E2.3 Investigate diffraction, refraction, polarization, and interference of light

Ontario Grade 12 Physics

The Wave Nature of Light

Diffraction, refraction, polarization, and interference

Light can travel through a uniform material in straight lines, but its behaviour changes when it meets an opening, a boundary, or a polarizing filter. In this lesson, the system is light and the optical components it meets. For diagrams, take the direction from the source toward a screen as positive along the main path. Angles are measured from the normal, an imaginary line perpendicular to a boundary or slit. Light’s direction and wave travel are vectors; wavelength, frequency, and intensity are scalars. A ray diagram shows direction, while a wave diagram can show wavefronts and their spacing. These models help organize observations; they are not the observations themselves.

What you will learn

Describe how observations and controlled investigations can reveal diffraction, refraction, polarization, and interference of light.
Use wave relationships and ray models to explain what happens to light at openings, boundaries, and polarizing filters.
Calculate a refraction angle or a light-wave pattern using Grade 12 algebra and trigonometry.
Distinguish a proposed investigation from measured evidence.

From wave basics to diffraction and interference

A wave transfers energy without carrying matter from place to place overall. In a wave diagram, crests are regions of maximum displacement. The distance between adjacent crests is the wavelength, written as

\lambda

. Frequency,

f

, is the number of cycles passing a point each second. Its SI unit is the hertz, or cycles per second. For light in a vacuum, speed, wavelength, and frequency are related by

c=f\lambda

, where

c

is about

3.00\times10^8\ \mathrm{m/s}

. Light travels more slowly in transparent materials.

Diffraction is the spreading of a wave as it passes through an opening or around an obstacle. The spreading is most noticeable when the opening width is similar to the wavelength. A narrow slit can produce a broad pattern on a screen. A much wider slit produces less spreading. This comparison is a qualitative prediction; observations must be made with an actual setup before claiming a measured result.

Interference occurs when overlapping waves combine. Where their displacements reinforce, the result is constructive interference; where they reduce one another, it is destructive interference. For two slits separated by distance

d

, bright fringes occur when the path difference is an integer number of wavelengths. In a far-screen setup, the bright-fringe positions can be estimated with

d\sin\theta=m\lambda

. Here,

m

is an integer fringe number and

\theta

is the angle from the central direction.

A proposed investigation can use a laser, a double slit, and a screen. Keep the slit spacing and screen position fixed, and observe the bright and dark bands. Record actual positions and apparatus details before using the data. A simulation can help predict the pattern, but its output is simulated evidence, not a completed physical measurement. Never look directly into a laser beam.

c=f\lambda

Diffraction is wave spreading at an opening or obstacle.
Interference patterns have bright regions from reinforcement and dark regions from cancellation.
A wave model predicts patterns; a physical investigation is needed to collect measured evidence.

Refraction at a boundary

Refraction is a change in the direction of light as it enters a different material at an angle. The frequency does not change at the boundary. Because the speed changes, the wavelength changes too. A ray diagram should show the boundary, the normal, the incoming ray, and the refracted ray. Measure both angles from the normal, not from the surface.

The refractive index,

n

, compares the speed of light in a vacuum with its speed in a material. It has no units. Snell’s law relates the indices and angles on the two sides of a boundary. If light enters a material with a higher refractive index, it bends toward the normal. If it enters a lower-index material, it bends away from the normal.

A suitable investigation uses a narrow light ray directed at a clear block. Mark the boundary and normal, then measure the incident and refracted angles with a protractor. Repeat with different incident angles and record the actual readings. Careful alignment and angle measurement matter. Do not present suggested values as measured data.

n_1\sin\theta_1=n_2\sin\theta_2

The reference line for refraction angles is the normal.
Frequency remains constant across a boundary, while speed and wavelength can change.
Refractive index is dimensionless.

Polarization and observations

Polarization describes the direction of vibration of a transverse wave. Light from an ordinary lamp has vibrations in many directions perpendicular to its direction of travel. A polarizing filter transmits light vibrating along one preferred direction. After one filter, the transmitted light is polarized. Turning a second filter changes how much light passes through it.

