9.3 Refraction

Refraction is a fundamental optical phenomenon that occurs when light changes direction as it passes between different media. It explains why objects appear bent in water, how lenses focus light, and forms the basis for many optical technologies. From corrective lenses to fiber optic communications, refraction principles come up throughout Physics II and beyond.

Fundamentals of Refraction

Definition of Refraction

Refraction happens when light waves change direction as they cross from one medium into another. The reason light bends is that its speed changes when it enters the new medium. If light hits the boundary at an angle (rather than straight on), one side of the wavefront slows down before the other, causing the whole wave to pivot.

• The bending occurs at the interface between two materials with different optical densities.
• The amount of bending depends on the angle of incidence and the refractive indices of both media.
• Light bends toward the normal when entering a denser medium (slower speed) and away from the normal when entering a less dense medium (faster speed).

Snell's Law

Snell's law gives you the quantitative relationship between the angle going in and the angle coming out:

$n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$

• $n_1$ and $n_2$ are the refractive indices of the first and second media.
• $\theta_1$ is the angle of incidence (measured from the normal to the surface).
• $\theta_2$ is the angle of refraction (also measured from the normal).

Using Snell's law step by step:

1. Identify which medium the light starts in ($n_1$) and which it enters ($n_2$).
2. Measure or identify the angle of incidence $\theta_1$ from the normal.
3. Plug into $\sin(\theta_2) = \frac{n_1}{n_2} \sin(\theta_1)$.
4. Take the inverse sine to find $\theta_2$.

For example, light hitting water ($n = 1.33$) from air ($n = 1.00$) at $30°$ gives $\sin(\theta_2) = \frac{1.00}{1.33} \sin(30°) = 0.376$, so $\theta_2 \approx 22.1°$. The light bends toward the normal, as expected when entering a denser medium.

Index of Refraction

The index of refraction ($n$) measures how much a material slows down light compared to vacuum:

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

• $c$ is the speed of light in vacuum ($3.0 \times 10^8$ m/s).
• $v$ is the speed of light in the material.

Since light always travels slower in a material than in vacuum, $n$ is always ≥ 1. Some common values:

A higher index means light travels slower in that material and bends more when entering it from air.

Refraction at Interfaces

Air-Water Interface

When light passes from air into water, it slows down and bends toward the normal. This is why objects underwater appear closer to the surface than they really are. The apparent depth is always less than the actual depth, which you can estimate using $d_{\text{apparent}} = d_{\text{actual}} \times \frac{n_{\text{air}}}{n_{\text{water}}}$.

When light travels the other direction (water to air), total internal reflection becomes possible at the critical angle of approximately $48.6°$.

Air-Glass Interface

Light bends more sharply at an air-glass interface than at air-water because glass has a higher refractive index (typically 1.5 to 1.9, depending on composition). This stronger bending is what makes glass so useful for lenses, prisms, and other optical components found in microscopes, telescopes, and cameras.

Total Internal Reflection

Total internal reflection (TIR) occurs when light tries to pass from a denser medium to a less dense one and hits the interface at an angle greater than the critical angle. Instead of refracting through, all the light reflects back.

The critical angle is found from:

$\sin(\theta_c) = \frac{n_2}{n_1}$

where $n_1 > n_2$ (light must be in the denser medium).

For water to air: $\sin(\theta_c) = \frac{1.00}{1.33} = 0.752$, giving $\theta_c \approx 48.6°$. Any angle of incidence above this results in complete reflection.

TIR is the principle behind fiber optic cables, and it also explains why diamonds sparkle so intensely (diamond's high $n = 2.42$ gives a small critical angle of about $24.4°$, trapping light inside through many internal reflections).

Optical Phenomena

Mirages

Mirages are caused by refraction through layers of air at different temperatures. Hot air near the ground has a lower density (and lower refractive index) than the cooler air above, so light gradually bends upward as it passes through these layers.

• Inferior mirages form when the ground is very hot. Light from the sky bends upward near the surface, creating the illusion of water on a road.
• Superior mirages occur when cooler air sits beneath warmer air (common in arctic regions). Distant objects can appear elevated or even inverted.

Rainbows

Rainbows result from refraction, internal reflection, and dispersion of sunlight inside water droplets.

• The primary rainbow appears at about $42°$ from the antisolar point (the point directly opposite the sun from your perspective). Colors go from red on the outside to violet on the inside.
• The secondary rainbow appears at about $51°$ with the color order reversed, and it's dimmer because the light undergoes two internal reflections.
• You'll only see a rainbow with the sun behind you and water droplets in front of you.

Dispersion of Light

Dispersion occurs because a material's refractive index varies slightly with wavelength. Shorter wavelengths (blue/violet) have a higher refractive index than longer wavelengths (red), so blue light bends more than red light when passing through glass or water.

This wavelength-dependent bending is what separates white light into a spectrum in a prism, and it's the same mechanism that produces rainbow colors. Dispersion is also used in spectroscopy to analyze the composition of light sources.

Refraction in Lenses

Convex vs. Concave Lenses

Convex (converging) lenses are thicker at the center than at the edges. They bend parallel light rays inward so they meet at a focal point on the other side.

• Used in magnifying glasses, cameras, and correcting farsightedness (hyperopia).

Concave (diverging) lenses are thinner at the center. They spread parallel light rays outward so they appear to originate from a focal point on the same side as the incoming light.

• Used in correcting nearsightedness (myopia) and in certain telescope designs.

Combining convex and concave lenses in a system can correct optical aberrations like chromatic aberration.

Focal Length and the Thin Lens Equation

The focal length ($f$) is the distance from the center of a lens to the point where parallel rays converge (for a converging lens) or appear to diverge from (for a diverging lens).

