinc(πbx/λz) .

Whereas, for the double slit one gets:

Where b is the slit width and d is the separation between the slits.

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