10.5 Intereference of Light Waves and Young's Experiment
If two sodium lamps illuminate two pinholes \(S_{1}\) and \(S_{2}\) , the intensities will add up and no interference fringes will be observed on the screen.
We will now discuss interference using light waves. If we use two sodium lamps illuminating two pinholes (Fig. 10.11) we will not observe any interference fringes. This is because of the fact that the light wave emitted from an ordinary source (like a sodium lamp) undergoes abrupt phase hanges in times of the order of \(10^{–10}\) seconds. Thus the light waves coming out from two independent sources of light will not have any fixed phase relationship and would be incoherent, when this happens, as discussed in the previous section, the intensities on the screen will add up.
Young’s arrangement to produce interference pattern.
The British physicist Thomas Young used an ingenious technique to “lock” the phases of the waves emanating from \(S_{1}\) and \(S_{2}\) . He made two pinholes \(S_{1}\) and \(S_{2}\) (very close to each other) on an opaque screen [Fig. 10.12(a)]. These were illuminated by another pinholes that was in turn, lit by a bright source. Light waves spread out from S and fall on both \(S_{1}\) and \(S_{2}\) . \(S_{1}\) and \(S_{2}\) then behave like two coherent sources because light waves coming out from \(S_{1}\) and \(S_{2}\) are derived from the same original source and any abrupt phase change in S will manifest in exactly similar phase changes in the light coming out from \(S_{1}\) and \(S_{2}\) . Thus, the two sources \(S_{1}\) and \(S_{2}\) will be locked in phase; i.e., they will be coherent like the two vibrating needle in our water wave example [Fig. 10.8(a)].
Thus spherical waves emanating from \(S_{1}\)and \(S_{2}\) will produce interference fringes on the screen GG′, as shown in Fig. 10.12(b). The positions of maximum and minimum intensities can be calculated by using the analysis given in Section 10.4 where we had shown that for an arbitrary point P on the line GG′ [Fig. 10.12(b)] to correspond to a maximum, we must have
\[ \mathrm{S}_{2} \mathrm{P}-\mathrm{S}_{1} \mathrm{P}=n \lambda ; \quad n=0,1,2 \ldots \]
Now, \[ \left(S_{2} P\right)^{2}-\left(S_{1} P\right)^{2}=\left[D^{2}+\left(x+\frac{d}{2}\right)^{2}\right]-\left[D^{2}+\left(x-\frac{d}{2}\right)^{2}\right]=2 x d \] where \(S_{1}\) \(S_{2}\) = d and OP = x . Thus \[ \mathrm{S}_{2} \mathrm{P}-\mathrm{S}_{1} \mathrm{P}=\frac{2 x d}{\mathrm{~S}_{2} \mathrm{P}+\mathrm{S}_{1} \mathrm{P}} \]
If x, d is much lesser than D then negligible error will be introduced if \(S_{2}\) P + \(S_{1}\) P (in the denominator) is replaced by 2D. For example, for d = 0.1 cm, D = 100 cm, OP = 1 cm (which correspond to typical values for an interference experiment using light waves), we have \[ \begin{aligned} \mathrm{S}_{2} \mathrm{P}+\mathrm{S}_{1} \mathrm{P}=&\left[(100)^{2}+(1.05)^{2}\right]^{1 / 2}+\left[(100)^{2}+(0.95)^{2}\right]^{1 / 2} \\ & \approx 200.01 \mathrm{~cm} \end{aligned} \] Thus if we replace \(S_{2}\) P + \(S_{1}\) P by 2 D, the error involved is about 0.005%. In this approximation, Eq. (10.16) becomes \[ \mathrm{S}_{2} \mathrm{P}-\mathrm{S}_{1} \mathrm{P} \approx \frac{x d}{D} \]
Hence we will have constructive interference resulting in a bright region when \( \frac{x d}{D}=n \lambda \) [Eq. (10.15)]. That is,
\[ x=x_{n}=\frac{n \lambda D}{d} ; \mathrm{n}=0,\pm 1,\pm 2, \ldots \]
On the other hand, we will have destructive interference resulting in a dark region when \( \frac{x d}{D}=\left(n+\frac{1}{2}\right) \lambda \) that is
\[ x=x_{\mathrm{n}}=\left(n+\frac{1}{2}\right) \frac{\lambda D}{d} ; n=0,\pm 1,\pm 2 \]
Thus dark and bright bands appear on the screen, as shown in Fig. 10.13. Such bands are called fringes. Equations (10.18) and (10.19) show that dark and bright fringes are equally spaced and the distance between two consecutive bright and dark fringes is given by
\[ \beta=x_{n+1}-x_{n} \text { or } \beta=\frac{\lambda D}{d} \] expression for the fringe width
Computer generated fringe pattern produced by two point source \(S_{1}\) and \(S_{2}\) on the screen GG′ (Fig. 10.12); (a) and (b) correspond to d = 0.005 mm and 0.025 mm, respectively (both figures correspond to D = 5 cm and λ = 5 × 10 –5 cm.) (Adopted from OPTICS by A. Ghatak, Tata McGraw Hill Publishing Co. Ltd., New Delhi, 2000.)
Obviously, the central point O (in Fig. 10.12) will be bright because \(S_{1}\) O = \(S_{2}\) O and it will correspond to n = 0 [Eq. (10.18)]. If we consider the line perpendicular to the plane of the paper and passing through O [i.e., along the y-axis] then all points on this line will be equidistant from \(S_{1}\) and \(S_{2}\) and we will have a bright central fringe which is a straight line as shown in Fig. 10.13. In order to determine the shape of the interference pattern on the screen we note that a particular fringe would correspond to the locus of points with a constant value of \(S_{2}\) P – \(S_{1}\) P. Whenever this constant is an integral multiple of λ, the fringe will be bright and whenever it is an odd integral multiple of λ/2 it will be a dark fringe. Now, the locus of the point P lying in the x-y plane such that \(S_{2}\) P – \(S_{1}\) P (= ∆) is a constant, is a hyperbola. Thus the fringe pattern will strictly be a hyperbola; however, if the distance D is very large compared to the fringe width, the fringes will be very nearly straight lines as shown in Fig. 10.13.
We end this section by quoting from the Nobel lecture of Dennis Gabor* The wave nature of light was demonstrated convincingly for the first time in 1801 by Thomas Young by a wonderfully simple experiment. He let a ray of sunlight into a dark room, placed a dark screen in front of it, pierced with two small pinholes, and beyond this, at some distance, a white screen. He then saw two darkish lines at both sides of a bright line, which gave him sufficient encouragement to repeat the experiment, this time with spirit flame as light source, with a little salt in it to produce the bright yellow sodium light. This time he saw a number of dark lines, regularly spaced; the first clear proof that light added to light can produce darkness. This phenomenon is called interference. Thomas Young had expected it because he believed in the wave theory of light.
We should mention here that the fringes are straight lines although \(S_{1}\) and \(S_{2}\) are point sources. If we had slits instead of the point sources (Fig. 10.14), each pair of points would have produced straight line fringes resulting in straight line fringes with increased intensities.
Photograph and the graph of the intensity distribution in Young’s double-slit experiment