The thin lens equation relates focal length to object and image distances:

$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$

• $f$ = focal length
• $d_o$ = distance from the object to the lens
• $d_i$ = distance from the lens to the image

Sign conventions matter here: for converging lenses, $f$ is positive. For diverging lenses, $f$ is negative. A negative $d_i$ means the image is virtual (on the same side as the object).

Shorter focal lengths produce greater magnification but a narrower field of view.

Lens-Maker's Equation

The lens-maker's equation connects a lens's focal length to its physical shape and material:

$\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$

• $n$ = refractive index of the lens material
• $R_1$ = radius of curvature of the first surface (the one light hits first)
• $R_2$ = radius of curvature of the second surface

This equation is what lens designers use to choose the right combination of glass type and surface curvature to achieve a desired focal length.

Applications of Refraction

Eyeglasses and Contact Lenses

Corrective lenses work by adjusting the path of light before it enters the eye so that images focus properly on the retina.

• Convex lenses correct farsightedness by converging light rays slightly before they reach the eye.
• Concave lenses correct nearsightedness by diverging light rays so the focal point moves back onto the retina.
• Cylindrical lenses correct astigmatism, where the eye focuses light differently along different axes.

Microscopes and Telescopes

Both instruments use combinations of lenses to magnify objects.

• Compound microscopes use an objective lens (short focal length, near the specimen) and an eyepiece lens to achieve high magnification of tiny objects.
• Refracting telescopes use a large objective lens to gather light from distant objects and an eyepiece to magnify the image.
• Reflecting telescopes use mirrors instead of (or in addition to) lenses for the objective, which avoids chromatic aberration and allows for larger apertures.

Fiber Optic Communication

Fiber optic cables transmit data as pulses of light through thin glass or plastic fibers. The light stays inside the fiber because of total internal reflection: as long as the light hits the fiber walls at an angle greater than the critical angle, it bounces along the length of the fiber with very little loss.

This technology enables high-speed, long-distance data transmission and is the backbone of modern internet infrastructure. Fiber optics are also used in medical imaging tools like endoscopes.

Wave Theory of Refraction

Huygens' Principle

Huygens' principle states that every point on a wavefront acts as a source of secondary spherical wavelets. The new wavefront is the surface tangent to all those wavelets.

This provides an intuitive explanation for refraction: when a wavefront hits an interface at an angle, the part that enters the slower medium first produces wavelets that travel a shorter distance. The wavefront pivots, changing direction. Huygens' principle correctly predicts the same result as Snell's law.

Wavefronts and Ray Diagrams

Wavefronts are surfaces of constant phase (like the crests of a wave). Rays are arrows drawn perpendicular to wavefronts, showing the direction of propagation.

At an interface, refraction causes wavefronts to change both direction and spacing. The spacing decreases in a denser medium because the wavelength shortens (frequency stays the same, but speed drops). Ray diagrams are the practical tool you'll use most often to trace light paths through lenses and interfaces.

Phase Velocity vs. Group Velocity

• Phase velocity is the speed at which individual wave crests move: $v_p = \frac{\omega}{k}$.
• Group velocity is the speed at which the overall envelope of a wave packet travels: $v_g = \frac{d\omega}{dk}$.

In a non-dispersive medium, these two are the same. In a dispersive medium (where $n$ depends on wavelength), they differ. The group velocity is the one that determines how fast energy and information actually travel through the medium.

Refraction in Everyday Life

Swimming Pool Depth Illusion

Pools always look shallower than they are. Light bending away from the normal as it exits the water makes the bottom appear closer to the surface. The apparent depth is roughly $\frac{3}{4}$ of the actual depth (since $\frac{n_{\text{air}}}{n_{\text{water}}} \approx \frac{1}{1.33} \approx 0.75$). This can genuinely lead to misjudging water depth, so it's worth being aware of.

Apparent Bending of Objects

A straight stick partially submerged in water appears to bend at the surface. Light from the submerged portion refracts as it exits the water, shifting the apparent position of that part of the stick. The degree of apparent bending depends on your viewing angle and the refractive index difference between the two media.

Atmospheric Refraction

Earth's atmosphere has layers of varying density, which gradually bend light from celestial objects. This makes stars and planets appear slightly higher in the sky than they actually are, with the effect strongest near the horizon.

Atmospheric refraction adds roughly 0.5° of apparent elevation near the horizon. This means you can still see the sun for a short time after it has geometrically set below the horizon. It also explains why the sun looks flattened (oval-shaped) at sunrise and sunset: the bottom edge is refracted more than the top edge.

Advanced Concepts

Gradient-Index Optics

Most lenses rely on curved surfaces to bend light, but gradient-index (GRIN) materials have a refractive index that varies continuously throughout the material. Light curves smoothly through the material without needing a curved surface. GRIN lenses are used in specialized fiber optics, compact camera systems, and optical waveguides.

Metamaterials and Negative Refraction

Metamaterials are artificially engineered structures with optical properties not found in nature. Some metamaterials can exhibit a negative refractive index, meaning light bends to the same side of the normal as the incoming ray (opposite to normal refraction).

This opens up possibilities like superlenses that can resolve details smaller than the wavelength of light (beating the usual diffraction limit) and theoretical applications like invisibility cloaking.

Nonlinear Optical Effects

In most situations, a material's response to light is proportional to the light's intensity. At very high intensities (like those from lasers), the response becomes nonlinear, producing effects such as:

• Second-harmonic generation: doubling the frequency of light (used to create green laser pointers from infrared lasers).
• Optical Kerr effect: the refractive index itself changes with light intensity.

These effects find applications in laser technology, optical switching, and telecommunications.