Geometric and Physical Optics

Big Picture

When light's wavelength is much smaller than the objects it encounters, we can treat it as rays traveling in straight lines — this is geometric optics. Reflection, refraction, and image formation follow simple geometric rules. When light encounters structures comparable to its wavelength, wave effects dominate — interference, diffraction, and polarization reveal that light is fundamentally a wave.

ELI-10: Explain It Like I'm 10

Light normally travels in straight lines — like a laser pointer beam. That is why shadows have sharp edges and why mirrors and lenses can focus light into images. But light is secretly a wave, and when it squeezes through tiny gaps or encounters very small patterns, it bends and spreads out in ways straight lines cannot explain. A soap bubble's rainbow colors and the way a CD shimmers are wave effects — interference.

15.1 Reflection and Refraction

Core Ideas

Law of reflection: angle of incidence = angle of reflection (θ_i = θ_r), measured from the normal.

Refraction: light bends when crossing a boundary between media with different speeds.

Index of refraction: n = c/v. For vacuum, n = 1; for water, n ≈ 1.33; for glass, n ≈ 1.5.

Snell's law: n₁sinθ₁ = n₂sinθ₂

Light bends toward the normal when entering a medium with higher n (slower). It bends away from the normal when entering a lower-n medium (faster).

Worked Example 1: Snell's Law (Air to Water)

Problem: Light enters water (n = 1.33) from air at 30° from the normal. Find the refraction angle.

Solution: 1.00 sin30^°= 1.33 sinθ₂ → sinθ₂ = 0.5/1.33 = 0.376 → θ₂ ≈ 22^°. The light bends toward the normal.

Worked Example 2: Snell's Law with Critical Angle (Water to Air)

Problem: A diver shines a flashlight upward from underwater (n₁ = 1.33) toward the air (n₂ = 1.00). (a) Find the critical angle for the water–air interface. (b) The flashlight beam hits the surface at 55° from the normal. Does the light escape into the air or undergo total internal reflection?

Solution:

(a) Critical angle: sinθ_c = n₂n₁ = 1.001.33 = 0.752 → θ_c = arcsin(0.752) ≈ 48.8^°.

(b) At 55°, the incident angle exceeds the critical angle (θ_i = 55^°> 48.8^°= θ_c). The light undergoes total internal reflection — it bounces back down into the water. No light escapes. If the diver angles the beam to 40° (less than θ_c), Snell's law gives: 1.33 sin40^°= 1.00 sinθ₂   →  sinθ₂ = 1.33 × 0.643 = 0.855   →  θ₂ ≈ 58.7^° The light escapes, bending away from the normal into the air.

ELI-10: Explain It Like I'm 10

Light slows down in water and glass. When light hits the surface at an angle, one side of the wave slows down first — like a car's wheels hitting mud on one side, causing it to swerve. That swerve is refraction. A straw in a glass of water looks bent because light from the underwater part changes direction at the surface on its way to your eye.

15.2 Total Internal Reflection

Core Idea

When light travels from a higher-n to lower-n medium, beyond a critical angle θ_c, all light reflects — none transmits.

sinθ_c = n₂n₁  (n₁ > n₂)

Applications: fiber optics (light trapped inside glass fiber by repeated total internal reflection), endoscopes, diamond brilliance (high n → small θ_c → light bounces many times inside).

ELI-10: Explain It Like I'm 10

Look up from underwater toward the surface at a shallow angle. Instead of seeing the sky, you see a mirror — the water surface reflects everything. This is total internal reflection. Light trying to escape the water at too flat an angle gets bounced back down. Fiber-optic cables use this to send light signals around the world: the light bounces endlessly inside a hair-thin glass strand.

15.3 Mirrors and Lenses

Core Ideas

Mirror/lens equation: 1f = 1d_o + 1d_i

Magnification: m = -d_id_o = h_ih_o (negative m means inverted image).

