Refraction and Lenses

Overview

Refraction is the bending of light as it passes from one medium to another. Lenses use refraction to form images and are essential components in cameras, eyeglasses, microscopes, and telescopes.

Index of Refraction

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

Where:

• $c$ = speed of light in vacuum ($3 \times 10^8$ m/s)
• $v$ = speed of light in medium
• $n \geq 1$ ($n = 1$ for vacuum)

Common Values

Material	$n$
Vacuum	1.00
Air	1.0003
Water	1.33
Glass	1.5
Diamond	2.42

Snell's Law

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

Where:

• $n_1, n_2$ = indices of refraction
• $\theta_1$ = angle of incidence
• $\theta_2$ = angle of refraction

Bending Direction

• From less dense to more dense ($n_1 < n_2$): bends toward normal
• From more dense to less dense ($n_1 > n_2$): bends away from normal

Total Internal Reflection

Occurs when light goes from denser to less dense medium at angles greater than critical angle.

Critical Angle

$$\sin(\theta_c) = \frac{n_2}{n_1} \quad \text{(for } n_1 > n_2\text{)}$$

Applications: fiber optics, prisms, diamond brilliance

Lenses

Converging (Convex) Lens

• Thicker in the middle
• Positive focal length
• Converges parallel rays to focal point

Diverging (Concave) Lens

• Thinner in the middle
• Negative focal length
• Diverges parallel rays (appear to come from focal point)

Lensmaker's Equation

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

Where $R_1, R_2$ are radii of curvature of the surfaces.

Thin Lens Equation

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

Same as mirror equation!

Magnification

$$m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}$$

Same conventions as mirrors:

• $m > 0$: upright
• $m < 0$: inverted
• $\lvert m \rvert > 1$: enlarged

Converging Lens - Image Formation

Object Position	Image Position	Image Type
Beyond 2F	Between F and 2F	Real, inverted, diminished
At 2F	At 2F	Real, inverted, same size
Between F and 2F	Beyond 2F	Real, inverted, enlarged
At F	At infinity	—
Inside F	Same side	Virtual, upright, enlarged

Diverging Lens - Image Formation

• Always virtual
• Always upright
• Always diminished
• Between lens and F (same side as object)

Power of a Lens

$$P = \frac{1}{f}$$

Unit: Diopter (D) = m⁻¹

• Positive $P$: converging lens
• Negative $P$: diverging lens

Lenses in Combination

Close Together

$$P_{\text{total}} = P_1 + P_2 + P_3 + \cdots$$ $$\frac{1}{f_{\text{eq}}} = \frac{1}{f_1} + \frac{1}{f_2} + \cdots$$

Separated by Distance $d$

Use image from first lens as object for second.

Ray Diagrams for Lenses

Converging Lens Rules

1. Parallel ray → refracts through F (far side)
2. Ray through F (near side) → refracts parallel
3. Ray through center → passes straight through

Diverging Lens Rules

1. Parallel ray → refracts as if from F (near side)
2. Ray toward F (far side) → refracts parallel
3. Ray through center → passes straight through

Examples

Example 1: Snell's Law

Light goes from air to glass ($n = 1.5$) at 45°.

$$\sin(\theta_2) = \frac{n_1 \sin(\theta_1)}{n_2} = \frac{1 \times \sin(45°)}{1.5} = \frac{0.707}{1.5} = 0.471$$ $$\theta_2 = 28.1°$$

Example 2: Critical Angle

Find critical angle for glass-air interface ($n_{\text{glass}} = 1.5$).

$$\sin(\theta_c) = \frac{n_{\text{air}}}{n_{\text{glass}}} = \frac{1}{1.5} = 0.667$$ $$\theta_c = 41.8°$$

Example 3: Converging Lens - Real Image

A lens ($f = 20$ cm) has an object at $d_o = 60$ cm.

$$\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} = \frac{1}{20} - \frac{1}{60} = \frac{3}{60} - \frac{1}{60} = \frac{2}{60}$$ $$d_i = 30 \text{ cm (real, opposite side)}$$ $$m = -\frac{30}{60} = -0.5 \text{ (inverted, diminished)}$$

Example 4: Converging Lens - Virtual Image

Same lens, object at $d_o = 10$ cm (inside F).

$$\frac{1}{d_i} = \frac{1}{20} - \frac{1}{10} = \frac{1}{20} - \frac{2}{20} = -\frac{1}{20}$$ $$d_i = -20 \text{ cm (virtual, same side)}$$ $$m = -\frac{-20}{10} = 2 \text{ (upright, magnified)}$$

Example 5: Diverging Lens

A lens ($f = -15$ cm) has an object at $d_o = 30$ cm.

$$\frac{1}{d_i} = \frac{1}{-15} - \frac{1}{30} = -\frac{2}{30} - \frac{1}{30} = -\frac{3}{30}$$ $$d_i = -10 \text{ cm (virtual, same side)}$$ $$m = -\frac{-10}{30} = 0.33 \text{ (upright, diminished)}$$

Example 6: Lens Combination

Two lenses ($f_1 = 10$ cm, $f_2 = 20$ cm) are in contact.

$$P = \frac{1}{0.1} + \frac{1}{0.2} = 10 + 5 = 15 \text{ D}$$ $$f = \frac{1}{15} \text{ m} = 6.67 \text{ cm}$$

Example 7: Vision Correction

A nearsighted person can't see beyond 100 cm. What lens corrects to infinity?

Object at infinity should form image at 100 cm (far point):

$$\frac{1}{\infty} + \frac{1}{-1} = \frac{1}{f}$$ $$f = -1 \text{ m} = -100 \text{ cm}$$ $$P = -1 \text{ D (diverging lens)}$$

Sign Convention Summary

Quantity	Positive	Negative
$d_o$	Object in front	Object behind
$d_i$	Image behind (real)	Image in front (virtual)
$f$	Converging lens	Diverging lens
$R$	Center on opposite side	Center on same side