DIFFRACTION GRATINGS
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1. Introduction and theory.
Diffraction Grating is optical device used to learn the different wavelengths or colors contained in a beam of light. The device usually consists of thousands of narrow, closely spaced parallel slits (or grooves). Because of interference the intensity of the light getting pass through the slits depends upon the direction of the light propagation. There are selected directions at which the light waves from the different slits interfere in phase and in these directions the maximums of the light intensity are observed. These selected directions depend upon wavelength, and so the light beams with different wavelength will propagate in different directions. The condition for maximum intensity is the same as that for the double slit or multiple slits, but with a large number of slits the intensity maximum is very sharp and narrow, providing the high resolution for spectroscopic applications. The peak intensities are also higher and depend proportionally to the second power of amount of the slits illuminated.
In the beginning let us consider the diffraction from double slit, which consists of two parallel 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 I_{0} 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 I_{0} in the center of diffraction pattern will be proportional to b^{2}. 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 >> d^{2}/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
Next, we shall consider the diffraction grating, which consists of N parallel slits. In this case the light waves from every slit will interfere each other producing the interference fringes as shown in figure. Because of diffraction the distribution of the light intensity behind of every slit will not be isotropic (see figure for diffraction at one slit). For diffraction gratings both these effects take place, so the resultant intensity of the light on the screen is described by the equation:
The first multiplier of the equation describes the Fraunhofer diffraction on one slit and the second multiplier describes the interference from N point sources.
It is seen from the figure that d·sinj is the path length difference D between the rays emitted by the slits. If it is equal to the integer number, then the oscillations will interfere in phase magnifying each other. Therefore, we can write the equation for the main maximums of interference pattern: d·sinj= ml, where m = 0, 1, 2,…
2. Set of diffraction gratings DG-10.
We produce the set of diffraction gratings DG-10 and can send it to you by mail. Specifications are given below. You can order diffraction grating DG-10 using KAGI online payment processing system.
The vertical axis is normalized to the intensity of the light at the center of the screen. Actually, for the single and double slit experiments the intensity of the light on the screen depends proportionally to the second power of the width of the slit. So, particularly, for the slits of 5 pixels wide the intensity on the screen will be 16 times less than for the slits of 20 pixels wide. For multi-slit gratings the intensity of the light on the screen will be N^{2} times bigger as compared to one slit (where N is number of illuminated slits) and it also depends proportionally to second power of the width of every slit. Technical specifications: Dimensions: 148 x 95 mm (every
grating 10 x 10 mm) |
Click on the appropriate diffraction grating to see the graph for the intensity of the light on the screen. |
The gratings in rows 1, 3,
4 are vertical strips marked as d/b,
where d is period of the grating in pixels, b
is the width of the transparent strips. The gratings in row 2 are
superposition of vertical and horizontal strips. Row 5 consists
the single slits and rows 6 and 7 consist of double slits. |
Gratings in the Row 1 consist of
alternating transparent and black strips of equal width. Illuminating these
gratings by laser light we can see the diffraction pattern on the screen behind
of them. We see that all gratings in row 1 produce the same amount of the
interferometric fringes. This occurs because the total width of diffraction
pattern depends upon the width of every slit as 1/b and the frequency of
interferometric fringes depend proportionally the period of grating d. For all
gratings in row 1 d/b=2, so we can see the same amount of fringes. Gratings in Row 2 are superposition of vertical and horizontal strips. We can see
two-dimensional diffraction pattern in this case. The width of all slits in
Row 3 is equal to 1 pixel, while the period of these gratings is varying
from 2 to 10 pixels (1 pixel = 1/4000 inch in our case). In this case the total
width of diffraction pattern is constant, while the frequency of interferometric
fringes is different. The intensity of the diffraction fringes at screen behind
the grating 10/1 is rather small because of small width of slits in this grating.
For Row 4 the period of all gratings is the same and equal to 10 pixel,
while the widths of the slits are different and equal to 1 - 5 pixels. We can
see in experiment that the frequency of interference fringes is the same for all
gratings in this row, but the total widths of diffraction pattern are different.
