Geometric Optics, Vision and Optical Instruments
68 Dispersion: The Rainbow and Prisms
Learning Objectives
- Explain how dispersion separates white light into its component wavelengths.
- Describe why different wavelengths of light refract by different amounts.
- Discuss useful applications and unwanted effects of dispersion in optical systems.
A rainbow appearing against a dark stormy sky is one of the most familiar examples of light being separated into color. The same effect occurs when white light passes through a glass prism or reflects repeatedly inside a diamond. In each case, light of different wavelengths follows slightly different paths.
Visible light includes a continuous range of wavelengths. Longer visible wavelengths are perceived as red, while shorter visible wavelengths are perceived as violet. Between them lie orange, yellow, green, and blue. Although these color names are useful, there are no sharp boundaries between them; the visible spectrum changes continuously from one wavelength to the next.
White light contains a mixture of many visible wavelengths. When these wavelengths are separated so that they travel in different directions, the result is called dispersion. More generally, dispersion occurs whenever the behavior or speed of a wave depends on its wavelength. In optics, this usually means that different wavelengths are refracted by slightly different amounts.
Dispersion
Dispersion is the separation of light into its component wavelengths because different wavelengths travel or refract differently in a material.
Dispersion is not limited to visible light. Other electromagnetic waves, sound waves, and water waves can also disperse when their speed depends on wavelength or frequency. In optical systems, dispersion may be useful when we want to analyze the wavelengths present in light, but it can also be undesirable when it causes colors or signals to spread apart.
Making Connections: Dispersion in Healthcare
Many medical instruments use dispersion to separate light into wavelengths and identify substances by how they absorb or emit light. Spectrophotometers analyze blood and tissue samples, pulse oximeters compare the absorption of red and infrared light, and fluorescence instruments detect specific molecules by the wavelengths they emit.
However, dispersion can also reduce image sharpness or cause light pulses to spread as they travel through optical fibers. Engineers must therefore control dispersion carefully in endoscopes, microscopes, imaging systems, and fiber-optic communication networks.
Dispersion occurs because the index of refraction of a material depends slightly on the wavelength of light. Although we often treat the index of refraction as a single value for a material, each wavelength actually travels at a slightly different speed. In most transparent materials, shorter wavelengths (violet and blue light) have slightly larger indices of refraction than longer wavelengths (red light). As a result, violet light slows down more and bends more strongly than red light when it enters or leaves the material.
This wavelength dependence explains why a prism separates white light into a spectrum of colors. As each wavelength refracts by a slightly different amount, the colors spread apart instead of continuing along the same path. The order of the colors produced by a prism is the same as that observed in a rainbow.
Making Connections: Dispersion Beyond Visible Light
Dispersion is a general property of waves, not just visible light. It occurs whenever the speed of a wave depends on its wavelength or frequency. Sound waves, water waves, and many types of electromagnetic waves can all exhibit dispersion under the right conditions.
Scientists often use dispersion to learn about materials and biological tissues. For example, measuring how different wavelengths travel through tissue helps improve medical imaging techniques, while astronomers study the dispersion of radio waves from distant stars to determine the properties of the matter between them. In optical instruments, engineers must sometimes minimize dispersion to preserve sharp images, while in other applications they intentionally use it to separate wavelengths for analysis.
Table 68.1 shows that the index of refraction is not exactly the same for every wavelength. In every material listed, the index of refraction increases slightly as the wavelength decreases. Consequently, violet light bends more than blue light, blue bends more than green, and red light bends the least.
A prism therefore separates white light into its component colors. Although the differences in refractive index are small, they accumulate as light enters and exits the prism, producing the familiar spectrum from red to violet.
Rainbows are produced by the same physical principles. Sunlight entering a raindrop is refracted as it enters the water, reflected from the back surface of the drop, and refracted again as it exits. Because each wavelength bends by a different amount, the emerging light is separated into different colors.
Every raindrop sends only one color toward a particular observer. The complete rainbow is formed because millions of different raindrops send different wavelengths toward the observer's eye at the appropriate angles. This is why the rainbow appears as an arc centered opposite the Sun.
Applications of Dispersion
Dispersion can be either useful or undesirable, depending on the application.
- Useful: Spectrometers separate light into its component wavelengths to identify chemicals, analyze blood samples, detect biomarkers, and study biological tissues. Dispersion also produces the brilliant sparkle of gemstones by separating colors.
- Undesirable: In lenses, excessive dispersion causes chromatic aberration, producing colored fringes around images. In optical fibers, different wavelengths travel at slightly different speeds, causing pulses of light to spread out over long distances and limiting communication rates.
- Engineering solutions: Many medical imaging systems, microscopes, telescopes, cameras, and endoscopes use specially designed combinations of lenses or lasers with narrow wavelength ranges to minimize unwanted dispersion while preserving image quality.
