Light Waves
3. Refraction and Lenses
Learning outcomes
- I can explain why light bends when entering a new medium.
- I can distinguish between converging and diverging lenses.
- I can construct simple ray diagrams for lenses.
- I can describe how refractive index affects light.
- I can identify applications of lenses.
Refraction and Lenses
Refraction is the change in direction of light when it passes from one transparent medium into another and its speed changes.
For example, light may travel from:
- air into water
- air into glass
- glass into air
- water into air
Because light travels at different speeds in different materials, its direction may change at the boundary between them.
This behaviour allows lenses to focus or spread light and is the basis of technologies such as:
- glasses
- contact lenses
- cameras
- microscopes
- telescopes
- magnifying glasses
- projectors
Why Light Refracts
Light travels fastest in a vacuum.
In transparent materials such as glass or water, light travels more slowly because of its interaction with the material.
When light crosses a boundary and its speed changes, its direction can also change.
This change in direction is called:
refraction
The amount of bending depends on:
- the materials involved
- the angle at which the light reaches the boundary
- the wavelength of the light
The Normal
As with reflection, refraction angles are measured from an imaginary line called the normal.
The normal is:
perpendicular to the boundary
at the point where the light ray crosses it.
Angles are measured between:
ray and normal
not between the ray and the surface.
Angle of Incidence
The angle of incidence, θᵢ, is the angle between:
incident ray and normal
The incident ray is the incoming ray approaching the boundary.
Angle of Refraction
The angle of refraction, θᵣ, is the angle between:
refracted ray and normal
The refracted ray is the ray travelling through the second medium.
Unlike reflection:
θᵢ and θᵣ are not generally equal.
Bending Toward the Normal
When light enters a medium in which it travels more slowly, the ray normally bends:
toward the normal
For example:
air → glass
Light slows down.
Therefore, for an angled ray:
angle of refraction < angle of incidence
The refracted ray is closer to the normal.
Bending Away from the Normal
When light enters a medium in which it travels faster, the ray normally bends:
away from the normal
For example:
glass → air
Light speeds up.
Therefore:
angle of refraction > angle of incidence
for angles below the critical angle.
A Useful Refraction Rule
A useful summary is:
faster → slower: toward the normal
slower → faster: away from the normal
This rule works when light strikes the boundary at an angle.
Light Travelling Along the Normal
Suppose light strikes a boundary directly along the normal.
Then:
θᵢ = 0°
The light changes speed when it enters the new medium, but it does not change direction.
Therefore:
a change in speed does not always produce visible bending.
The direction changes only when the light enters the new medium at an angle to the normal.
What Changes During Refraction?
When light passes into a different transparent medium:
speed can change
wavelength can change
direction can change
but:
frequency remains constant
This is an important distinction.
The source determines the frequency of the light, so crossing a boundary does not change its frequency.
Frequency and Wavelength
The wave equation is:
v = fλ
where:
v = wave speed
f = frequency
λ = wavelength
When light enters a medium where it travels more slowly:
- v decreases
- f remains constant
- λ decreases
Therefore:
slower light → shorter wavelength in that medium
When light speeds up again:
wavelength increases
Refractive Index
The refractive index, n, describes how much slower light travels through a material compared with its speed in a vacuum.
It is calculated using:
n = c/v
where:
n = refractive index
c = speed of light in vacuum
v = speed of light in the material
Since refractive index is a ratio of two speeds, it has:
no unit
Understanding Refractive Index
A material with a larger refractive index causes light to travel more slowly.
Therefore:
larger n → lower light speed
For example, if:
n = 1.5
then light travels through the material at:
v = c/n
v = (3.00 × 10⁸)/1.5
v = 2.00 × 10⁸ m/s
Light is therefore travelling considerably more slowly than it does in a vacuum.
Typical Refractive Indices
Approximate values for visible light include:
| Medium | Refractive Index |
|---|---|
| Vacuum | 1.000 |
| Air | ≈ 1.0003 |
| Water | ≈ 1.33 |
| Typical glass | ≈ 1.5 |
| Diamond | ≈ 2.42 |
Exact values can depend on wavelength and material composition.