In Row 5 there are vertical single slits. The widths of these slits are
different and equal to 5 - 80 pixels. For 5 pixel slit the width of diffraction
pattern is maximal, but the intensity on the screen is minimal because only the
small part of optical radiation incident on the slit can pass through it. In
Row 6 and 7 there are double slits, which consists of two closely situated
slits. In Row 6 the separation between the slits is constant and equals
50 pixels, while the widths of these slits are different. For Row 7 the
width of every slit is constant and equals 10 pixels, but the distances between
them are different and equal to 20-40 pixels. The diffraction pattern for double
slit is about the same as for one slit with the same width, but in case of
double slit we can see the appearance of interferometric fringes. The period of
these fringes depends upon the distance between the slits.
The diffraction can be observed more easily if you:
3. Spectrum analysis using diffraction grating.
In the next experiment we shall explore the light which consists of the waves of different frequencies. In this case the angle of diffraction depends on wavelength of the light and hence instead of single interferometric lines the spectrums will appear in different orders. This property of diffraction grating can be used for investigation of the spectrums of the light of different optical sources.
In the figure above we can see the classical experimental setup for observation of Frounghofer diffraction. We used luminescent lamp B and diffraction grating GD. Light of lamp B passes through a narrow slot situated in the focus of the lens L1. As a result the parallel beam of the light is formed behind of the lens. Then this light is incident on a transmitting diffraction grating DG. Because of interference the flat waves with different wavelengths appears at the output of diffraction grating. Light with the wavelength λ will propagate in the direction φ in accordance with the equation dsinφ = mλ, where m is the positive integer, which has a sense of the spectrum order. m is equal to difference of the optical paths of the light from two adjacent slits related to the wavelength. So, in the first order the optical path difference equals λ. The light expanded in a spectrum is incident then to lens L2, which focus it to screen S. In the centre of the screen we can see the white line, which corresponds to image of the slot in zero order of spectrum. Then, up and down of the screen there are colored strips, which correspond a spectral composition of the light. The repeated groups of lines is interference in the first, second, etc. orders. Quality of the grating is defined by the resolution: R=λ/δλ=mN, where λ is the wavelength, δλ is the minimal difference in wavelengths of the lines, which can be resolved, m is the order of spectrum, N is the number of the slots used for interference (which are inside the light spot).
Spectrum components of the light can be observed even without the lenses. We shall need a photo camera PH and diffraction grating DG. Let us focus the camera on the remote slit illuminated by lamp B as shown in the figure above and put the diffraction grating in front of objective of the camera. One white line and many color lines will appear on the matrix of the camera (or on photo-film), which correspond to spectral components of the lamp radiation. The spectrums repeat in higher orders, but do not overlap each other. We analyzed the spectrum of the luminescent day-light lamp. Spectrum consists of many separate lines (you can see a photo below). We easily distinguish two green lines, which differ by the wavelength at about 4 nm, and see also the set of fine red and blue lines in spectrum. Resolution of such a system was measured to be about 600 in the first order of spectrum. It was limited by the size of the objective which is equal to size of the grating participating in interference.
Let us take a photo of the candle flame through the diffraction grating DG-2/1-4000. We can see that the image is multiplexing. The central image of the flame coincides with the one without the grating while the images in higher orders are dissolved in the spectrums like in rainbow. The spectrum of the candle flame is continuous so we can not see the discrete lines.
We accepts the orders and can send to you by mail the diffraction gratings DG-2/1-4000 or DG-4/2-4000 of any size. Digit 4000 in the labeling of the grating means that the resolution of the grating is 4000 dpi. 1 pixel = 6.35 microns. Grating consists of alternating black and white (transparent) lines on a thin transparent film. Period of the grating is 2 or 4 pixels, the width of the line is 1 or 2 pixels. This corresponds to 2/1 and 4/2 in the labeling of the grating. The price for 1 sq.inch of grating is $5. Minimal order is $20. Contact us to order this diffraction grating.
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