Interactive Exploration: Dispersion with a Prism
White light contains a continuous range of wavelengths. Because different wavelengths refract by slightly different amounts, a prism separates white light into a spectrum of colors. In this simulation, explore how the refractive index depends on wavelength and investigate how prisms produce dispersion.
Switch to the Prism option and shine white light through the prism. Then compare the behavior of different materials and observe how changing the angle of incidence affects the emerging spectrum. Relate your observations to the wavelength-dependent indices of refraction discussed in this chapter.
Guided Exploration
As you interact with the simulation, try to answer the following questions:
- Switch to the Prism view and shine white light through the prism. Which color bends the most? Which bends the least?
- How does the order of the colors compare with the visible spectrum shown in Figure 68.2?
- Replace the prism with different materials. Does the amount of dispersion increase or decrease? What does this tell you about the material's refractive index?
- Change the angle at which the light enters the prism. Does the order of the colors change? Does the amount of separation change?
- Switch back to a single-color beam. Why is there no dispersion when only one wavelength is present?
- Based on your observations, explain why prisms produce spectra, why rainbows form in water droplets, and why lasers experience much less dispersion than white light.
After completing the exploration, compare your observations with the concepts presented in this chapter. Dispersion occurs because the refractive index of a material depends on wavelength. Shorter wavelengths, such as violet light, generally refract more strongly than longer wavelengths, such as red light. This wavelength dependence allows prisms to separate white light into its component colors and explains natural phenomena such as rainbows as well as the operation of many scientific and medical optical instruments.
Section Summary
- Dispersion is the separation of light into its component wavelengths because the index of refraction depends on wavelength.
- In most transparent materials, shorter visible wavelengths have larger indices of refraction and bend more strongly than longer wavelengths.
- Rainbows form through a combination of refraction, internal reflection, and dispersion of sunlight inside water droplets.
- Dispersion is useful in spectrometers and other instruments that analyze the wavelengths present in light.
- Dispersion can also be undesirable because it may blur images, create colored fringes, or cause light pulses to spread as they travel through optical fibers.
Problems & Exercises
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- Using Table 68.1, calculate the ratio of the speed of red light to the speed of violet light in diamond.
- Calculate the same ratio for polystyrene.
- Which material is more dispersive? Explain how your calculated ratios support your answer.
- A beam of white light travels from air into water at an incident angle of [latex]75.0^\circ[/latex]. At what angles are the red (660 nm) and violet (410 nm) components refracted?
- By how much do the critical angles for red light (660 nm) and violet light (410 nm) differ for a diamond surrounded by air?
- A narrow beam containing yellow light (580 nm) and green light (550 nm) travels from polystyrene into air and strikes the boundary at an incident angle of [latex]30.0^\circ[/latex].
- What is the angle between the two colors after they emerge into the air?
- How far must the rays travel after leaving the polystyrene for their separation to reach 1.00 mm?
- A parallel beam containing orange light (610 nm) and violet light (410 nm) travels from fused quartz into water. The beam strikes the boundary at an incident angle of [latex]60.0^\circ[/latex]. What is the angle between the two colors in the water?
- A ray of 610-nm light travels from air into fused quartz at an incident angle of [latex]55.0^\circ[/latex]. At what incident angle must 470-nm light enter flint glass to have the same angle of refraction?
- A narrow beam containing red light (660 nm) and blue light (470 nm) travels from air through a flat piece of crown glass that is 1.00 cm thick and then returns to air. The beam strikes the glass at an incident angle of [latex]30.0^\circ[/latex].
- At what angles do the red and blue rays emerge from the glass?
- What is the distance between the red and blue rays when they emerge?
- A narrow beam of white light enters an equilateral prism made of crown glass at an incident angle of [latex]45.0^\circ[/latex], as shown in Figure 68.7. At what angles, [latex]\theta_{\text{R}}[/latex] and [latex]\theta_{\text{V}}[/latex], do the red (660 nm) and violet (410 nm) components emerge from the prism?
Figure 68.7: White light enters an equilateral crown-glass prism at an incident angle of [latex]45.0^\circ[/latex]. Dispersion causes the red and violet components to emerge at different angles, labeled [latex]\theta_{\text{R}}[/latex] and [latex]\theta_{\text{V}}[/latex].
Glossary
- dispersion
- the separation of light into its component wavelengths because different wavelengths travel or refract differently in a material
- rainbow
- a continuous spectrum of colors produced when sunlight is refracted, internally reflected, and dispersed by water droplets in the atmosphere
- chromatic aberration
- an optical distortion in which different wavelengths are focused at different positions, producing colored fringes or a blurred image
the separation of light into its component wavelengths because different wavelengths travel or refract differently in a material
a continuous spectrum of colors produced when sunlight is refracted, internally reflected, and dispersed by water droplets in the atmosphere
an optical distortion in which different wavelengths are focused at different positions, producing colored fringes or a blurred image
