Optics

Light is an electromagnetic wave that can travel through a vacuum. It carries energy in the form of oscillating electric and magnetic fields perpendicular to each other and to the direction of travel.

• Light rays travel in straight lines through a uniform medium.
• Light slows down when it enters a medium denser than vacuum — this causes refraction.
• The full electromagnetic spectrum extends far beyond visible light: radio waves, microwaves, infrared, ultraviolet, X-rays, and gamma rays.

Law of Reflection

Plane Mirrors

A flat mirror forms an image with the following properties:

• Virtual — the image cannot be projected onto a screen; it appears to be behind the mirror.
• Upright — the image is the same orientation as the object.
• Same size — magnification = 1.
• Same distance behind mirror as the object is in front of it.
• Laterally inverted — left and right are swapped.

Specular vs. Diffuse Reflection

Type	Surface	Result	Example
Specular	Very smooth	Parallel rays stay parallel after reflection — clear image	Mirror, calm water
Diffuse	Rough / matte	Rays scatter in many directions — no clear image	Paper, painted wall

Types of Curved Mirrors

Type	Shape	Effect on rays	Focal length
Concave	Curves inward (like a cave)	Converging — brings parallel rays to a focus in front	f > 0
Convex	Curves outward	Diverging — spreads parallel rays as if from a focus behind	f < 0

Mirror Equation

Sign Conventions

Quantity	Positive (+)	Negative (−)
d᷹ (object distance)	Object in front of mirror (always)	Virtual object (rare)
dᵢ (image distance)	Real image — forms in front of mirror	Virtual image — appears behind mirror
f (focal length)	Concave mirror	Convex mirror
m (magnification)	Upright image	Inverted image

When light passes from one medium to another, it changes speed. This change in speed causes the ray to bend — this bending is called refraction.

Index of Refraction

Medium	Index n	Speed of light
Vacuum	1.000	3.00 × 10&sup8; m/s
Air	~1.000	~3.00 × 10&sup8; m/s
Water	1.33	2.26 × 10&sup8; m/s
Glass (typical)	1.50	2.00 × 10&sup8; m/s
Diamond	2.42	1.24 × 10&sup8; m/s

Snell's Law

• Angles are always measured from the normal (not the surface).
• Going into a denser medium (higher n): ray bends toward the normal (angle decreases).
• Going into a less dense medium (lower n): ray bends away from the normal (angle increases).

When light travels from a dense medium to a less dense medium (e.g., glass to air), there is a maximum angle beyond which light cannot escape — it is completely reflected back inside the medium.

Real-World Applications

• Optical fibres — light is trapped inside a glass core by TIR, allowing data to travel long distances with minimal loss.
• Diamonds — n = 2.42 gives a critical angle of only ~24.4°, so most light reflects internally many times, creating the characteristic sparkle.
• Mirages — hot air near the ground has a lower refractive index than cooler air above, bending light upward and creating the illusion of water.
• Endoscopes — used in medicine to see inside the body using optical fibre bundles.

Types of Lenses

Type	Also called	Effect on rays	Focal length
Converging	Convex lens	Brings parallel rays to a real focal point on the far side	f > 0
Diverging	Concave lens	Spreads rays as if from a virtual focal point on the same side as the object	f < 0

Thin Lens Equation

Sign Conventions for Lenses

Quantity	Positive (+)	Negative (−)
d᷹	Object on incoming side (always)	Virtual object
dᵢ	Real image — on opposite side from object	Virtual image — on same side as object
f	Converging (convex) lens	Diverging (concave) lens
m	Upright image	Inverted image

To locate an image with a ray diagram, draw any two of the three principal rays. Where they intersect (or where their extensions intersect) is where the image forms.

Three Principal Rays — Converging Lens

• Ray 1: Parallel to the principal axis → after the lens, passes through the far focal point F′.
• Ray 2: Through the optical centre of the lens → passes straight through, no bending.
• Ray 3: Through the near focal point F → after the lens, travels parallel to the principal axis.

Three Principal Rays — Concave Mirror

• Ray 1: Parallel to the principal axis → reflects through the focal point F.
• Ray 2: Through the focal point F → reflects parallel to the principal axis.
• Ray 3: Through the centre of curvature C (= 2F) → reflects straight back through C.

Summary of Image Types by Object Position (Converging Lens)

Object position	Image type	Orientation	Size
Beyond 2F	Real	Inverted	Reduced
At 2F	Real	Inverted	Same size
Between F and 2F	Real	Inverted	Enlarged
At F	No image	—	At infinity
Inside F	Virtual	Upright	Enlarged (magnifying glass)

Mistake	What to do instead
Measuring angles from the surface	Always measure θ from the normal (perpendicular to surface). Add 90° if you start from the surface.
Forgetting sign conventions	Real images: dᵢ > 0. Virtual images: dᵢ < 0. Concave f > 0. Convex f < 0.
Writing n = v/c instead of n = c/v	n = c/v. Since v < c in any medium, n is always ≥ 1.
Confusing mirror and lens equations	They are the same equation: 1/f = 1/d᷹ + 1/dᵢ. Just watch the sign conventions.
Negative magnification means smaller	Negative m means the image is inverted, not necessarily smaller. |m| < 1 means reduced; |m| > 1 means enlarged.
TIR going from less dense to dense	TIR only occurs going from dense to less dense (θ > θ₃). It cannot happen going from air into glass.
Using Snell's Law with surface angles	Snell's Law uses angles from the normal: n₁ sinθ₁ = n₂ sinθ₂.