A diffraction grating consists of many equally spaced slits or grooves. When monochromatic light falls on the grating, waves from adjacent slits interfere. Strong principal maxima occur only at specific angles where the path difference between neighboring slits is an integer multiple of the wavelength.
The grating condition for principal maxima is
\[
d\sin\theta_m = m\lambda,
\]
where \(d\) is the slit spacing, \(\lambda\) is the wavelength, and \(m=0,\pm1,\pm2,\dots\) is the diffraction order. Solving for the angle gives
\[
\sin\theta_m = \frac{m\lambda}{d}.
\]
A real maximum exists only if
\[
\left|\frac{m\lambda}{d}\right|\le 1.
\]
For the sample values
\[
d=1\,\mu\text{m},\qquad \lambda=500\,\text{nm},\qquad m=1,
\]
we first convert units:
\[
d=1\times 10^{-6}\text{ m},\qquad \lambda=5.00\times 10^{-7}\text{ m}.
\]
Then
\[
\sin\theta_1=\frac{\lambda}{d}=\frac{5.00\times10^{-7}}{1.00\times10^{-6}}=0.5.
\]
Therefore
\[
\theta_1=\sin^{-1}(0.5)=30^\circ.
\]
This is the expected first-order diffraction angle.
The angular intensity pattern of an ideal \(N\)-slit grating is much sharper than the double-slit case. The grating interference factor is
\[
I = I_0\left(\frac{\sin(N\beta/2)}{\sin(\beta/2)}\right)^2,
\]
where
\[
\beta = \frac{2\pi d\sin\theta}{\lambda}.
\]
The numerator produces very narrow strong maxima when all slit contributions add in phase, while the denominator controls the spacing between the interference fringes.
Increasing the number of illuminated slits \(N\) makes the principal maxima narrower and brighter relative to the surrounding regions. This is why gratings are excellent tools for spectroscopy. Closely spaced wavelengths that would blur together in a simple prism can often be resolved by a grating.
A standard measure of grating performance is the resolving power
\[
R=\frac{\lambda}{\Delta\lambda}=Nm.
\]
This formula shows that resolution improves when more slits are illuminated or when one observes a higher diffraction order.
Diffraction gratings appear in many familiar situations: laboratory spectrometers, compact-disc rainbow reflections, optical sensors, and wavelength-division systems in photonics. The same mathematics also underlies more advanced devices such as blazed gratings and echelle gratings.
This calculator uses the ideal grating model. It computes maxima angles, checks whether the requested order exists, estimates the normalized intensity pattern, and reports the resolving power. The graph and animation help visualize how a slit array spreads light into distinct diffraction orders.