Notice that a higher refractive index corresponds to a lower speed of light in the material.
Worked Example: Speed in Water
The refractive index of water is approximately:
n = 1.33
Use:
n = c/v
Rearrange:
v = c/n
v = (3.00 × 10⁸)/1.33
v ≈ 2.26 × 10⁸ m/s
Light travels through water at approximately:
2.26 × 10⁸ m/s
Optical Density
A material with a higher refractive index is sometimes described as being optically denser.
This does not necessarily mean it has greater mass density.
Optical density refers to how the material affects the speed of light.
Therefore:
optical density and physical density are not the same thing.
Snell's Law
The amount of refraction can be calculated using Snell's law:
n₁ sin θ₁ = n₂ sin θ₂
where:
n₁ = refractive index of the first medium
n₂ = refractive index of the second medium
θ₁ = angle in the first medium
θ₂ = angle in the second medium
All angles are measured from the normal.
Even when a calculation is not required, Snell's law explains why changing refractive index changes the amount of bending.
Worked Example: Air to Glass
Suppose light travels from air into glass.
Use:
n₁ = 1.00
n₂ = 1.50
θ₁ = 30°
n₁ sin θ₁ = n₂ sin θ₂
1.00 × sin 30° = 1.50 × sin θ₂
0.50 = 1.50 sin θ₂
sin θ₂ = 0.333
Therefore:
θ₂ ≈ 19.5°
The angle decreased from 30° to approximately 19.5°.
The light bent:
toward the normal
as expected.
Refraction in Everyday Life
Refraction produces many familiar effects.
Examples include:
- a straw appearing bent in water
- the bottom of a swimming pool appearing shallower
- objects underwater appearing displaced
- prisms separating colours
- lenses focusing light
- shimmering above hot roads
- atmospheric optical effects
These effects occur because light changes speed as it moves between materials or regions with different optical properties.
Apparent Depth
An object underwater can appear closer to the surface than it actually is.
Light travelling from:
water → air
speeds up and bends away from the normal.
Your brain normally assumes that light travelled toward your eyes in a straight line.
As a result, the object appears to be at a different position.
This is why a swimming pool can appear shallower than its true depth.
Lenses
A lens is a transparent optical object with one or more curved surfaces that refracts light.
The two major types are:
- converging lenses
- diverging lenses
Their shapes cause light rays to refract in predictable ways.
Converging Lenses
A converging lens is usually thicker in the centre than at the edges.
It is also commonly called a:
convex lens
Parallel light rays passing through a converging lens are refracted toward the principal axis.
Ideally, they meet at a point called the:
principal focus
or:
focal point
Diverging Lenses
A diverging lens is usually thinner in the centre than at the edges.
It is also commonly called a:
concave lens
Parallel rays passing through a diverging lens spread apart.
The rays appear to have originated from a focal point on the same side of the lens as the incoming light.
Comparing Converging and Diverging Lenses
| Property | Converging Lens | Diverging Lens |
|---|---|---|
| Common name | Convex lens | Concave lens |
| Typical shape | Thick centre | Thin centre |
| Effect on parallel rays | Converges them | Diverges them |
| Focus | Real focus for parallel incoming rays | Virtual focus |
| Image possibilities | Real or virtual | Usually virtual for a real object |
| Image size | Smaller, same, or larger | Smaller |
| Common uses | Magnifiers, cameras, eyes | Vision correction, optical systems |
Principal Axis
The principal axis is an imaginary line passing through the optical centre of a lens.
Ray diagrams are usually constructed around this line.
Important positions include:
- optical centre
- focal point F
- sometimes 2F
For a thin lens, there is a focal point on each side.
Focal Point
The focal point, F, is a point associated with the refraction of rays through the lens.
For a converging lens:
parallel incoming rays converge at F on the far side.
For a diverging lens:
parallel rays spread apart as though they originated from F on the incoming side.
Focal Length
The focal length, f, is the distance between:
optical centre and focal point
A strongly curved lens generally has greater optical power and a shorter focal length than a less strongly curved lens made from the same material.
The refractive index of the lens material also affects its focusing ability.
