Diffraction gratings Knowledge-Based Study

Study notes

1. What is a Diffraction Grating?

   A diffraction grating is an optical component consisting of a periodic structure of a very large number of equally spaced, closely positioned parallel slits or reflective grooves. While Young's double-slit experiment demonstrates basic interference using two slits, a diffraction grating typically incorporates hundreds or thousands of lines per millimeter.

   The distance between the centers of two adjacent slits is called the grating spacing or slit separation, denoted by $d$. If a grating has $N$ lines (or grooves) per unit length (such as per meter or per millimeter), the grating spacing is calculated as:

   $$d = \frac{1}{N}$$

   For example, a grating labeled with 500 lines per millimeter has a slit spacing of:

   $$d = \frac{1\text{ mm}}{500} = 2 \times 10^{-3}\text{ mm} = 2 \times 10^{-6}\text{ m}$$

   Condition for Constructive Interference

   When monochromatic light of wavelength $\lambda$ is incident normally on a transmission grating, each slit acts as a coherent source of cylindrical secondary wavelets (according to Huygens' principle). Waves emerging from adjacent slits at an angle $\theta$ relative to the original beam direction travel different distances.

   The path difference between rays from two adjacent slits is:

   $$\Delta L = d \sin\theta$$

   Constructive interference across all slits occurs simultaneously when this path difference is an integer multiple of the wavelength. This yields the fundamental diffraction grating equation:

   $$d \sin\theta = m\lambda \quad \text{where } m = 0, \pm 1, \pm 2, \dots$$

   • $m = 0$ corresponds to the central maximum (zero-order maximum), where all waves travel identical path lengths and exit undeflected ($\theta = 0$).
   • $m = \pm 1, \pm 2, \dots$ are the first-order, second-order, and higher-order principal maxima.

   Because $\sin\theta$ cannot exceed 1, the total number of observable diffraction orders is strictly limited by the physical constraint:

   $$|m| \le \frac{d}{\lambda}$$

   Why Gratings Produce Sharp Spectral Lines

   In a double-slit setup, interference fringes vary smoothly in brightness across a broad angular range. In contrast, a diffraction grating produces extremely sharp, intense, and narrow principal maxima separated by wide dark regions.

   As the total number of illuminated slits increases, light completely cancels out at angles just slightly off the principal maximum direction because wavelets from widely separated slits destructively interfere. As a result, the angular width of each maximum decreases proportionally to $1/N$, producing fine lines suitable for high-precision spectroscopic measurement.

   Angular Dispersion

   Angular dispersion ($D$) quantifies the angular separation between two spectral lines differing by a unit wavelength:

   $$D = \frac{\mathrm{d}\theta}{\mathrm{d}\lambda} = \frac{m}{d \cos\theta}$$

   Dispersion increases with higher diffraction orders ($m$) and smaller slit separations ($d$).

   Resolving Power of a Grating

   The ability of an optical grating to distinguish two closely spaced wavelengths $\lambda$ and $\lambda + \Delta\lambda$ is characterized by its resolving power ($R$):

   $$R = \frac{\lambda}{\Delta\lambda} = N_{\text{total}} \cdot m$$

   Here, $N_{\text{total}}$ is the total number of illuminated grating lines and $m$ is the spectral order. According to the Rayleigh criterion, two adjacent spectral peaks are just resolved when the central maximum of one falls on the first minimum of the other.

   Diffraction of Polychromatic Light

   When white light illuminates a grating:

   • The zero-order maximum ($m = 0$) remains white because all wavelengths satisfy $\theta = 0$.
   • For $m \ge 1$, because $\sin\theta = m\lambda / d$, longer wavelengths diffract at greater angles than shorter wavelengths. Consequently, red light ($\approx 700\text{ nm}$) diffracts more strongly than violet light ($\approx 400\text{ nm}$), spreading white light into a continuous rainbow spectrum with violet closest to the center and red furthest away.
   • Higher-order spectra (such as $m = 2$ and $m = 3$) may partially overlap when the angular deflection of blue wavelengths in the higher order matches that of red wavelengths in the lower order.