1647205

Phys 222

EXPERIMENT 4

diffraction and interference

Purpose: The aim of doing this experiment was to examine diffraction and interference effects of light passing through various apertures, and use the diffraction patterns obtained by single and double slit apertures to find the wavelength of the light source used. Theory: We know that light can be described by two theories, namely the particle theory and the wave theory of light, each having its own experimental proofs. In this experiment, we examine the interference and diffraction phenomena of light, both of which can be described by the wave theory of light. While interference is just the superposition of waves, diffraction is also any deviation from geometrical optics that results from the obstruction of a wavefront of light. In other words, diffraction is considering the double-slit experiment by taking into account the width of the slit openings, too. Another way of distinguishing between interference and diffraction is to consider the interfering beams in diffraction phenomena as originating from a continuous distribution of sources, whereas the interfering beams in interference phenomena as originating from a discrete number of sources. This way of treatment of interference and diffraction is a result of Huygens’ principle which states that every point of a given wavefront of light can be considered a source of secondary spherical wavelets. Hence, superposition occurs between these secondary waves emitted from different parts of the wavefront, taking into account both their amplitudes and phases. Diffraction effects can also be classified according to the mathematical approximations used in calculations. In the case of the light source and the observation screen being very far from the slit, relative to the slit width, the incident and diffracted waves are assumed to be plane and the diffraction type is called Fraunhofer, or far-field diffraction. In this case, as the viewing screen is moved relative to the aperture, the size of the diffraction pattern changes, but not the shape. We are going to use this kind of approximation in this experiment. We should keep in mind that the Huygens’ principle used to find the diffraction relations is itself an approximation. When calculating the single-slit Fraunhofer diffraction a rectangular aperture with a length much larger than its width is considered. In this case the intensity of the light reaching the screen at point P, at an angle θ is given by: Is=I0(sin2αα2)

where

α=12kasinθ=πasinθλ

In the above relations I0 is the intensity at the middle of the central maxima and a is the slit width. Hence, by taking the limit as →0, we observe that this pattern attains its maximum at θ=0. Similarly, equating =mπ, we obtain the minima of the pattern and we get the following relation for this case: nλ=asinθ

where n=1,2,3,… For small angles we can make the sinθ=tanθ approximation and, calling L the distance between the slit and the screen, we can get y=Lsinθ, where y is the distance from the central maximum to the observation point. For this case, we conclude that on the screen, the irradiance is a maximum at θ=0, hence y=0, and it drops to zero at values of y such that y=λLa . Therefore, we can find λ using this relation. (Here, y is the average distance between adjacent minima). When we regard the double-slit diffraction we see that we have to do with two different terms, one of which belongs to the interference pattern, and the other to the diffraction pattern. If we ignore the effect of the slit widths, we get the intensity of the pattern given by only the interference term as I=4I0cos2β, where β=(πbλ)sinθ. Here, θ is the angle of observation and b is the slit separation. Nevertheless, since the intensity from a single slit depends on the angle θ through diffraction, we should take into account the diffraction pattern, too. Now, the intensity is given by: I=4I0(sin2αα2)cos2β

In this case ...