The transmission axes are the directions each filter passes. When the axes are parallel, transmission is greatest. When they are at right angles, an ideal second filter blocks the light transmitted by the first. At intermediate angles, the transmitted intensity changes. For ideal filters and already polarized incoming light, the relationship is

I=I_0\cos^2\theta

, where

I_0

is the intensity before the second filter and

\theta

is the angle between the axes. Intensity is power per area and is measured in watts per square metre.

A proposed investigation uses two filters and a light source. Rotate one filter while keeping the source and detector fixed. Record detector readings and filter angles. The readings would be measured evidence if collected with real equipment; a predicted curve or simulated reading is not a measurement. This investigation demonstrates that light’s transverse vibrations have a direction that a filter can select.

I=I_0\cos^2\theta

A polarizer selects one vibration direction.
The angle in the intensity relationship is between the filter axes.
At a right angle between ideal polarizers, transmitted intensity is zero.

Choosing and checking a model

Use a wave description when spreading, overlapping patterns, or polarization are central. Use a ray diagram to track direction at a boundary. These are complementary Grade 12 models: the diagram should match the question being investigated.

Before calculating, identify the light path, the boundary or slit arrangement, the known quantities, and the unknown. Keep angles measured from the stated reference line. Convert distances to metres when using SI relationships. Check that both sides of an equation have compatible units. A refractive index or a trigonometric ratio has no units; a wavelength or distance must be reported in metres. Finally, ask whether the result agrees with the setup: for example, a ray entering a higher-index material should bend toward the normal.

Choose a ray or wave model to suit the observation.
State the reference line for every angle.
Check units, direction, and physical reasonableness before accepting a result.

Worked example

Finding a refraction angle

A ray travels from air into a clear material. The incident angle is

40.0^\circ

and the material has refractive index

1.50

. Find the refracted angle.

Define the setup
The system is the ray at the air–material boundary. The normal is the reference direction for both angles. Take the incident ray toward the boundary as the positive travel direction. The known values are $n_1=1.00$ , $\theta_1=40.0^\circ$ , and $n_2=1.50$ . The unknown is $\theta_2$ .
Apply Snell’s law
Use Snell’s law because the ray crosses between two materials. The refractive indices and sine values are dimensionless.
$n_1\sin\theta_1=n_2\sin\theta_2$
Substitute and solve
Rearrange for the sine of the refracted angle, substitute the known values, and take the inverse sine.
$\theta_2=\sin^{-1}\!\left(\frac{1.00\sin40.0^\circ}{1.50}\right)=25.4^\circ$

Answer: The refracted angle is

25.4^\circ

from the normal, toward the normal.

Check: The angle is smaller than the incident angle, as expected for entry into a higher-index material. The answer is an angle, so it has no physical unit.

Worked example

Estimating a double-slit wavelength

Two slits are separated by

0.250\ \mathrm{mm}

. The first bright fringe is observed at

0.600^\circ

from the central maximum. Estimate the wavelength.

Define the setup
The system is light passing through two slits and forming a pattern. The central direction is the reference direction, and the first bright fringe has $m=1$ . The known slit spacing is $d=2.50\times10^{-4}\ \mathrm{m}$ and the angle is $0.600^\circ$ . The unknown is wavelength, $\lambda$ .
Use the bright-fringe condition
For a double slit, bright fringes occur when the path difference is an integer number of wavelengths. Use the stated angle and fringe number.
$d\sin\theta=m\lambda$
Substitute and solve
Convert the slit spacing to metres and solve for wavelength. The angle is dimensionless inside the sine, so the result has the units of $d$ .
$\lambda=\frac{(2.50\times10^{-4}\ \mathrm{m})\sin(0.600^\circ)}{1}=2.62\times10^{-6}\ \mathrm{m}$

Answer: The estimated wavelength is

2.62\times10^{-6}\ \mathrm{m}

.

Check: The units reduce to metres. The calculation follows the specified first bright fringe. This value is an estimate based on the stated setup and model, not a claimed measured wavelength for a real source.

