Q11Engineering Physics

A plane transmission diffraction grating is a highly precise optical component that acts as a 'super prism', capable of separating light into its constituent wavelengths with exceptional resolution. It is constructed by ruling thousands of perfectly parallel, incredibly fine, equidistant scratches on an optically flat, transparent glass plate using a diamond point. The scratches act as opaque barriers (scattering light randomly), while the untouched transparent spaces between them function as perfectly parallel optical slits.

The Grating Element

Let the width of each transparent slit be denoted as '$a$' and the width of each opaque ruling be '$b$'. The distance between the corresponding points of two consecutive slits is $(a+b)$. This fundamental distance is known as the grating element or grating constant. If a grating has $N$ lines ruled per inch, the grating element is $(a+b)=\frac{1\text{ inch}}{N}=\frac{2.54\text{ cm}}{N}$.

Formation of Principal Maxima (The Grating Equation)

When a monochromatic plane wavefront of wavelength $\lambda$ falls normally upon the grating surface, every single transparent slit acts as an independent source of secondary Huygens wavelets. These wavelets diffract (bend) outwards in all possible directions. A convex lens is placed behind the grating to focus these parallel diffracted rays onto a screen.

Consider rays diffracting at a specific angle $\theta$ relative to the normal. To find the resulting intensity, we must evaluate the path difference between rays originating from exactly corresponding points in adjacent slits. By dropping a perpendicular from the edge of one slit to the diffracted ray of the next slit, simple geometry reveals that the path difference ($\Delta$) is: $\Delta =(a+b)\sin \theta$

According to the principle of superposition, these millions of diffracted rays will interfere constructively (reinforce each other perfectly) only if this path difference is an exact integer multiple of the wavelength $\lambda$. This leads to the fundamental Grating Equation: $(a+b)\sin \theta =n\lambda$

Where $n=0,1,2,\mathrm{3...}$ is the order of the principal maximum. When this condition is met, all slits contribute light perfectly in phase, producing extremely intense, needle-sharp bright fringes called principal maxima. For $n=0$, $\theta =0$, all wavelengths form a combined Central Maximum (white light). For higher orders ($n=1,\mathrm{2...}$), the diffraction angle $\theta$ depends strictly on the wavelength $\lambda$, meaning different colors are deflected at different angles, creating a highly dispersed, distinct spectrum.