Tuesday, 7 June 2011

Diffraction Grating

 A diffraction grating is a glass plate having a large number of close parallel equidistant slits mechanically rules on it. The transparent spacing between the scratches on the glass plate act as slits. A typical diffraction grating has about 400 to 5000 lines per centimeter. 

        In order to understand how a grating diffracts light, consider a parallel beam of monochromatic light illuminating the grating at normal incidence. A few of the equally spaced narrow slits are shown in the figure given below.
   Fig   Diffraction of light due to grating
        The distance between two adjacent slits is d, called grating element. Its value is obtained by dividing the length L of the grating by the total number N of the lines rules on it. The sections of wavefront that pass through the slits behave as source of secondary wavelets according to Huygen's principle.
     In the above figure, consider the parallel rays which after diffraction through the grating make an angle θ with AB, the normal to the grating. They are then brought to focus on the screen at P by a convex lens. If the path difference between ray 1 and 2 is one wavelength λ, they will reinforce each other at P. As the incident beam consists of parallel rays, the rays from any two consecutive slits will differ in path by λ when arrive at P. They will, therefore, interfere constuctively. Hence, the condition for constructive interference is that ab, the path difference between two consecutive rays, should be equal to λ, i.e.,
ab = λ 
From figure,
ab = dsinθ 
d being the grating element. Substituating the value of ab in the above equation we get,
dsinθ = λ 
        According to the above equation, when  θ = 0 i.e., along the direction of normal to the grating, the path difference between the rays coming out from the slits of the grating will be zero. So we will get a bright image in this direction. This is known as zero-order image formed by the grating.
        If we increase θ on either side of this direction, a value of θ will be arrived at which dsinθ will become λ and according to the dsinθ = λ, we will again get a bright image. This is known as the first-order image of the grating. In this way if we continue increasing θ, we will get the second, third, etc. images on either side of the zero-order image with dark regions in between. The second, third order bright images would occur according as dsinθ becoming equal to 2λ, 3λ, etc. Thus the equation dsinθ = λ can be written in more general form as,
dsinθ =nλ 
where n = 0,1,2,3, etc.
        However, if the incident light contains different wavelengths, the image of each wavelength for certain value of n is diffracted in a different directions. Thus, separate images are obtained corresponding to each wavelength or color. The above equation shows that the value of θ depends upon n, so the images of different colors are much separated in higher orders. 

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