Dispersion Knowledge-Based Study

Study notes

1. Dispersion is the optical phenomenon in which the phase velocity of a wave depends on its frequency or wavelength. In geometric optics, when polychromatic light—most commonly white light—enters a transparent material medium, it separates into its constituent spectral colors.

   White light from the Sun or an incandescent bulb is not a single entity; it is a superposition of electromagnetic waves spanning a continuous spectrum of wavelengths from roughly 380 nm (violet) to 750 nm (red). In a vacuum, all electromagnetic waves travel at the exact same universal speed, $c \approx 3.00 \times 10^8\text{ m/s}$. Consequently, no dispersion occurs in a vacuum.

   When light enters a material medium (such as glass, water, or diamond), its speed decreases to a value $v$, governed by the absolute refractive index $n$ of the medium:

   $$n = \frac{c}{v}$$

   For most transparent media in the visible spectrum, the refractive index is not constant across all wavelengths; rather, it decreases as wavelength increases. This behavior is called normal dispersion:

   • Violet light: Has the shortest visible wavelength, travels slowest in the medium, and encounters the largest refractive index ($n_{\text{violet}} > n_{\text{red}}$).
   • Red light: Has the longest visible wavelength, travels fastest in the medium, and encounters the smallest refractive index.

   By Snell's Law of Refraction ($n_1 \sin\theta_1 = n_2 \sin\theta_2$), a higher refractive index causes a greater bending of the ray toward the normal upon entering a denser medium.

   A triangular glass prism is the classic device used to observe dispersion experimentally. When a narrow beam of white light strikes an inclined face of a prism at an oblique angle:

   1. First Refraction (Air to Glass): The rays enter the glass. Because $n_{\text{violet}} > n_{\text{red}}$, violet light is refracted through a larger angle toward the normal than red light. The colors begin to separate inside the prism.
   2. Second Refraction (Glass to Air): The rays exit the second face into air, bending away from the normal. Because of the non-parallel geometry of the prism faces, the angular separation is enhanced rather than reversed.

   The overall angle of deviation ($\delta$) is greatest for violet light and smallest for red light. As a result, the emergent beam displays an ordered spectrum traditionally remembered as red, orange, yellow, green, blue, indigo, and violet (ROYGBIV), with red deviating least and violet deviating most from the original direction.

   A prominent natural consequence of dispersion is the primary rainbow, formed when sunlight interacts with roughly spherical airborne raindrops:

   • Refraction and Dispersion at Entry: Sunlight enters the droplet at an oblique angle, refracting and dispersing into component colors.
   • Internal Reflection: The dispersed rays strike the rear boundary of the droplet and undergo partial internal reflection.
   • Refraction at Exit: The rays exit through the front of the droplet back into the air, refracting once more and increasing the angular separation.

   Because the concentration of emerging rays depends on the angle of minimum deviation, red light emerges at an angle of approximately $42^\circ$ relative to the incoming sunlight, while violet light emerges at approximately $40^\circ$. An observer viewing millions of droplets with their back to the Sun perceives an arc with red along the outer edge and violet along the inner edge.