Sign conventions:

• f > 0: converging (concave mirror, converging lens). f < 0: diverging.
• d_o > 0: object on incoming side.
• d_i > 0: real image (light actually converges there). d_i < 0: virtual image.

Full Sign Convention Table

The table below gives the complete Cartesian sign convention used for mirrors and lenses. All distances are measured from the optical center (lens) or vertex (mirror) along the principal axis.

Assumptions: Paraxial Approximation and Thin Lens

The mirror and lens equations are derived under two key simplifying assumptions:

1. Paraxial approximation: All rays make small angles with the principal axis (sinθ ≈ θ ≈ tanθ). This means we only consider rays close to the axis. Rays at large angles suffer from spherical aberration — they do not all focus to the same point, blurring the image.

1. Thin-lens approximation: The lens thickness is negligible compared to its focal length and the object/image distances. In a thin lens, a ray passing through the center is not displaced sideways. Real lenses have finite thickness, requiring a more complex thick-lens formula, but the thin-lens equation is an excellent starting approximation for most simple optical systems.

These assumptions mean the equations are approximations — they work well for paraxial rays through thin lenses but break down for wide-angle optics and thick lenses used in high-end cameras and microscopes.

Ray Diagram Rules

Drawing ray diagrams helps visualize image formation. For lenses (converging or diverging), use three principal rays:

• Ray 1 (Parallel ray): A ray parallel to the principal axis refracts through the focal point on the far side (converging lens) or appears to come from the focal point on the near side (diverging lens).
• Ray 2 (Focal ray): A ray through (or toward) the near-side focal point emerges parallel to the principal axis.
• Ray 3 (Central ray): A ray through the center of the lens continues in a straight line without bending.

For mirrors, the three rays are analogous, replacing refraction with reflection:

• Ray 1 (Parallel ray): A ray parallel to the principal axis reflects through the focal point (concave) or appears to come from the focal point (convex).
• Ray 2 (Focal ray): A ray through (or toward) the focal point reflects parallel to the principal axis.
• Ray 3 (Central ray): A ray to the vertex reflects symmetrically (θ_i = θ_r).

Where any two rays intersect, the image forms. For virtual images, the rays diverge — extend them backward as dashed lines to find the virtual intersection.

Worked Example: Converging Lens

Problem: A 4.0 cm tall object is placed 30 cm in front of a converging lens with focal length f = 10 cm. Find: (a) the image distance, (b) the magnification, (c) the image height, and (d) describe the image (real/virtual, upright/inverted, larger/smaller).

Solution:

(a) Lens equation: 1d_i = 1f - 1d_o = 110 - 130 = 3 - 130 = 230 d_i = 302 = 15 cm d_i > 0 → real image, forms 15 cm behind the lens.

(b) Magnification: m = -d_id_o = -1530 = -0.50 |m| < 1 → image is smaller than the object.

(c) Image height: h_i = m · h_o = -0.50 × 4.0 = -2.0 cm The negative sign means the image is inverted.

(d) Description: Real, inverted, and half the size of the object. The image forms at 15 cm on the far side of the lens. A ray diagram would show the three principal rays converging at this point.

Verify with ray diagram: The object is at d_o = 30 cm = 3f (beyond 2f = 20 cm). According to the summary table, this produces a real, inverted, smaller image between f and 2f — confirmed (15 cm lies between 10 cm and 20 cm).

Summary Table

ELI-10: Explain It Like I'm 10

A magnifying glass (converging lens) makes things look bigger by bending light rays so they appear to come from a larger object behind the lens. A makeup mirror (concave mirror) does the same by curving inward. A car's side mirror (convex) curves outward to give a wider view, making cars look smaller and farther away ("objects in mirror are closer than they appear"). The lens equation is a simple formula that tells you exactly where the image forms and how big it will be.

15.4 Interference

Core Ideas

Young's double-slit experiment (1801) definitively proved light is a wave.

