Young's Double Slit Experiment - Division of the wavefronts

Thomas Young was the first who observed the interference of light in 1801. He placed monochromatic source of light in front of a single slit. In order to get a coherent light he placed another two slits very close to each other in front of the first one. After passing light rays from the two slits a pattern of alternate dark and bright fringes is formed on a screen placed parallel to slits at some distance.The central fringe was bright. From the interference of light we can easily calculate the wavelength of the light.
a

  Simple ray geometry of Young's double slit experiment  

      The above figure shows the experimental arrangement, similar to that devised by Young in 1801, for studying interference effect of light. A screen having two narrow slits is illuminated by a beam of monochromatic light. The portion of the wavefront incident on the slits behaves as a source of secondary wavelets (Huygen's principle). The secondary wavelets leaving the slits are coherent. Superposition of these wavelets result in a series of alternate bright and dark fringes which are observed on a second screen placed at some distance parallel to the first screen.

       Let us now consider the formation of bright and dark bands. As pointed out earlier the two slits behave as coherent sources of secondary wavelets. The wavelets arrive at the screen in such a way that at some points crests fall on crests and troughs on troughs resulting in constructive interference and bright fringe is formed. There are some points on the screen where crests meet troughs giving rise to destructive interference and dark fringe is thus formed.

  Fig 1.  Young's Double Slit Experiment for Interference of Light

    The bright fringes are termed as maxima and dark fringes as minima.    .

      In order to derive equation for the maxima and minima, an arbitrary point P is taken on the screen on one side of the central point O as shown in the figure given below.

  Fig 2.  Geometrical construction of Young's double slit experiment

         AP and BP are the paths of rays reaching P. The line AD drawn such that  AP = DP. The separation between the centres of the two slits is AB = d. The distance of the second screen from the slits is CO = L. The angle between CP and CO is θ. It can be proved that the angle BAD = θ by assuming that AD is nearly normal to BP. The path difference between the wavelets, leaving the slits and arriving at P, is BD. It is the number of wavelengths, contained within BD, that determines whether bright or dark fringe will appear at P. If the point P is to have bright fringe, the path difference BD must be an integral multiple of wavelength. Thus,

where,

m = 0,1,2,....

since BD = dsinθ so,

                            

          
          It is observed that each bright fringe on one side of O has symmetrically located bright fringe on the other side of O. The central bright fringe is obtained when m=0. If the point P is to have a dark fringe, the path difference BD must contain half-integral number of wavelengths. Mathematically,

Or

                               

      The first dark fringe, in this case, will obviously appear for m=0 and second dark for m=1. The interference pattern obtained in the Young's experiment is shown in the figure given below:

  Fig 3.  An interference pattern by monochromatic light in Young's double slit experiment

        Equation (1) and (2) can be utilized for determining the linear distance on the screen between adjacent bright or dark fringes. If the angle θ is small, then

     Now from Fig 2, tan θ = y/L, where y is the distance of the point P from O. If a bright fringe is observed at P, the, from equation (1), we get,

                                  

If P is to have dark fringe it can be proved that,

                            

        In order to determine the distance between two adjacent bright fringes on the screen, mth and (m+1)th fringe are considered.

For the mth bright fringe,

and for the (m+1)th bright fringe,

If the the distance between two adjacent bright fringes is ∆ y then,

Therefore,

                                    

       similarly, the distance between two adajacent dark fringes can be proved to be λ L/d. It is, therefore found that the bright and dark fringes are of equal width and are equally spaced.

      The equation (5) reveals that fringe spacing increases if red light (long wavelength) is used as compared to blue light (short wavelength). The fringe spacing varies directly with distance (L) between the slits and screen and inversely with the separation d of the slits.

       If the separation d between the two slits, the order m of a bright or dark fringe and fringe spacing ∆y is known, the wavelength λ of the light used for interference effect can be determined by equation (5).