INTERFERENCE
DR. U. N. PATEL
The phase relation between the waves emitted by two independent light sources rapidly changes with time and therefore they can never be coherent, though the sources are identical in all respects. However, if two sources are derived from a single source by some device, then any phase change occurring in one source is simultaneously accompanied by the same phase change in the other source. Therefore, the phase difference between the waves emerging from the two sources remains constant and the sources are coherent.
The techniques used for creating coherent sources of light can be divided into the following two broad classes.
(a) Wavefront splitting : One of the methods consists in dividing a light wavefront, emerging from a narrow slit, by passing it through two slits closely spaced side by side. The two parts of the same wavefront travel through different paths and reunite on a screen to produce fringe pattern. This is known as interference due to division of wavefront. This method is useful only with narrow sources. Young’s double slit, Fresnel’s double mirror, Fresnel’s biprism. Lloyd’s mirror, etc employ this technique.
(b) Amplitude splitting :
Alternately, the amplitude (intensity) of a light wave is divided into two parts, namely reflected and transmitted components, by partial reflection at a surface. The two parts travel through different paths and reunite to produce interference fringes. This is known as interference due to division of amplitude. Optical elements such as beam
splitters, mirrors are used for
achieving amplitude division.
Interference in thin films (wedge,
Newton’s rings etc). Michelson’s
interferometer etc interferometers utilize this method. This method requires extended source.
2. FRESNEL BIPRISM
Fresnel used a biprism to show interference phenomenon. The biprism consists of two prisms of very small refracting angles joined base to base. In practice, a thin glass plate is taken and one of its faces is ground and polished till a prism (Fig. 1 a) is formed with an obtuse angle of about 179° and two side angles of the order of 30’.
Fig. 1
When a light ray is incident on an ordinary prism, the ray is bent through an angle called the angle of deviation. As a result, the ray emerging out of the prism appears to have emanated from a source S’ located at a small distance above the real source, as shown in Fig. 1 (b). We say that the prism produced a virtual image of the source. A biprism, in the same way, creates two virtual sources S1 and S2, as seen in Fig. 1 (c). These two virtual sources are images of the same source S produced by refracting and are hence coherent.
2.1 EXPERIMENTAL ARRANGEMENT
The biprism is mounted suitably on an optical bench. An optical bench consists of two horizontal long rods, which are kept strictly parallel to each other and at the same level. The rods carry uprights on which the optical components are positioned. A monochromatic light source such as sodium vapour lamp illuminates a vertical slit S. Therefore, the slit S acts as a narrow linear monochromatic light source. The biprism is placed in such a way that its refracting edge is parallel to the length of the slit S. A single cylindrical wavefront impinges on both prisms. The top portion of wavefront is refracted
downward and appears to have emanated from the virtual image S1. The lower segment, falling on the lower part of the biprism, is refracted upward and appears to have emanated from the virtual source S2. The virtual source S1 and S2 are coherent (see fig. 2), and hence the light waves are in a position to
interfere in the region beyond the biprism.
Fig. 2
If a screen is held there, interference fringes are seen. In order to observe fringes, a micrometer eyepiece is used.
Theory :
The theory of the interference and fringe formation in case of Fresnel biprism is the same as described in for the double-slit. As the point O is equidistant from S1 and S2, the central bright fringe of maximum intensity occurs there. On both side of O, alternate bright and dark fringes, as shown in Fig. 2 (b), are produced.
The width of the dark or bright fringe is given by equ.
Where D(= a + b) is the distance of
the sources from the eye piece.
Fresnel biprism
Dark or bright fringes
2.2 DETERMINATION OF WAVELENGTH OF LIGHT
The wavelength of the light can be determined using the equ. . For using the relation, the values of β, D and d are to be measured. These measurements are done as follows.
Adjustments :
A narrow adjustable slit S, the biprism, and a micrometer eyepiece are mounted on the uprights and are adjusted to be at the same height and a straight line. The slit is made vertical and parallel to the refracting edge of the biprism by rotating it in its own plane. It is illuminated with the light from the monochromatic source. The biprism is moved along the optical bench till, on looking through it along the axis of the optical bench, two equally bright vertical slit images are seen. Then the eyepiece is moved till the fringes appear in the focal plane of the eyepiece.
