A multiple-slit system, often called a diffraction grating when the number of slits is large, produces a much sharper interference pattern than the ordinary two-slit experiment.
Each slit acts as a coherent source, and the waves from all slits superpose at the observation angle \(\theta\).
Because many phase contributions are added together, the main maxima become narrow and intense, while the minima between them can become very deep.
The central phase quantity is determined by the path difference between neighboring slits. For slit spacing \(d\), that path difference is
\[
d\sin\theta.
\]
Multiplying by \(2\pi/\lambda\) converts this into a phase difference, so a convenient grating phase variable is
\[
\beta = \frac{2\pi d\sin\theta}{\lambda}.
\]
In that notation, the \(N\)-slit interference factor is
\[
I = I_0\left[\frac{\sin(N\beta/2)}{\sin(\beta/2)}\right]^2.
\]
This is equivalent to the very common alternative notation
\[
\alpha = \frac{\pi d\sin\theta}{\lambda},
\qquad
I = I_0\left[\frac{\sin(N\alpha)}{\sin\alpha}\right]^2.
\]
Both forms are mathematically the same. The important point is that the intensity comes from the coherent sum of \(N\) equally spaced sources.
The principal maxima occur when all neighboring slit phase differences line up as integer multiples of \(2\pi\). That gives the standard grating condition
\[
d\sin\theta = m\lambda,
\qquad
m = 0,\pm1,\pm2,\dots
\]
or equivalently
\[
\sin\theta_m = \frac{m\lambda}{d}.
\]
These are the strongest bright peaks in the pattern. The central maximum corresponds to \(m=0\), so it is always at \(\theta=0\).
At a principal maximum, the numerator and denominator in the interference factor both approach zero, but the limit is finite:
\[
\left[\frac{\sin(N\beta/2)}{\sin(\beta/2)}\right]^2 \to N^2.
\]
This means the central peak and all other principal maxima have height
\[
I_{\max}=N^2 I_0,
\]
assuming equal slit amplitudes in this simplified model.
So when \(N\) increases, the main peaks become much taller and much narrower.
That is why gratings are so useful for wavelength separation and spectroscopy.
For the sample input
\(N=4\),
\(d=2\ \mu\text{m}\),
and
\(\lambda=500\ \text{nm}\),
first convert the spacing and wavelength:
\[
d = 2\times10^{-6}\ \text{m},
\qquad
\lambda = 500\times10^{-9}\ \text{m}.
\]
Then
\[
\frac{\lambda}{d} = \frac{500\times10^{-9}}{2\times10^{-6}} = 0.25.
\]
Therefore the principal maxima satisfy
\[
\sin\theta_m = 0.25\,m.
\]
So allowed orders are those for which \(|0.25m|\le1\), namely
\(m=0,\pm1,\pm2,\pm3,\pm4\).
The first principal maximum occurs at
\[
\sin\theta_1 = 0.25,
\qquad
\theta_1 = \sin^{-1}(0.25)\approx 14.48^\circ.
\]
This matches the probe angle often used in the sample preview.
The animation in the calculator shows a finite slit array, rays heading toward a selected probe angle, and a screen whose brightness varies with angle.
The right-side plot shows the normalized grating intensity relative to the central peak. This helps visualize one of the main physical differences between a small-\(N\) system and a true grating:
as \(N\) grows, the peaks become sharper and the contrast improves.
At more advanced university level, one usually combines the multiple-slit interference factor with a single-slit diffraction envelope. In that case the observed intensity becomes the product of an envelope term and the grating factor, and some principal maxima can even be missing.
But in the simplest pure-interference model, the central formulas to remember are
\[
\beta = \frac{2\pi d\sin\theta}{\lambda},
\qquad
I = I_0\left[\frac{\sin(N\beta/2)}{\sin(\beta/2)}\right]^2,
\qquad
d\sin\theta = m\lambda.
\]
Together, these describe where the main maxima occur and how sharply the multi-slit pattern is concentrated.
