8.4 Multi source inteference

N-source interference

Lets look back at two source interference again, this time using exponential notation: We will consider both sources to be in phase with the sameamplitude. Also, just like before we define $E_0$ to be at the point where the interference is occurring, and approximate that it is the same for both waves. We have $$\overrightarrow{E_1}=\overrightarrow{E_0}{e}^{i(k{r}_1-\omega t)}$$ $$\overrightarrow{E_2}=\overrightarrow{E_0}{e}^{i(k{r}_2-\omega t)}$$ or $$\overrightarrow{E_2}=\overrightarrow{E_0}{e}^{i(k(r_1+\Delta r)-\omega t)}$$ where $$\Delta r=d\sin \theta$$

So when we add the E fields together we get $${\displaystyle \begin{array}{c}\vec{E}=\overrightarrow{E_1}+\overrightarrow{E_2}\\ =\overrightarrow{E_0}{e}^{i(k{r}_1-\omega t)}(1+e^{i\delta })\end{array}}$$

Now if I consider the case of not 2 but N sources, all separated by a distance d, then this simplyextends $$\overrightarrow{E_1}=\overrightarrow{E_0}{e}^{i(k{r}_1-\omega t)}$$ $$\overrightarrow{E_2}=\overrightarrow{E_0}{e}^{i(k{r}_2-\omega t)}$$ $$\overrightarrow{E_3}=\overrightarrow{E_0}{e}^{i(k{r}_3-\omega t)}$$ $$\text{.}$$ $$\text{.}$$ $$\text{.}$$ $$\overrightarrow{E_N}=\overrightarrow{E_0}{e}^{i(k{r}_N-\omega t)}$$ then $r_2=r_1+d\sin \theta$

$r_3=r_1+2d\sin \theta$

and so on,

or $r_2=r_1+\Delta r$

$r_3=r_1+2\Delta r$

So now when we add the E fields up we get $$\vec{E}=\overrightarrow{E_0}{e}^{i(k{r}_1-\omega t)}(1+e^{i\delta }+e^{2i\delta }+e^{3i\delta }+\dots +e^{(N-1)i\delta })$$ or rewriting $$$$

Now following is a general property of geometric series: $$\sum _{n=0}^{N-1}{x}^n=\frac{1-x^N}{1-x}$$

So now we get $$\vec{E}=\overrightarrow{E_0}{e}^{-i\omega t}{e}^{ik{r}_1}\frac{1-e^{i\delta N}}{1-e^{i\delta }}$$ or $$\vec{E}=\overrightarrow{E_0}{e}^{-i\omega t}{e}^{ik{r}_1}\frac{e^{i\delta N}-1}{e^{i\delta }-1}$$ or $$\vec{E}=\overrightarrow{E_0}{e}^{-i\omega t}{e}^{ik{r}_1}\frac{e^{i\delta N/2}(e^{i\delta N/2}-e^{-i\delta N/2})}{e^{i\delta /2}(e^{i\delta /2}-e^{-i\delta /2})}$$ or $$\vec{E}=\overrightarrow{E_0}{e}^{-i\omega t}{e}^{i(k{r}_1+(N-1)\delta /2)}\frac{\sin (\delta N/2)}{\sin (\delta /2)}$$ Now we can define $kR\equiv k{r}_1+(N-1)\delta /2)$ , which makes sense, this rephrases the equation in terms of the distance fromthe middle of the array of sources. So the equation becomes $$\vec{E}=\overrightarrow{E_0}{e}^{i(kR-\omega t)}\frac{\sin (\delta N/2)}{\sin (\delta /2)}$$

To find the irradiance, lets simplify things by taking the real part of this $$\vec{E}=\overrightarrow{E_0}\cos (kR-\omega t)\frac{\sin (\delta N/2)}{\sin (\delta /2)}$$

Then $$I=\epsilon c<E^2>=\epsilon c\frac{E_0^2}{2}\frac{{\sin }^2(\delta N/2)}{{\sin }^2(\delta /2)}$$ or $$I=I_0\frac{{\sin }^2(\delta N/2)}{{\sin }^2(\delta /2)}$$

Look at the plots, which show what the function $\frac{{\sin }^2(\delta N/2)}{{\sin }^2(\delta /2)}$ looks like for $N=100$ and $N=20$ . The height of the first principle maximum is equal to $N^2$ . This is because as

$\theta \to 0$ then $\delta \to 0$ (recall $\delta =kd\sin \theta$ ) Then $$\underset{\delta \to 0}{lim}\frac{{\sin }^2(\delta N/2)}{{\sin }^2(\delta /2)}\to \frac{{(\delta N/2)}^2}{{(\delta /2)}^2}=N^2$$ or at $\theta =0$ $$I=N^2{I}_0$$

It is also interesting to note that the first maxima become narrower as $N$ becomes larger.

Principle maxima occur when $$\delta /2=n\pi$$ or $$kd\sin {\theta }_{max}=2n\pi n=0,1,2,3$$ or $$\frac{2\pi }{\lambda }d\sin {\theta }_{max}=2n\pi$$ or $$\sin {\theta }_{max}=\frac{n\lambda }{d}$$

Minima occur when the numerator vanishes but the denominator does not: $$N\delta /2=n\pi n=1,2,3\dots \frac{n}{N}\ne integer$$ $$kd\sin \theta =2n\pi /N$$ $$\frac{2\pi }{\lambda }d\sin \theta =2n\pi /N$$ or minima occur at $$\sin \theta =\frac{n\lambda }{Nd}n=1,2,3\dots \frac{n}{N}\ne integer$$ There are secondary maxima between the minima that are away from a principle maximum.

This gives us an insight into phase array radar and interferometric radio telescopes. Suppose you have a series or radar antennas in a row. Then youintroduce a phase shift $\epsilon$ between each oscillator, then you get $$\delta =kd\sin \theta +\epsilon$$ and principle maxima will occur at $$d\sin {\theta }_{max}=n\lambda -\epsilon /k$$ Concentrating on the principal maximum we see that we can adjust the direction of theprinciple maximum simple by adjusting $\epsilon$ . In a modern phase array radar in fact a dome of antennas are used and thesituation is a bit more complicated but certainly a tractable problem with the help of a computer. So these radars have computers adjusting the phases of thevarious antennas to point the radar beam where desired - which can be much more rapidly scanned than a rotating parabolic antenna for example. We cansee that if you increase the number of antennas, then you will get a more narrowly collimated beam.