02 January 2016

Diffraction

Introduction

The effects of diffraction of light were first carefully observed and characterized by Francesco Maria Grimaldi. The results of Grimaldi's observations were published in 1665.Diffraction is the slight bending of light as it passes around the edge of an object. The amount of bending depends on the relative size of the wavelength of light to the size of the opening. If the opening is much larger than the light's wavelength, the bending will be almost unnoticeable. However, if the two are closer in size or equal, the amount of bending is considerable, and easily seen with the naked eye.

In the atmosphere, diffracted light is actually bent around atmospheric particles -- most commonly, the atmospheric particles are tiny water droplets found in clouds. Diffracted light can produce fringes of light, dark or colored bands. An optical effect that results from the diffraction of light is the silver lining sometimes found around the edges of clouds or coronas surrounding the sun or moon.

The phenomenon in which light waves bend around corners of an object and their spreading into the geometrical shadow of an object is called diffraction.

Classification of diffraction phenomena

Diffraction phenomenon is broadly divided into two groups, based on the method adopted to study diffraction effects. They are 1) Fresnel class of diffraction 2) Fraunhofer class of diffraction.



S.No.
Fresnel Diffraction
Fraunhofer Diffraction
1
Source and screen are at finite distance from the obstacle/aperture.
Both the source and the screen are at infinite distance from the obstacle/aperture.
2
Spherical/Cylindrical wave front falls on the obstacle/aperture.
Plane wave front falls on the obstacle/aperture.
3
No lenses are used in Fresnel diffraction.
Converging lenses are used to study Fraunhofer diffraction.
4
Waves falling on the obstacle/aperture will not be in the same phase.
Waves falling on the obstacle/aperture have the same phase.
5
Fresnel diffraction is general case of diffraction, which reduces to Fraunhofer case when the source and screen are at infinite distance from the obstacle/aperture.
Fraunhofer diffraction is a particular case of diffraction with source and screen at infinity from the obstacle/aperture.


Diffraction Grating

An arrangement consisting of a large number of equidistant parallel narrow slits of equal width separated by equal opaque portions is known as a diffraction grating.

The plane transmission grating is a plane sheet of transparent material on which opaque rulings are made with a fine diamond pointer. The modern commercial form of grating contains about 6000 lines per centimetre.

The rulings act as obstacles having a definite width ‘b’ and the transparent space between the rulings act as slit of width ‘a’. The combined width of a ruling and a slit is called grating element (e). Points on successive slits separated by a distance equal to the grating element are called corresponding points.

Theory

MN represents the section of a plane transmission grating. AB, CD, EF … are the successive slits of equal width a and BC, DE … be the rulings of equal width b. Let e = a + b.


Let a plane wave front of monochromatic light of wave length λ be incident normally on the grating. According to Huygen’s principle, the points in the slit AB, CD … etc act as a source of secondary wavelets which spread in all directions on the other side of the grating.
Let us consider the secondary diffracted wavelets, which makes an angle θ with the normal to the grating.
The path difference between the wavelets from one pair of corresponding points A and C is CG = (a + b) sin θ. It will be seen that the path difference between waves from any pair of corresponding points is also (a + b) sin θ
The point P1 will be bright, when
(a + b) sin θ = m λ where m = 0, 1, 2, 3............

In the undiffracted position θ = 0 and hence sin θ = 0.
(a + b) sin θ = 0, satisfies the condition for brightness for m = 0. Hence the wavelets proceeding in the direction of the incident rays will produce maximum intensity at the centre O of the screen. This is called zero order maximum or central maximum.
If (a + b) sin θ1 = λ, the diffracted wavelets inclined at an angle θ1 to the incident direction, reinforce and the first order maximum is obtained.
Similarly, for second order maximum, (a + b) sin θ2 = 2λ
On either side of central maxima different orders of secondary maxima are formed at the point P1, P2.
In general, (a + b) sin θ = m λ is the condition for maximum intensity, where m is an integer, the order of the maximum intensity.
sin θ = Nmλ
where N = 1/(a+ b) , gives the number of grating element or number of lines per unit width of the grating.
When white light is used, the diffraction pattern consists of a white central maximum and on both sides continuous coloured images are formed.
In the undiffracted position, θ = 0 and hence sin θ = 0. Therefore sin θ = Nmλ is satisfied for m= 0 for all values of λ. Hence, at O all the wavelengths reinforce each other producing maximum intensity for all wave lengths. Hence an undispersed white image is obtained.

