Young’s Double Slit Experiment – Interference of Light (YDSE)

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In 1801, Thomas young first demonstrated experimentally the phenomenon of interference of light. The experimental setup consists of a single source ’S’ from which light rays travel and incident on two pinholes S₁, and S₂, as shown in Figure. Now S₁, and S₂, act as two coherent sources. The distance (d) between S₁, and S₂, is of order of the wavelength of light and emit spherical wave fronts. According to Huygen’s wave theory, each point on these spherical wave fronts acts as secondary sources. The emitted secondary wavelets by the two sources overlap each other and forming interference patterns.

When the crest of one wave front overlaps with crest of another wave front (or trough of one wave front with trough of another wave front), constructive interference takes place. Similarly, when the crest of one wave front overlaps with the trough of second wave front, destructive interference pattern takes place. Hence all the points corresponding to constructive interference form bright bands and all points corresponding to destructive interference form dark bands in the interference pattern.

Derivation of the intensity of fringe

Let \( y_1 \) is the displacement produced at point P due to source \( S_1 \). It is given by

\[ y_1 = a_1 \sin \omega t \tag{1} \]

where \( a_1 \) is the amplitude of wave train given by \( S_1 \) and \( \omega \) is its angular frequency. Similarly, let \( y_2 \) be the displacement produced by \( S_2 \) at point P due to source \( S_2 \). It is given by

\[ y_2 = a_2 \sin (\omega t + \delta) \tag{2} \]

According to the principle of superposition, the resultant displacement at P is given by

\[ y = y_1 + y_2 = a_1 \sin \omega t + a_2 \sin (\omega t + \delta) \tag{3} \] \[ y = a_1 \sin \omega t + a_2 \sin (\omega t) \cos \delta + a_2 \cos (\omega t) \sin \delta \tag{4} \] \[ a_1 + a_2 \cos (\delta) = R \cos (\theta) \tag{5} \] \[ a_2 \sin \delta = R \sin (\theta) \tag{6} \]

Where \( R \) and \( \theta \) are new constants. Squaring and adding equations 5 and 6, we get

\[ R^2 \sin^2 \theta + R^2 \cos^2 \theta = (a_1 + a_2 \cos \delta)^2 + (a_2 \sin \delta)^2 \tag{7} \] \[ R^2 = a_1^2 + a_2^2 + 2 a_1 a_2 \cos \delta \tag{8} \]

Hence resultant intensity at point P is,

\[ I = R^2 = a_1^2 + a_2^2 + 2 a_1 a_2 \cos \delta \tag{9} \]

i) Condition for maximum intensity

\( I = I_{max} \) intensity at P is max, when \( \cos \delta = 1 \Rightarrow \delta = 2n\pi \)

Since, phase difference

\[ \delta = \frac{2\pi}{\lambda} \text{ (path difference)} \tag{10} \] \[ \frac{2\pi}{\lambda} (S_2P - S_1P) = 2n\pi \tag{11} \] \[ I_{max} = a_1^2 + a_2^2 + 2a_1a_2 = (a_1 + a_2)^2 \tag{12}\]

So the intensity at point P is maximum when path difference between two rays reaching point P is equal to n times of \( \lambda \), where \( n = 0, 1, 2, 3, \ldots \)

ii) Condition for minimum intensity

Intensity at P is minimum \( I = I_{min} \), where \( \cos \delta = -1 \Rightarrow \delta = (2n + 1)\pi \)

\[ \delta = \frac{2\pi}{\lambda} \text{ (path difference)} \tag{13} \] \[ \frac{2\pi}{\lambda} (S_2P - S_1P) = (2n + 1)\pi \tag{14} \] \[ S_2P - S_1P = (2n + 1) \frac{\lambda}{2} \tag{15} \] \[ I_{min} = a_1^2 + a_2^2 - 2a_1a_2 = (a_1 - a_2)^2 \tag{16} \]

So intensity at point P is minimum, when the path difference between two rays reaching P is odd number times of \( \lambda / 2 \).

Derivation of fringe width \( (\beta) \)

Let us consider a point P on the screen at a distance X from the center point O. From right angle \( S_1PQ \)

\[ (s_{1P})^2 = (s_1Q)^2 + (QP)^2 \tag{17} \] \[ y_1^2 = (x - d/2)^2 + D^2 \tag{18} \]

From right angle \( S_2PR \),

\[ (s_{2P})^2 = (s_2R)^2 + (RP)^2 \tag{19} \] \[ y_2^2 = (x + d/2)^2 + D^2 \tag{20} \]

Subtracting

\[ y_2^2 - y_1^2 = \big[ (x + d/2)^2 + D^2 \big] - \big[ (x - d/2)^2 + D^2 \big] \tag{21} \]

According to \( a^2 - b^2 = (a + b)(a - b) \)

\[ (y_2 - y_1)(y_2 + y_1) = 2xd \tag{22} \] \[ y_2 - y_1 = \frac{2xd}{y_2 + y_1} \tag{23} \]

Since the distance between slits and screen is large, \( S_2P + S_1P = 2D \), i.e., \( y_1 + y_2 = 2D \)

\[ y_2 - y_1 = \frac{xd}{D} \tag{24} \]

The path difference is

\[ \frac{xd}{D} \]

For bright fringe, the path difference is \( S_2P - S_1P = n \lambda \)

\[ \frac{xd}{D} = n \lambda \Rightarrow x = \frac{n \lambda D}{d} \tag{25} \]

The distance of first bright fringe from 'O' is given by, for \( n = 1 \),

\[ x_1 = \frac{\lambda D}{d} \]

The distance of second bright fringe, for \( n = 2 \),

\[ x_2 = \frac{2 \lambda D}{d} \]

The distance of third bright fringe, for \( n = 3 \),

\[ x_3 = \frac{3 \lambda D}{d} \]

The distance between the adjacent bright fringes is given by

\[ x_2 - x_1 = \frac{2 \lambda D}{d} - \frac{\lambda D}{d} = \frac{\lambda D}{d} \tag{26} \]

The spacing between consecutive dark fringes is equal to

\[ \frac{\lambda D}{d} \]

Hence fringe width,

\[ x_2 - x_1 = \frac{\lambda D}{d} = \beta \]

Hence the wavelength of the source of light \( \lambda \) is given by

\[ \lambda = \frac{\beta d}{D} \]