Bright fringes (constructive): dsinθ= mλ, m = 0, 1, 2, …

Dark fringes (destructive): dsinθ= (m + 12)λ

Thin-film interference: Light reflects from both the top and bottom surfaces of a thin film. The two reflections interfere. This produces the rainbow colors in soap bubbles, oil slicks, and anti-reflective coatings on glasses.

Worked Example 3: Double-Slit Fringe Spacing

Problem: In a Young's double-slit experiment, the slit separation is d = 0.20 mm and the screen is L = 2.0 m from the slits. Green light of wavelength λ= 550 nm illuminates the slits. Find: (a) the angular position of the first-order (m = 1) bright fringe, and (b) the spacing between adjacent bright fringes on the screen.

Solution:

(a) Bright fringe condition: dsinθ= mλ sinθ₁ = 1 × 550 × 10^-90.20 × 10^-3 = 5.5 × 10^-72.0 × 10^-4 = 2.75 × 10^-3 θ₁ ≈ 0.00275 rad ≈ 0.158^°

(b) For small angles, sinθ ≈ tanθ ≈ y/L, where y is the distance from the central maximum on the screen. The fringe spacing Δy is the distance between adjacent bright fringes: y_m = mλLd Δy = y_m+1 - y_m = λLd = 550 × 10^-9 × 2.00.20 × 10^-3 = 1.10 × 10^-62.0 × 10^-4 = 5.5 × 10^-3 m = 5.5 mm

Each bright fringe is separated from its neighbors by 5.5 mm on the screen. The central maximum (m = 0) sits at y = 0; the first-order bright fringe is at y = 5.5 mm; the second-order at y = 11.0 mm.

Worked Example 4: Thin-Film Interference

Problem: A soap bubble film (n_film = 1.33) surrounded by air (n_air = 1.00) has thickness t = 300 nm. White light strikes the film at near-normal incidence. For visible light reflected from the bubble, which wavelengths undergo constructive interference? Visible range: 400–700 nm.

Solution: At near-normal incidence, the path difference between the two reflected rays (top surface and bottom surface) is approximately 2t. Two key effects determine the phase:

1. Phase change on reflection: When light reflects off a medium of higher n, it undergoes a 180° (half-wavelength) phase shift. The top reflection (air → film, 1.00 → 1.33) gains a half-wavelength shift. The bottom reflection (film → air, 1.33 → 1.00) has no phase shift (going to lower n).

1. Path difference inside the film: 2t.

The net phase difference for constructive interference accounts for the half-wavelength shift: 2t = (m + 12)λn_film   →  2 n_film t = (m + 12)λ

Solve for λ: λ= 2 n_film tm + 12 = 2 × 1.33 × 300m + 0.5 = 798m + 0.5 nm

Try integer values of m:

• m = 0: λ= 798 / 0.5 = 1596 nm — infrared, not visible.
• m = 1: λ= 798 / 1.5 = 532 nm — green, visible.
• m = 2: λ= 798 / 2.5 = 319 nm — ultraviolet, not visible.

Only λ ≈ 532 nm (green light) constructively interferes in the visible range. The reflected bubble appears green — and as the film thins, the enhanced color shifts, producing the shimmering rainbow effect we see in soap bubbles.

ELI-10: Explain It Like I'm 10

Shine a laser through two tiny slits and you do not see two bright lines on the screen — you see a whole pattern of bright and dark stripes. The light waves from each slit travel different distances to each point on the screen. Where the waves arrive in sync (crest meets crest), they add up to bright. Where they arrive out of sync (crest meets trough), they cancel to dark. This interference pattern could only happen if light is a wave. Soap bubbles shimmer with rainbow colors because light bouncing off the front and back of the thin soap film interferes, reinforcing some colors and canceling others.

15.5 Diffraction

Core Ideas

Waves bend around obstacles and spread through openings. Significant diffraction occurs when the aperture size is comparable to the wavelength.

Single-slit diffraction minima: asinθ= mλ, m = 1, 2, 3, … (where a is slit width).