Why a Converging Lens Converges Light
A converging lens has curved surfaces.
As light enters and leaves the lens, it refracts at both boundaries.
The combined effect bends rays toward the principal axis.
A ray passing through the centre of a thin lens is often treated as travelling approximately straight in simple ray diagrams.
Parallel rays are brought toward a common focus.
Why a Diverging Lens Diverges Light
The surfaces of a diverging lens cause parallel rays to bend away from the principal axis.
After passing through the lens, the rays spread apart.
If these rays are extended backward, they appear to originate from a common focal point.
Because the rays do not actually meet there, this is a:
virtual focus
Principal Rays for a Converging Lens
Simple lens diagrams use standard rays.
Parallel ray
A ray parallel to the principal axis refracts through the far focal point.
Central ray
A ray passing through the optical centre of a thin lens continues approximately straight.
Focal ray
A ray travelling through the near focal point emerges parallel to the principal axis.
Usually, two correctly drawn rays are sufficient to locate an image.
Constructing a Converging-Lens Ray Diagram
To construct a simple ray diagram:
- Draw the principal axis.
- Draw the lens.
- Mark F on both sides.
- Mark 2F if needed.
- Draw the object as an upright arrow.
- Draw a ray from the top of the object parallel to the axis.
- Refract that ray through the far F.
- Draw a second ray through the optical centre.
- Continue it approximately straight.
- Locate where the rays intersect.
- Draw the image at that position.
- Determine the image's size, orientation, position, and type.
Converging Lens: Object Beyond 2F
If an object is placed beyond 2F:
the image forms:
- between F and 2F
- real
- inverted
- smaller
This arrangement is similar to the way some optical systems form reduced real images.
Converging Lens: Object at 2F
If the object is at 2F:
the image forms at approximately 2F on the opposite side.
It is:
- real
- inverted
- same size
Converging Lens: Object Between F and 2F
If the object is between F and 2F:
the image forms beyond 2F.
It is:
- real
- inverted
- enlarged
This arrangement can be useful when an enlarged real image is required.
Converging Lens: Object at F
If the object is at the focal point, emerging rays are approximately parallel.
In the ideal thin-lens model, the image is effectively at a very large distance.
Converging Lens: Object Inside F
If the object is between the lens and F, the refracted rays diverge.
Their backward extensions meet on the same side of the lens as the object.
The image is:
- virtual
- upright
- enlarged
This is how a magnifying glass is used.
Converging Lens Image Summary
| Object Position | Image Position | Type | Orientation | Size |
|---|---|---|---|---|
| Beyond 2F | Between F and 2F | Real | Inverted | Smaller |
| At 2F | At 2F | Real | Inverted | Same |
| Between F and 2F | Beyond 2F | Real | Inverted | Larger |
| At F | Very far away | Limiting case | Inverted | Very large |
| Inside F | Same side as object | Virtual | Upright | Larger |
The image produced by a converging lens therefore depends strongly on the object's distance from the lens.
Principal Rays for a Diverging Lens
For a diverging lens:
Parallel ray
A ray parallel to the principal axis refracts outward as though it came from the near focal point.
Central ray
A ray passing through the optical centre continues approximately straight.
The refracted rays do not actually meet.
Extend them backward using dashed lines.
The backward extensions locate the virtual image.
Images Formed by Diverging Lenses
For an ordinary real object, a diverging lens produces an image that is:
- virtual
- upright
- smaller
- on the same side of the lens as the object
This is true for ordinary object positions in front of a single diverging lens.
Real and Virtual Images
A real image occurs where light rays actually converge.
A real image can usually be projected onto a screen.
A virtual image occurs where light rays only appear to originate.
The rays do not actually meet at the virtual image location.
Converging lenses can produce:
- real images
- virtual images
Diverging lenses normally produce virtual images for real objects.
Determining Image Characteristics
When describing a lens image, consider:
Type
Real or virtual?
Orientation
Upright or inverted?
Size
Larger, smaller, or the same size?
Position
Where is it relative to the lens and focal points?
These four characteristics provide a useful complete description.
Cameras
A camera uses a converging lens or lens system to form a real image on a sensor.
