AP Physics 2: Chromatic Dispersion

When white light passes through a prism, it fans out into a stunning rainbow of colors. This phenomenon, called chromatic dispersion, is far more than a simple visual effect; it is a fundamental optical principle with profound implications. Understanding dispersion explains everything from the beauty of a rainbow to the frustrating color fringes in cheap binoculars and is critical for designing precision optical systems like cameras, telescopes, and microscopes.

The Mechanism of Dispersion: Why Light Splits

At its core, dispersion occurs because the speed of light in a material depends on its wavelength. In a vacuum, all colors of light travel at the same speed, $c$, approximately $3.00\times 10^8$ m/s. However, when light enters a transparent medium like glass or water, it slows down. The extent of this slowing is quantified by the index of refraction, $n$, defined as $n=c/v$, where $v$ is the speed of light in the material.

Crucially, $n$ is not a constant for a given material; it varies with the wavelength ($\lambda$) of the incident light. This variation is described by dispersion relations. For most transparent materials in the visible spectrum, $n$ decreases as wavelength increases. This means that shorter wavelength light (violet, blue) slows down more and bends more sharply upon entering a material than longer wavelength light (red, orange). This wavelength-dependent bending, or refraction, is the direct cause of chromatic dispersion. A common empirical model for this relationship is Cauchy's formula: $n(\lambda )\approx A+\frac{B}{{\lambda }^2}$, where $A$ and $B$ are material-specific constants, clearly showing that $n$ increases as $\lambda$ decreases.

Prisms: The Classic Demonstration

A prism provides the clearest example of dispersion in action. As a parallel beam of white light strikes the first face of the prism, each wavelength component refracts at a slightly different angle due to the differing indices of refraction. Inside the prism, the paths diverge further. When these separated rays exit the second face, they refract again, increasing the angular spread. The result is a spectrum projected on a screen.

The amount of spreading, or angular dispersion, depends on two key factors: the dispersive power of the prism material and the prism's apex angle. Materials with a steeper variation of $n$ with $\lambda$ (like flint glass) produce greater dispersion than those with a gentler variation (like crown glass). The apex angle controls the total deviation; a larger angle typically leads to greater overall bending and separation, though the relationship is not perfectly linear due to the geometry of refraction.

The Physics of Rainbows

A rainbow is nature's masterpiece of dispersion, created by the combined effects of refraction, dispersion, and internal reflection within spherical water droplets. Sunlight enters a raindrop, refracts and disperses at the air-water interface, reflects off the inside back surface of the droplet, and refracts and disperses again as it exits. Because the exiting light is concentrated around a specific angle relative to the original sunlight (about 42° for the primary red bow and 40° for violet), your eye sees each color coming from droplets at a slightly different elevation in the sky, forming the familiar arc. The secondary rainbow, which appears fainter and with colors reversed, involves two internal reflections inside the droplet.

Quantifying Dispersion: Angular Separation

You can calculate the angular separation between two specific colors after passing through a prism. This requires applying Snell's Law, $n_1\sin {\theta }_1=n_2\sin {\theta }_2$, at both interfaces for each wavelength, using the appropriate $n(\lambda )$ for the prism material. For small apex angles or near-minimum deviation, the deviation angle $\delta$ for a given color can be approximated by $\delta \approx (n-1)A$, where $A$ is the prism apex angle in radians. The angular separation $\Delta \theta$ between two colors (e.g., red and blue) is then roughly proportional to the difference in their indices: $\Delta \theta \approx A\cdot (n_{blue}-n_{red})$. A precise calculation requires a stepwise geometric application of Snell's Law.

Engineering Solutions: Achromatic Lenses

Dispersion is not just for creating spectra; it's also a major problem in lens-based optics, where it causes chromatic aberration. In a simple converging lens, blue light focuses closer to the lens than red light, resulting in a focal point that varies with color. This produces colored fringes and blurry images. To correct this, optical engineers design achromatic doublets (achromats).

An achromat combines two lenses made from different types of glass (e.g., crown and flint) with different dispersive powers. The lenses are cemented together or placed in close contact. The converging lens (crown glass) is paired with a weaker diverging lens (flint glass). The design goal is to ensure the combined lens has the same focal length for two specific wavelengths (often red and blue), thereby bringing those colors to the same focus and dramatically reducing the color fringe. This is achieved by carefully choosing the shapes and materials so that the dispersion of the flint lens counteracts the dispersion of the crown lens, while their combined refractive power still provides the desired net convergence.

Common Pitfalls

1. Confusing Cause and Effect: A common misconception is that the prism "adds color" to the light. Remember, white light is a mixture of all visible colors; the prism merely separates them due to their inherent physical differences. Dispersion is an analysis, not a creation.
2. Misapplying Snell's Law: When solving prism problems, students often use a single index of refraction $n$ for white light. You must use the correct $n$ for each wavelength or color you are tracking. Snell's Law applies independently to each wavelength component.
3. Overgeneralizing the n vs. $\lambda$ Relationship: While $n$ decreases with increasing $\lambda$ for most visible light in normal materials, this trend is not universal across all electromagnetic spectra. In certain wavelength ranges or for some exotic materials, anomalous dispersion can occur where the trend reverses. For AP Physics 2, assume the normal trend holds in the visible spectrum unless stated otherwise.
4. Ignoring the Role of Internal Reflection in Rainbows: It's easy to think a rainbow is just dispersion in a single refraction. The iconic arc shape and color order are only possible because of the specific geometry enforced by the internal reflection within the spherical droplet. Without it, you would see a much different, diffuse spectrum.

Summary

• Chromatic dispersion is the separation of white light into its constituent colors caused by the wavelength-dependent index of refraction of a material; shorter wavelengths (violet/blue) refract more than longer ones (red).
• Prisms demonstrate dispersion by refracting light at two interfaces, with the angular spread governed by the material's dispersive power and the prism's apex angle. Rainbows form from dispersion, internal reflection, and refraction within spherical water droplets.
• The angular separation between colors can be calculated by applying Snell's Law with the appropriate indices of refraction for each wavelength at each interface of an optical element.
• In lens systems, dispersion causes chromatic aberration, which is corrected using achromatic doublets—compound lenses that use two materials with different dispersion to bring two specific wavelengths to a common focal point.