Diffraction grating: Many parallel slits produce sharp, well-separated bright fringes. Used in spectroscopy to analyze light by wavelength.

Key insight: Smaller aperture → wider diffraction pattern. This limits the resolution of microscopes and telescopes. The blurring of distant point sources (like stars) is due to diffraction at the telescope aperture.

Worked Example 5: Single-Slit Diffraction

Problem: Red light (λ= 650 nm) passes through a single slit of width a = 0.050 mm and falls on a screen L = 1.5 m away. Find: (a) the angular position of the first minimum (m = 1), and (b) the width of the central bright fringe on the screen.

Solution:

(a) Single-slit minima: asinθ= mλ sinθ₁ = 1 × 650 × 10^-90.050 × 10^-3 = 6.5 × 10^-75.0 × 10^-5 = 0.013 θ₁ ≈ 0.013 rad ≈ 0.74^°

(b) The central bright fringe extends from the first minimum on one side (m = 1) to the first minimum on the other side (m = -1). For small angles, the distance on the screen from the center to the first minimum is: y₁ = L tanθ₁ ≈ Lθ₁ = 1.5 × 0.013 = 0.0195 m = 19.5 mm

The total width of the central maximum: Δy_central = 2y₁ = 2 × 19.5 = 39.0 mm

Compare this to a slit twice as wide (a = 0.10 mm): sinθ₁ = 0.0065, so y₁ = 9.75 mm and the central fringe shrinks to 19.5 mm. Smaller slit → wider central maximum. This is the fundamental trade-off that limits resolution: narrowing the slit increases the field of view but reduces the ability to distinguish closely spaced sources.

ELI-10: Explain It Like I'm 10

Light does not always go straight. When it passes through a very small hole, it fans out in all directions — just like water waves spreading after passing through a gap in a wall. The smaller the hole, the more the light spreads. This is why telescopes need large mirrors — to minimize this spreading (diffraction) and see fine detail. A CD or DVD acts like a diffraction grating: the microscopic grooves split white light into its rainbow colors.

15.6 Polarization

Core Ideas

Polarization describes the orientation of the electric field in an EM wave. Unpolarized light (like sunlight) has random orientation.

Linear polarization: the electric field oscillates in a single plane.

Malus's law (intensity after a polarizer): I = I₀cos²θ, where θ is the angle between the transmission axis and the incident polarization.

Brewster's angle: At a specific incident angle, reflected light is completely polarized parallel to the surface. tanθ_B = n₂/n₁.

Polarization by scattering: Blue sky is partially polarized — Rayleigh scattering selects specific polarization directions.

Worked Example 6: Malus's Law

Problem: Unpolarized light of intensity I₀ = 100 W/m² passes through two polarizing filters. The first filter's transmission axis is vertical. The second filter's transmission axis is rotated 60° from the vertical. What is the intensity of light emerging from the second filter?

Solution:

Step 1 — First polarizer: Unpolarized light passing through an ideal polarizer has its intensity halved (only the component parallel to the transmission axis passes): I₁ = I₀2 = 1002 = 50 W/m² The light after the first polarizer is vertically polarized.

Step 2 — Second polarizer (Malus's law): The angle between the vertically polarized light and the second filter's axis is θ= 60^°: I₂ = I₁ cos²(60^°) = 50 × (0.5)² = 50 × 0.25 = 12.5 W/m²

Follow-up: If the two filters are crossed at 90°, the intensity drops to zero (I₂ = 50 cos² 90^°= 0). Inserting a third polarizer at 45° between them restores some light: I_{after 2nd} = 50 cos² 45^°= 25 W/m², I_{after 3rd} = 25 cos² 45^°= 12.5 W/m² This is a classic quantum-mechanics-like surprise: three filters transmit more light than two. It demonstrates that polarization is not a simple "blocking" phenomenon but a projection of the electric field onto successive axes.

