What is the maximum number of lines per centimeter a diffraction grating can have and produce a complete first-order spectrum for visible light?

The diffraction grating is made of equally spaced parallel lines, which can be considered as a slit. The lightwave when passes through these parallel slits result in interference patterns of different orders.

The number of slits is inversely proportional to the slit separation, of the value $\mathbf{d}$. So, for the sake of getting the maximum number of lines per centimeter, we are supposed to minimize the value $\mathbf{d}$. The equation of the constructive interference:

$\mathbf{\theta sin\lambda }$ reveals that the smaller the value $\mathbf{d}$, the smaller will be $\mathbf{\lambda }$.

Then, to produce a complete first-order spectrum for visible light, we are supposed to evaluate the minimum $\mathrm{\lambda }$has to belocalid="1654054017267" $780\mathrm{nm}$.

The value must be localid="1654054020898" $\mathrm{m}=1,\mathrm{and}90.0^{°}\mathrm{in}\mathrm{\theta sin\lambda }$, and we get,

localid="1654245378786" $$\begin{gathered} \mathrm{d}=\frac{\mathrm{m\lambda }}{\mathrm{sin\theta }} \\ =\frac{1\times 780\times {10}^{‐9}\mathrm{m}}{\sin \left(90.0^{°}\right)} \\ =7.8\times {10}^{‐7} \end{gathered}$$

The number of lines per cm is then evaluated as:

$$\begin{gathered} \mathrm{n}=\frac{1\mathrm{cm}}{\mathrm{d}} \\ =\left(\frac{1\mathrm{cm}}{7.8\times {10}^{‐7}\mathrm{m}}\right)\left(\frac{1\mathrm{m}}{100\mathrm{cm}}\right) \\ =1.28\times {10}^4 \end{gathered}$$

Therefore, the maximum number of lines is $1.28\times {10}^4$lines per centimeter.