Worked example

Predicting transmission through a second polarizer

Polarized light of intensity

8.0\ \mathrm{W/m^2}

reaches a second ideal polarizer whose axis is

60^\circ

to the first. Find the transmitted intensity.

Define the setup
The system is polarized light reaching two ideal filters. The first filter’s transmission axis is the reference direction. The known intensity before the second filter is $I_0=8.0\ \mathrm{W/m^2}$ , and the angle between axes is $60^\circ$ . The unknown is transmitted intensity $I$ .
Apply the intensity relationship
For already polarized light passing through an ideal second filter, use the squared cosine of the angle between the axes.
$I=I_0\cos^2\theta$
Substitute
The cosine of $60^\circ$ is $0.50$ . Squaring this factor gives one quarter of the incoming intensity.
$I=(8.0\ \mathrm{W/m^2})\cos^2(60^\circ)=2.0\ \mathrm{W/m^2}$

Answer: The predicted transmitted intensity is

2.0\ \mathrm{W/m^2}

.

Check: Intensity keeps its SI unit, watts per square metre. The result is less than the incoming intensity, which is reasonable for a second filter at an angle.

Common mistakes and how to avoid them

Measuring refraction angles from the surface.

Correction: Measure both incident and refracted angles from the normal, the line perpendicular to the boundary.

Saying that light’s frequency changes when it enters another material.

Correction: The frequency stays the same at the boundary. The speed and wavelength change.

Treating every dark fringe as evidence that no light reached the setup.

Correction: In an interference pattern, a dark region can result from destructive interference where overlapping waves reduce one another.

Using the angle from the first polarizer instead of the angle between the two axes.

Correction: In the intensity relationship, use the angle between the transmission axes.

Describing predicted or simulated values as experimental measurements.

Correction: Label predictions and simulations clearly. Report measurements only when they have actually been collected.

Lesson summary

Diffraction is spreading at an opening or obstacle, and it is strongest when the opening is comparable in size to the wavelength.
Interference produces bright and dark regions through reinforcement and cancellation of overlapping waves.
Refraction changes light’s direction when its speed changes at a boundary; measure angles from the normal.
Polarizing filters select a vibration direction, and the angle between filter axes affects transmitted intensity.
An investigation must distinguish observations collected with equipment from predictions or simulations.

Check your understanding

Question 1

A light ray enters a material with a higher refractive index. What happens to its direction, if it strikes the boundary at an angle?

It bends toward the normal.
It bends away from the normal.
It continues unchanged because frequency is constant.
correctIndex": 0,"explanation":"A higher refractive index means a lower light speed in the material. The ray bends toward the normal when entering it at an angle."}]},

Show answer and explanation

It bends toward the normal.

A higher refractive index means a lower light speed in the material. The ray bends toward the normal when entering it at an angle.

Question 2

Two ideal polarizers have axes at right angles. What does the second polarizer transmit from light already polarized by the first?

All the incoming intensity
Half the incoming intensity
Zero intensity
correctIndex": 2,"explanation":"At a $90^\circ$ angle, the squared cosine is zero, so an ideal second polarizer transmits no light from the polarized input."}]},

Show answer and explanation

Zero intensity

At a

90^\circ

angle, the squared cosine is zero, so an ideal second polarizer transmits no light from the polarized input.

Key terms

Diffraction: The spreading of a wave as it passes an opening or obstacle.
Interference: The combining of overlapping waves, which can reinforce or reduce the resulting displacement.
Refraction: A change in the direction of light as it enters a different material at an angle.
Polarization: The direction of vibration of a transverse wave.
Normal: An imaginary line perpendicular to a surface at the point where a ray meets it.
Refractive index: The ratio of the speed of light in a vacuum to its speed in a material.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Physics (SPH4U), expectation E2.3. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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E2.3 · Investigate diffraction, refraction, polarization, and interference of light

Ontario Grade 12 Physics

Diffraction, refraction, polarization, and interference

What you will learn

From wave basics to diffraction and interference

Refraction at a boundary

Polarization and observations

Choosing and checking a model

Finding a refraction angle

Estimating a double-slit wavelength

Predicting transmission through a second polarizer

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Key terms

Continue through SPH4U

About this lesson and its review