Light from an object enters the camera.
The lens refracts the rays so that they converge on the image sensor.
The optical image formed on the sensor is:
- real
- typically inverted
- often reduced
Electronic processing then allows the image to be stored and displayed in the familiar orientation.
The Human Eye
The human eye also uses refraction.
Light entering the eye is refracted mainly by:
- the cornea
- the lens
The optical system focuses light onto the:
retina
The retina contains light-sensitive cells that convert incoming light into electrical signals.
The brain processes these signals to produce vision.
Focusing in the Eye
The eye must focus light from objects at different distances.
The eye's lens can change shape.
This changes its focusing power.
This process is called:
accommodation
For nearby objects, greater focusing power is generally required.
For distant objects, less focusing power is required.
Corrective Lenses
Some vision problems occur when light does not focus correctly on the retina.
Corrective lenses alter the path of incoming light before it enters the eye.
Different lens types are used for different vision conditions.
Myopia
Myopia, or short-sightedness, occurs when distant objects are focused in front of the retina when the eye is relaxed.
A diverging lens can be used to spread incoming light slightly before it enters the eye.
The eye then focuses the light farther back, onto the retina.
Hyperopia
Hyperopia, or long-sightedness, occurs when the eye would focus light from a near object behind the retina without sufficient accommodation.
A converging lens can help by beginning the convergence before light enters the eye.
This helps the eye focus the image onto the retina.
Magnifying Glasses
A magnifying glass uses a:
converging lens
The object is placed:
inside the focal length
The lens produces a:
- virtual
- upright
- enlarged image
This makes small details appear larger.
Microscopes
Optical microscopes use combinations of converging lenses.
A simple compound microscope contains:
- an objective lens
- an eyepiece lens
The lenses work together to produce a highly magnified view of a small object.
Microscopes allow us to observe structures that are too small to see clearly with the unaided eye.
Refracting Telescopes
A refracting telescope uses lenses to collect and focus light from distant objects.
The objective lens gathers light and forms an image.
An eyepiece then allows the observer to view the image at greater angular magnification.
Telescopes demonstrate how refraction can be used to investigate extremely distant objects.
Projectors
A projector uses a converging lens system to produce an enlarged real image on a screen.
Because a screen must receive actual converging light rays, the projected image must be:
real
The positions of the object, lens, and screen are adjusted so that the image is sharply focused.
Refractive Index and Lens Design
The amount of bending produced by a lens depends partly on:
- lens shape
- surface curvature
- refractive index of the lens material
- refractive index of the surrounding medium
A lens made from a material with a different refractive index may require a different shape to achieve the same focal length.
Engineers therefore choose optical materials carefully.
Lenses in Different Surroundings
A lens does not behave only according to its own refractive index.
What matters is also the refractive index of the surrounding medium.
For example, a glass lens behaves differently when surrounded by:
- air
- water
- another transparent material
This is because refraction depends on the change in refractive index across each boundary.
Dispersion in Lenses
The refractive index of a material can vary slightly with wavelength.
Therefore, different colours may refract by slightly different amounts.
This can cause coloured edges in images produced by simple lenses.
This effect is called:
chromatic aberration
Optical engineers can reduce it by combining lenses made from different materials.
Total Internal Reflection
When light travels from a material with higher refractive index toward one with lower refractive index, it bends away from the normal.
At sufficiently large angles, the light may no longer leave the higher-index material.
Instead, it is completely reflected internally.
This is called:
total internal reflection.
It is important in technologies such as:
- optical fibres
- some prisms
- endoscopes
- telecommunications
Optical Fibres
An optical fibre is a thin transparent structure designed to guide light.
Light can be kept inside the fibre through total internal reflection under appropriate conditions.
Optical fibres are used in:
- internet communication
- telephone networks
- medical endoscopes
- sensors
Although an optical fibre is not simply a lens, it demonstrates how refractive index and refraction principles can be engineered to control light.
Lens Power
The strength of a lens can be described by its optical power.
Lens power is related to focal length:
P = 1/f
when f is measured in metres.
The unit of lens power is the:
dioptre (D)
For example, a converging lens with:
f = 0.50 m
has:
P = 1/0.50
P = +2.0 D
Converging lenses have positive power.