ELI-10: Explain It Like I'm 10

Light is a transverse wave — it wiggles sideways to its direction of travel. Most light wiggles in all sideways directions randomly. A polarizing filter (like good sunglasses) acts like a picket fence — it only lets through light wiggling in one direction. Rotate a second filter and you can completely block all light. Polarized sunglasses reduce glare because light reflected off water or roads is horizontally polarized. Some animals (bees, octopuses) can see polarization patterns in the sky that humans cannot.

15.7 Common Misconceptions

Optics has several persistent misconceptions. Here are the most common and their corrections:

1. "Light always travels in straight lines." Not always. Light travels in straight lines in a homogeneous medium — but it bends at interfaces (refraction) and spreads around obstacles (diffraction). The ray model is an approximation that breaks down at small scales.

1. "A larger aperture lets in more light, so it should give better resolution." Larger apertures do collect more light, but they also improve resolution by reducing diffraction. Resolution and light-gathering are linked through diffraction physics.

1. "Converging lenses always produce magnified images." Only when the object is inside the focal point (like a magnifying glass). When the object is beyond 2F, a converging lens produces a smaller image — as in a camera.

1. "Virtual images are 'not real' and cannot be photographed." Virtual images are real optical phenomena — your eye (or a camera) can focus them because the diverging rays behave as if they originated from the virtual image location. You photograph yourself in a mirror every day.

1. "Covering half a lens blocks half the image." Covering half a lens reduces brightness (fewer rays contribute) but the entire image still forms because rays from every point on the object pass through every part of the lens. The image gets dimmer, not cropped.

1. "Interference and diffraction are different phenomena." They are manifestations of the same wave superposition principle. Diffraction is interference of wavelets from the same wavefront (Huygens' principle); two-slit interference is superposition of waves from separate sources. Both use path difference arguments.

1. "Polarized light is a special kind of light." All light is polarized at each instant — but most light sources emit waves with randomly and rapidly changing polarization directions, making it unpolarized on average. A polarizer simply selects one orientation.

1. "The critical angle means no light at all crosses the boundary." Total internal reflection is total in the geometric optics limit. In reality, an evanescent wave penetrates a short distance (∼wavelength) into the lower-index medium. This enables near-field optical effects and frustrated total internal reflection.

Topic Summary

• Reflection: θ_i = θ_r. Refraction: Snell's law n₁sinθ₁ = n₂sinθ₂.
• Total internal reflection when n₁ > n₂ and θ> θ_c. Basis of fiber optics.
• Mirrors and lenses: 1/f = 1/d_o + 1/d_i, m = -d_i/d_o. Ray tracing determines image properties.
• Interference: double-slit proves wave nature. Constructive: dsinθ= mλ. Thin films.
• Diffraction: single-slit minima at asinθ= mλ. Smaller aperture = more spreading.
• Polarization: Malus's law. Brewster's angle. Polarized sunglasses.

Essential Equations

Concept Check

1. Why does a pool appear shallower than it actually is?
2. A converging lens produces a real, inverted image. If you cover the top half of the lens, what happens to the image?
3. Why do oil slicks on water appear colorful?
4. A single slit is narrowed. Does the central bright fringe get wider or narrower? Why?
5. Two polarizers are oriented at 90° to each other, transmitting zero light. A third polarizer is inserted between them at 45°. Does any light get through? Explain.
6. Light in glass (n = 1.50) strikes the glass–air interface at 45°. Does it escape? (Hint: compare to θ_c.)
7. A double-slit experiment uses blue light (λ= 450 nm) instead of red light (λ= 650 nm). Does the fringe spacing increase, decrease, or stay the same? Why?
8. A thin film of oil (n = 1.45) on water (n = 1.33) appears dark at a particular location for a given wavelength. What is the minimum thickness that produces this destructive interference?

Open Educational References

• OpenStax, College Physics, Chapters 25–27: Geometric Optics, Wave Optics
• OpenStax, University Physics, Volume 3, Chapters 1–4: Optics