As θ increases, (a + b) sin θ first passes through λ/2 values for all colours from violet to red and hence darkness results. As θ further increases, (a + b) sin θ passes through λ values of all colours resulting in the formation of bright images producing a spectrum from violet to red. These spectra are formed on either side of white, the central maximum.
Determination of wavelength of light (Normal Incidence)























The wavelength of white light can be determined using a plane diffraction grating and a spectrometer. The preliminary adjustments of the spectrometer are made. The slit of the collimator is illuminated by a mercury vapour lamp and the slit and the collimator are suitably adjusted to receive a narrow, vertical image of the slit. The telescope is turned to receive the direct ray, so that the vertical slit coincides with the vertical cross wire. The telescope reading is taken.
The grating is mounted on the prism table. The telescope is now moved exactly through 90° and fixed in that position. The prism table is rotated until the reflected image coincides with the point of intersection of the crosswires. Now the angle of incidence is 45°. The prism table is fixed in this position. The vernier disc is released and rotated through 45° so that the light from the collimator is normally incident on the grating.
The telescope is now released and turned to receive the direct image. It is gradually moved to the left so that the first order diffracted image (Violet to Red) is seen. After making the point of intersection of the cross a wire to coincide with the image, the telescope reading is taken. The telescope is then moved to the right and the reading corresponding to the first order image is taken as before. Half the difference between the two readings gives the angle of diffraction θ.
sin θ = Nmλ 
For the first order m=1. The number of rulings per unit length N is provided by the manufacturer. It can also be determined using light of known wavelength. Hence the wavelength λ can be calculated.
The readings are tabulated as shown below:
Order
Colour
Vernier Reading
θ
λ
Left
Right
I






II







Zone Plate

A zone plate is a transparent plate divided into half period zones and designed so as to cut off light through even numbered or odd numbered zones.

A zone plate is an optical device, which works on the principle of Fresnel’s zone. In Fresnel’s zones, the effect at a point due to alternate zones cancels each other. If the effect of either even or odd zones is blocked at that point, then the net effect due to alternate zoneshaving a path difference of λhas a maximum value. A plate, which allows only one set of alternate zones of the wavefront to pass through it, is called a zone plate.

The radius of the zones is proportional to square root of n, where n = 1, 2, 3, etc. This property isused in the construction of a zone plate. On a drawing sheet, concentric circles are drawn with the radii proportional to square root of natural numbers as shown in the following figure.

The alternate zones are painted black. A photograph of the pattern is taken. We get on developed negative a reduced pattern. This negative forms the zone plate. In this plate alternate zones are transparent and allow light and the remaining alternate zones act as opaque region. There are two types of zone plates. i) Positive zone plate and ii) Negative zone plate as shown in figure (1a) and (1b).

(1) Positive zone plate: A zone plate in which odd zones are transparent and even zones are opaque is known as a positive zone plate.

(2) Negative zone plate: A zone plate in which even zones are transparent and odd zones are opaque is known as a negative zone plate.


References:

http://ww2010.atmos.uiuc.edu/(Gh)/guides/mtr/opt/mch/diff.rxml

http://www.readorrefer.in/article/Diffraction-grating-explanation-with-Theory_568/

Optics – R.K.Agrawal, Garima Jain, Rekha Sharma, Krishna Prakashan Media(P) Ltd, Meerut, First edition 2006.

http://veerashaivacollege.org/images/stories/Diffraction_New2.pdf