Diverging lenses have negative power.
Worked Example: Lens Power
A converging lens has:
f = 0.25 m
Calculate its power.
Use:
P = 1/f
P = 1/0.25
P = +4.0 D
The positive sign indicates a converging lens.
Comparing Mirrors and Lenses
Mirrors and lenses can both control light, but they operate differently.
Mirrors
primarily use:
reflection
Lenses
primarily use:
refraction
Both can:
- converge light
- diverge light
- form images
- magnify objects
- be used in optical instruments
This is why similar ray-diagram ideas appear in both topics.
A Complete Ray-Diagram Example
Suppose an object is placed beyond 2F in front of a converging lens.
Draw a ray from the top of the object:
parallel to the principal axis
After passing through the lens, it refracts through the far focal point.
Draw another ray:
through the optical centre
It continues approximately straight.
The two rays intersect on the opposite side of the lens between F and 2F.
Therefore, the image is:
- real
- inverted
- smaller
- between F and 2F
The ray diagram allows all four image characteristics to be determined.
Choosing the Correct Lens
When given an unfamiliar lens problem, first ask:
Does the lens bring parallel rays together?
→ converging lens
Does the lens spread parallel rays apart?
→ diverging lens
Then determine:
- object position
- focal point
- principal axis
- ray directions
- image location
Finally describe:
- type
- orientation
- size
- position
A Reliable Strategy for Refraction Problems
When light crosses a boundary:
- Identify the two media.
- Draw the boundary.
- Draw the normal.
- Identify the incident ray.
- Determine whether light speeds up or slows down.
- If it slows, predict bending toward the normal.
- If it speeds up, predict bending away from the normal.
- Remember that frequency remains constant.
- Use n = c/v if speed or refractive index is required.
- Use Snell's law if angles must be calculated.
- Check that all angles are measured from the normal.
- Interpret the result physically.
A Reliable Strategy for Lens Ray Diagrams
For a simple lens diagram:
- Draw the principal axis.
- Draw the lens.
- Mark F on both sides.
- Mark 2F when useful.
- Draw the object.
- Start rays from the top of the object.
- Use a parallel ray.
- Use a central ray or focal ray.
- Draw the refracted rays accurately.
- Extend rays backward with dashed lines if necessary.
- Locate the image.
- State whether it is real or virtual.
- State whether it is upright or inverted.
- Compare its size with the object.
- Check whether the result makes sense for that lens type and object position.
Common Mistakes
Thinking refraction means reflection
Reflection sends light back from a boundary.
Refraction involves light passing into another medium and changing speed, often changing direction.
Measuring angles from the surface
Refraction angles are measured from the normal.
Thinking light always bends when entering another material
A ray travelling along the normal changes speed but does not change direction.
Thinking light bends toward the normal whenever it enters any new medium
It bends toward the normal when it enters a medium where it travels more slowly.
Thinking frequency changes during refraction
Frequency remains constant across the boundary.
Thinking wavelength stays constant
If wave speed changes while frequency remains constant, wavelength changes.
Thinking higher refractive index means faster light
It means the opposite:
higher n → lower speed
Confusing converging and diverging lenses
A converging lens brings parallel rays together.
A diverging lens spreads parallel rays apart.
Thinking all converging-lens images are real
A converging lens produces a virtual image when the object is inside its focal length.
Thinking a magnifying glass forms an enlarged real image on the object side
When used normally, it forms an enlarged virtual image.
Drawing virtual rays as actual light rays
Backward extensions used to locate virtual images should be shown as dashed lines.
Thinking optical density is the same as mass density
They describe different physical properties.
Key Terms
Refraction — change in direction of light associated with a change in speed as it enters a different medium.
Incident ray — incoming light ray approaching a boundary.
Refracted ray — light ray travelling through the second medium after crossing a boundary.
Normal — imaginary line perpendicular to a boundary at the point where a ray strikes it.
Angle of incidence — angle between the incident ray and the normal.
Angle of refraction — angle between the refracted ray and the normal.
Refractive index — measure of how much light slows in a material compared with its speed in vacuum.
