Chain of sources

Figure 12.10: Chain of coherent dipoles

Consider N dipole oscillators arranged along a linear chain as shown in Figure 12.10, all emitting radiation with identical amplitude and phase. How much radiation will a distant observer at an angle $\theta$ receive? If $\tilde E_0 , \tilde E_1, \tilde E_2, ...,~{\rm and~} \tilde{E}_{N-1}$ are the radiations from the $0$th, $1$st, $2$nd, ...,and the $N-1$th oscillator respectively, $\tilde E_1$ is identical to $\tilde E_0$ except for a phase difference as it travels a shorter path. We have

$$\tilde E_1 = \tilde E_0 ~ e^{i2 \alpha }$$ (12.12)

where $2 \alpha = \frac{2 \pi}{\lambda} d \sin \theta$ is the phase difference that arises due to the path difference. Similarly $\tilde E_2 = \left[ e^{i2 \alpha } \right]^2 \tilde E_0$. The total radiation is obtained by summing the contributions from all the sources and we have

$$\tilde E =\sum \limits^{N-1}_{n=0} \tilde E_n = \sum \limits^{N-1}_{n=0} \left[ e^{i 2 \alpha } \right]^n \tilde E_0$$ (12.13)

This is a geometric progression, on summing this we have

$$\tilde E =$$ (12.14)

This can be simplified further

$$\tilde E = \tilde E_0 \frac{e^{iN\alpha}}{e^{i \alpha }} \frac{e^... ... \alpha }} = \tilde E_0 e^{i (N-1)\alpha} \frac{\sin (N \alpha )}{\sin(\alpha)}$$ (12.15)

which gives the intensity to be

$$I = 0.5 \tilde E {\tilde E}^* = I_0 \frac{\sin^2 (N \alpha)}{\sin^2(\alpha)}$$ (12.16)

where

$$\alpha = \frac{\pi d \sin \theta }{\lambda}$$ (12.17)

Figure 12.11: Intensity pattern for chain of dipoles

Plotting the intensity as a function of $\alpha$ (Figure 12.11) we see that it has a value $I = N^2 I_0$ at $\alpha = 0$ . Further, it has the same value $I = I_0 N^2$ at all other $\alpha$ values where both the numerator and denominator are zero i.e $\alpha = m \pi ( m = 0,\pm 1, \pm 2 , \ldots )$ or

$$d \sin \theta = m \lambda ~~ ( m = 0,\pm 1,\pm 2 ,\pm 3 \ldots )$$ (12.18)

The intensity is maximum whenever this condition is satisfied These are referred to as the primary maxima of the diffraction pattern and $m$ gives the order of the maximum.

The intensity drops away from the primary maxima. The intensity becomes zero $N-1$ times between any two successive primary maxima and there are $N-2$ secondary maxima in between. The number of secondary maxima increases and the primary maxima becomes increasingly sharper (Figure 12.11) if the number of sources $N$ is increased. Let us estimate the width of the $m$th order principal maximum. The $m$th order principal maximum occurs at an angle $\theta_m$ which satisfies,

$$d \sin \theta_m = m \lambda \,.$$ (12.19)

If $\Delta \theta_m$ is the width of the maximum, the intensity should be zero at $\theta_m + \Delta\theta_m$ ie.

$$\frac{\pi d \sin (\theta_m + \Delta \theta_m )}{\lambda } = m \pi + \frac{\pi}{N}$$ (12.20)

which implies that

$$\sin ( \theta m + \Delta \theta_m) = \frac{\lambda}{d} \left( m+ \frac{1}{N} \right)$$ (12.21)

Expanding

$$\sin ( \theta_m + \Delta \theta_m)= \sin \theta_m \, \cos \Delta \theta_m + \cos \theta_m \, \sin \Delta \theta_m$$

and assuming that $\Delta \theta_m \ll 1$ we have

$$\Delta \theta_m ~ \cos \theta_m = \frac{\lambda}{d N}$$ (12.22)

which gives the width to be

$$\Delta \theta_m = \frac{\lambda}{d N \cos \theta_m}\,.$$ (12.23)

Thus we see that the principal maxima get sharper as the number of sources increases. Further, the $0$th order maximum is the sharpest, and the width of the maximum increases with increasing order $m$.

The chain of radiation sources serves as an useful model for many applications.