Intensity of an N-slit interference pattern

The displacement $u$ at position $\vec{r}$ caused by $N$ slits at positions $\vec{r}_j$ is,

$$u = \sum\limits_{j=1}^N \frac{A}{\sqrt{|\vec{r}-\vec{r}_j|}}\cos (k|\vec{r}-\vec{r}_j|-\omega t )$$

Here $|\vec{r}-\vec{r}_j|$ is the distance from slit $j$. The wavenumber $k$ is related to the wavelenth $\lambda$ by $k=\frac{2\pi}{\lambda}$ and the angular frequency $\omega$ is related to the period $T$ by $\omega=\frac{2\pi}{T}$. Consider the case where the slits are equally spaced in an interval $a$ which is indicated by the small blue dots on the left in the intensity map below.

If you focus on one location $\vec{r}$, the displacement executes simple harmonic motion with a certain amplitude. Simple harmonic motion is the real part of circular motion in the complex plane. Written in complex form, the oscillations are,

$$ \sum \limits_{j=1}^N \frac{A_j}{\sqrt{|\vec{r}-\vec{r}_j|}} e^{ i(k|\vec{r}-\vec{r}_j|-\omega t +\phi_j)} = \left( \sum \limits_{j=1}^N \frac{A_j}{\sqrt{|\vec{r}-\vec{r}_j|}} e^{ i(k|\vec{r}-\vec{r}_j| +\phi_j)}\right)e^{-i\omega t}.$$

The right side of this equation describes a phasor that moves in a circle in the complex plane with an amplitude $A$,

$$ A= \sum \limits_{j=1}^N \frac{A_j}{\sqrt{|\vec{r}-\vec{r}_j|}} e^{ i(k|\vec{r}-\vec{r}_j| +\phi_j)}.$$

The intensity is the square of the amplitude,

$$ I \propto A^*A = \left(\sum \limits_{j=1}^N \frac{A_j}{\sqrt{|\vec{r}-\vec{r}_j|}} \cos(k|\vec{r}-\vec{r}_j|+\phi_j)\right)^2+\left(\sum \limits_{j=1}^N \frac{A_j}{\sqrt{|\vec{r}-\vec{r}_j|}} \sin(k|\vec{r}-\vec{r}_j|+\phi_j)\right)^2.$$

The form below plots the logarithm of the intensity of the waves as a function of the position for $A_j=1$ and $\phi_j=0$.

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$N=$ 2

$\lambda=$ 0.3 [cm]

$a=$ 1 [cm]

$P_x=$ 2.8 [cm]

$P_y=$ 3.2 [cm]

There is a red point on the intensity pattern. The coordinates of this point are $P_x$ and $P_y$. To the left of the intensity pattern is a representation in the complex plane of the harmonic oscillations at the red point. The blue phasors represent the harmonic motion caused by the waves traveling from the slits. The red phasor is the sum of the blue phasors. The real part of the red phasor is the motion observed at the red point.

If the slits are spaced more closely than a wavelength, the interference forms a beam that radiates horizontally from the slits.