(ii) Determination of ‘d’ : (a) A convex lens of short focal length is placed between the slit and the eyepiece without disturbing their positions. The lens is moved back and forth near the biprism till a sharp pair of images of the slit is obtained in the field view of the eyepiece. The distance between the images is measured. Let it be denoted by d1.
Fig. 3
If u is the distance of the slit and v that of the eyepiece from the lens (Fig. 3 a.), then the magnification is
The lens is then moved to a position nearer to the eyepiece, where again a pair of images of the slit is seen. The distance between the two sharp images is again measured. Let it be d2. Again magnification is given by
Note that the magnification in one position is reciprocal of the magnification in the other position.
Multiplying the equations (1) and (2), we obtain
Using the values of β, d and D in the equation
, the wavelength λ can be computed.
(b) Alternatively, the value of d can be determined as follows. The deviation δ produced in the path of a ray by a thin prism is given by where α is the refracting angle of the prism. From the Fig. 2, it is seen that δ = θ/2. Since d is very small, we can also write d = a θ.
2.3 INTERFERENCE FRINGES WITH WHITE LIGHT
In the biprism experiment if the slit is illuminated by white light, the interference pattern consists of a central white fringe flanked on its both sides by a few coloured fringes and general illumination beyond the fringes. The central white fringe is the zero-order fringe.
with monochromatic light all the bright fringes are of the same colour and it is not possible to locate the zero-order fringe. Therefore, in order to locate the zero order fringe the byprism is to be illuminated by white light.
2.4 LATERAL DISPLACEMENT OF FRINGES
The biprism experiment can be used to determine the thickness of a given thin sheet of transparent material such as glass or mica. If a thin transparent sheet is introduced in the path of one of the two interfering beams, the fringe system gets displaced towards the beam in whose path the sheet is introduced. By measuring the amount of displacement, the thickness of the sheet can be determined.
Suppose S1 and S2 are the virtual coherent monochromatic sources. The point O is equidistant from S1 and S2, where we obtain the central bright finges. Therefore, the optical path S1O = S2O.
Fig. 4
Let a transparent plate G of thickness t and refractive index μ be introduced in the path of one of the beam (see Fig. 4). The optical path
lengths S1O and S2O are now equal and the central bright fringes shifts to P from O. The light waves from S1 to P travel partly in air and partly in the sheet G; the distance travelled in air is (S1P – t) and that in the sheet is t.
The optical path ΔS, P = (S1P – t) + μ t= S1P + (μ-1)t
The optical path ΔS, P = S2 P
The optical path difference at P is , since in the presence of the thin sheet, the optical path lengths S1P and S2P are equal and central zero fringe is obtained at P.
But according to the relation (From Young’s double slit experiment),
Where x is the lateral shift of the central fringe due to the introduction of the thin sheet.
3. LLOYD’S SINGLE MIRROR
In 1834, Lloyd devised an interesting method producing interference, using single mirror and using almost grazing incidence. The Lloyd’s mirror consists of a plane mirror about 30 cm in length and 6 to 8 cm in breath (See Fig. 1).
Fig. 1
It is polished on the front surface and blackened at the back to avoid multiple reflections. A cylindrical wavefront coming from a narrow slit S1 falls on the mirror which reflects a portion of the incident wavefront, giving rise to virtual image of the slit S2. Another portion of the wavefront proceeds directly from the slit S1 to the screen. The slits S1 and S2 acts as two coherent sources. Interference between direct and reflected waves occurs within the region of overlapping of the two beams and fringes are produced on the screen placed at a distance D from S1 in the shaded portion EF.
The points O is equidistant from S1 and S2.
Therefore, central (zero-order) fringe is expected to lie at O ( the perpendicular bisector of S1S2) and it is also expected to be bright. However it is not usually seen since the point O lies outside the region of interference (only the direct light and not the reflected light reaches O).
By moving the screen nearer to the mirror such that it comes into contact with the mirror, the point O can be just brought into the region of interference.
With white light the central fringe at O is expected to be white but in practice it is dark. The occurrence of dark fringe can be understood taking into the consideration of the phase change of π that light suffers when reflected from the mirror. The phase change leads to a path difference of λ/2 and hence destructive interference occurs there.
3.1. DETERMINATION OF WAVELENGTH
The fringe width is given by equ.
Thus, Measuring β, D and d, the wavelength λ can be determined.
Comparison between the fringes produced by biprism and Lloyd’s mirror :