Optical density — description related to how strongly a material reduces the speed of light.
Snell's law — relationship connecting refractive indices and angles of incidence and refraction.
Lens — transparent optical object that refracts light using curved surfaces.
Converging lens — lens that causes parallel rays to converge; commonly a convex lens.
Diverging lens — lens that causes parallel rays to diverge; commonly a concave lens.
Principal axis — reference line through the optical centre of a lens.
Optical centre — central reference point of a thin lens through which a ray is approximated as travelling straight.
Focal point (F) — point where parallel rays converge or from which diverging rays appear to originate.
Focal length (f) — distance between the optical centre and focal point.
Real image — image formed where actual light rays converge.
Virtual image — image formed where light rays appear to originate.
Lens power — measure of the focusing strength of a lens.
Dioptre (D) — unit of lens power.
Accommodation — adjustment of the eye's focusing power to view objects at different distances.
Myopia — vision condition in which distant objects are focused in front of the retina without correction.
Hyperopia — vision condition in which near objects would be focused behind the retina without sufficient accommodation.
Total internal reflection — complete reflection of light within a higher-index material under suitable conditions.
Chromatic aberration — colour distortion caused by different wavelengths being refracted by different amounts.
Key Takeaways
- Refraction occurs because light changes speed when it enters a different medium.
- Light bends toward the normal when it enters a medium where it travels more slowly.
- Light bends away from the normal when it enters a medium where it travels faster.
- A ray travelling along the normal does not change direction even though its speed may change.
- Angles of incidence and refraction are measured from the normal.
- During refraction, frequency remains constant.
- If speed changes, wavelength changes.
- Refractive index is calculated using n = c/v.
- A larger refractive index means a lower speed of light in the material.
- Snell's law relates refractive index to the angles of incidence and refraction.
- Everyday refraction explains why underwater objects can appear displaced.
- Lenses control light through refraction.
- Converging lenses bring parallel rays toward a focal point.
- Diverging lenses spread parallel rays apart.
- Converging lenses can produce real or virtual images depending on object position.
- Diverging lenses normally produce virtual, upright, reduced images for real objects.
- Ray diagrams can determine image position, type, orientation, and size.
- Magnifying glasses use converging lenses with the object inside the focal length.
- Cameras use lenses to form real images on sensors.
- The human eye uses refraction to focus light onto the retina.
- Diverging and converging corrective lenses can help correct different vision problems.
- Microscopes and refracting telescopes use combinations of lenses.
- Refractive index and lens curvature affect focusing strength.
- Total internal reflection allows light to be guided through optical fibres.
- Refraction and lenses are central to vision, medicine, communication, photography, astronomy, and many other optical technologies.
Check Your Understanding
- Explain why light can change direction when it enters a different transparent medium.
- Define the angle of incidence and angle of refraction. From which line are both angles measured?
- Light travels from air into glass at an angle to the normal. Explain what happens to its speed and direction.
- Light travels from glass into air at an angle below the critical angle. Explain why it bends away from the normal.
- A light ray travels directly along the normal from air into water. Its speed changes. Explain why its direction does not change.
- Describe what happens to the speed, frequency, and wavelength of light when it enters a material where it travels more slowly.
- A transparent material has a refractive index of 1.50. Calculate the speed of light in the material using c = 3.00 × 10⁸ m/s.
- Explain the difference between a converging lens and a diverging lens in terms of shape and their effect on parallel light rays.
- Describe how to construct a simple ray diagram for a converging lens.
- An object is placed beyond 2F in front of a converging lens. Describe the position, type, orientation, and size of the image.
- An object is placed between F and a converging lens. Describe the resulting image and explain why this arrangement is useful in a magnifying glass.
- Describe the image normally produced by a diverging lens for a real object.
- Explain how a camera and the human eye both use refraction and converging optical systems to form images.
- Explain why refractive index is important when engineers design lenses for cameras, microscopes, telescopes, glasses, or other optical instruments.
- A student says, "A lens bends light simply because its surfaces are curved." Improve this explanation by describing the roles of changing light speed, refractive index, refraction at boundaries, and lens shape.