# Optics Series Lecture, Lecture – IX.

“Young’s Double Slit Experiment. Coherent Sources and Conditions of Interference” This lecture was delivered on 14th February in a lecture session of 1 and 1/2 hours. This lecture was delivered to Physics elective students. At a later date this was delivered as a lecture to honors students as well. The web-version differs slightly from class delivered lectures, in that: any particular idea is explained without reference to what level it must cater to. That means in class lecture will modulate depending on the actual level of student body and their response. An honors student body who would find a particular discourse difficult will be supplied with further simplified versions of the concepts, verbatim. An elective students body which is well prepared would have no problems grasping the fundamentals at a purported level. Its a happy scenario if that is indeed the case.

The concurrent lecture is particularly divided into two parts. The first part pertains to what are coherent sources and what are the sustainable conditions for interference, for such to be observed. The second part leads us to describe in requisite detail the phenomenon of Young’s double slit interference. Note that we have already discussed the phenomenon of interference in our lecture-VII. We will only passively mention that there are two kinds of interference the so called wave-front-splitting and the amplitude splitting interference. Later on we will discuss any required details of both kinds. Before we do so we will have several interference phenomenon lectures from both types. Young’s double slit interference is an example of the wave-front splitting interference. What happens here is there are two primary or secondary coherent sources and two separate waves interfere at a given observation vantage. Another example of wave-front splitting interference is Fresnel’s bi-prism set-up which we will study soon, in an imminent lecture. For amplitude splitting interference only one wave produces the interference patterns, because the wave amplitude is partially reflected and partially transmitted — or refracted, and both channels meet up somewhere. Just to mention it for the time being, Newton’s Ring Interference patterns are example of amplitude splitting interference. Later we will study the details of all sorts of interference phenomena such as the ones just mentioned.

## coherent sources andconditions for interference:

Let us now discuss the first part of our lecture. Let us for the time being define coherence as the attribute of a light source such that there is no arbitrary and unexpected changes in the phases of different light waves such that when these waves at an observation point meet, we can apply the results of our interference analysis that we discussed in lecture-VII. Accordingly the coherence of light sources exists for certain amount of time, after which the coherence is lost. This is known as temporal coherence. If this time is denoted as Δt then light wave moves a distance of Δs = c × Δt and Δs is known as spatial coherence, as this is the distance for which coherence exists in the light wave. Coherence is necessary for light-waves, for us to observe the resultant interference.

1. Stable interference pattern is produced when two wave sources have same frequency.
Any large difference in the frequency destroys the coherence and therefore spoils the interference pattern. Large frequency difference results in rapid time-dependent phase difference, I12 averages to zero. Recall I12 as the interference term, in our description of interference in lecture-VII.
2. For white light there is interference among each color component.
Red interferes with red, blue interferes with blue and so on. An overall white interference pattern is observable.
3. The clearest interference pattern is produced,
when waves have nearly equal amplitude.
4. Two sources need to be in phase with each other,
for observation of fringe pattern.
5. As long as initial phase difference between two sources is constant,
an interference pattern which is shifted, is produced. Such sources are known as coherent sources.
6. The visibility of interference pattern is quantized: $\inline&space;\fn_cs&space;\boxed{v=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}}$
where the maximum and minimum irradiance of the two sources are related to the irradiance of the individual sources by the following, as discussed in lecture-VII$\inline&space;\fn_cs&space;\boxed{I_{max}=(\sqrt{I_1}+\sqrt{I_2})^2}$$\inline&space;\fn_cs&space;\boxed{I_{min}=(\sqrt{I_1}-\sqrt{I_2})^2}$$\inline&space;\fn_cs&space;v=\bigg[&space;\frac{2(I_1+I_2)}{4\sqrt{I_1I_2}}&space;\bigg]^{-1}$$\inline&space;\fn_cs&space;\boxed{v=\frac{2\sqrt{I_1I_2}&space;}{I_1+I_2}}$ when both sources have the same irradiance that is, I1 = I2 this would mean Imin = 0 and v = 1. The visibility is perfect or 100%. Therefore the amplitudes need be equal as we stated in point 3 above.

Let us now discuss the differences between coherent and incoherent sources. Let us also define formally, the coherence length and coherence time, that we just discussed above. Coherence Time: is the time interval during which light wave has sinusoidal purity and there are no random phase changes. Its given by: Δtc. Coherence Length: is the distance along the direction of propagation of light wave, over which temporal phase coherence exists in the wave, given by: L= c × Δtc.

## Young’s Double Slit Experiment.

Now we are in a position to discuss the interesting results of interference in a Young’s Double Slit Experiment set-up. Grimaldi was a famous Physicist and astronomer, who was deceased shortly before Galileo was born. Grimaldi had hoped to produce interference of light waves emanating from Sun by letting the light pass through a small hole. That way it would have produced two different wave-fronts emanating from the 2ndary source (the hole) acting like a primary source, a premonition about the wave-front splitting interference effect, so to say. But the set up resulted in a null effect. He did not observe any interference. Later Young modified his set up slightly to produce the expected interference patterns of light waves. The small change that Young, so inquisitive though, adopted is because of one of the primary reason for production of interference pattern as an observable. Namely Coherence. The sun is a vast region in the cosmos as far as we are concerned. This naturally means it does not have the required coherence, the primary condition for interference patterns, as light rays come from extremely long and uncorrelated points. Thus the small hole does not guarantee the desirable coherence.

Thomas Young, more than 100 years later, recognized the shortcoming that had clad Grimaldi’s understanding of the situation and brought a small change to the system. He introduced two small pinholes after a short distance to the primary hole. The first hole therefore acted like a primary and coherent source for the two secondary sources. This way while sun remained as an incoherent source to satisfy its own hubris, the secondary sources acted in tandem with the primary hole and provided a reasonable source of coherence, for the interference pattern to be observed. Let us now draw a diagram to depict this experiment famously known as Young’s Double Slit Experiment.

Let us consider two secondary coherent sources S1 and S2 which are distances r1 and r2 apart from point of observation P. The path difference of these two rays is given by S1 B = S1PS2P. Now S1B = r– r= a sin θ. This implies: r– r= aθ when angle θ is small. This also means θ = y/S. So: r– r= (a/S) y. Constructive interference occurs when path difference is integral multiple of λ. Thus: r– r= , where m is any integer; 0, ±1, ±2, ±3 … So height of observation point y can be given in terms of S: distance of plane of observation Σfrom screen of sources Σa , a: width between sources S1 and S2, m: order of interference, this is any integer and λ: the wavelength of the light sources. $\inline&space;\fn_cs&space;\boxed{y_m&space;\simeq&space;\frac{S}{a}m\lambda&space;}$Thus ym gives the position of the m-th order maxima or bright fringe, on the vertical axis. Since y = 0 for m = 0, we see a central bright fringe. The angular position of the m-th order bright fringe is given by: $\inline&space;\fn_cs&space;\boxed{\theta_m&space;\simeq&space;\frac{m\lambda&space;}{a}}$Two consecutive maxima are given by the integers m+1 and m. So, we can easily determine the spacing between bright fringes. $\inline&space;\fn_cs&space;y_{m+1}-y_m&space;\simeq\frac{S}{a}(m+1)\lambda-\frac{S}{a}m\lambda$$\inline&space;\fn_cs&space;\boxed{\Delta&space;y&space;=\frac{S}{a}\lambda&space;}&space;\,\,\,or&space;\,&space;\,\,\boxed{\beta_b=\frac{D\lambda}{d}}$where we have given the fringe spacing in alternative symbols used in some textbooks: βis fringe spacing for bright fringes, D is distance of separation between sources and screen of observation, d is slit separation. Similarly dark fringes or minima are formed when path difference is odd integer multiple of λ/2 or phase difference is odd integer multiple of π. We can thus write: $\inline&space;\fn_cs&space;y_m&space;\simeq&space;\frac{S}{a}\Big(m+\frac{1}{2}\Big)\lambda$$\inline&space;\fn_cs&space;y_{m+1}&space;\simeq&space;\frac{S}{a}\Big(m+\frac{3}{2}\Big)\lambda$So $\inline&space;\fn_cs&space;\boxed{\Delta&space;y&space;_{dark}&space;=\frac{S}{a}\lambda&space;}&space;\,\,\,or&space;\,&space;\,\,\boxed{\beta_d=\frac{D\lambda}{d}}$We see that the spacing between dark and bright fringes are the same and it remains constant even if we change the order of the interference: m.

When we studied interference due to two coherent plane harmonic waves, in lecture VII, for two beams having equal irradiance we derived the intensity distribution. $\inline&space;\fn_cs&space;\boxed{I=4\,I_0\,cos^2{\,\frac{\delta}{2}}}$The phase difference is δ = k (r– r2). By using r– r= (a/S) y we have:  $\inline&space;\fn_cs&space;\boxed{I=4\,I_0\,cos^2{\,\frac{ya\pi}{S\lambda}}}$In alternative symbols that we mentioned above: $\inline&space;\fn_cs&space;\boxed{I=4\,I_0\,cos^2{\,\frac{yd\pi}{D\lambda}}}$Let us plot this distribution as a function of y.