Optics: Reflection, Refraction, and Wave Phenomena

Chapter 22: Reflection and Refraction of Light

Corpuscular Theory of Light

Definition: In optics, the corpuscular theory of light states that light is made up of small discrete particles called "corpuscles" (little particles).
Properties of Corpuscles: * They travel in a straight line. * They possess a finite velocity. * They possess momentum.
Behavior: * Light moves in straight lines. * Light does not bend into the shadow.

The Ray Model of Light

Concept: Light very often travels in straight lines. We represent light using rays, which are straight lines emanating from an object.
Idealization: While the ray model is an idealization, it is extremely useful for geometric optics.
Pinhole Camera Example: * Light passes through a pinhole to form an image on the opposite side of a box. * The resulting image is always inverted. * This sketch demonstrates the practical application of the ray model.

The Nature and Study of Light

Dual Nature of Light: Light (electromagnetic radiation) can be described in two ways: the Wave Model and the Particle Model.
Wave Model: * Wavelengths of radiation vary significantly. * Radio waves: Can measure up to several kilometers long. * Gamma ray waves: Are less than a billionth of a centimeter long. * White light: Consists of several wavelengths corresponding to the colors of the rainbow.
Wave Relationships: * $\text{speed} = \text{wavelength} \times \text{frequency}$

The Photon Model of Light

Postulates: 1. Light consists of discrete, massless units called photons. A photon travels in a vacuum at the speed of light, which is exactly $3.00 \times 10^{8}\,m/s$ . 2. Each photon possesses energy defined by the equation: $E = hf$ , where $f$ is the frequency of the light and $h$ is a universal constant called Planck’s constant. 3. The value of Planck’s constant is: $h = 6.63 \times 10^{-34}\,J \cdot s$ . 4. The superposition of a sufficiently large number of photons displays the characteristics of a classical light wave.
Wave-Particle Duality: In some experiments light acts as a wave and in others it acts as a particle. Waves and particles are interpretations of light behavior.
Wave Packets: Photons are sometimes visualized as wave packets. These are electromagnetic waves with specific wavelengths and frequencies that remain discrete and fairly localized.

Stop to Think! (Photon Energy and Planck Units)

Question 1: The higher the photon energy, ________. * Answer: The shorter its wavelength ( $E = \frac{hc}{\lambda}$ ).
Question 2: Does a photon of red light have more energy or less energy than a photon of blue light? * Answer: Less energy (red has a longer wavelength/lower frequency).
Question 3: What is the energy of a photon with a wavelength of $400\,nm$ ? * Calculation: $E = \frac{hc}{\lambda} = \frac{(6.63 \times 10^{-34}\,J \cdot s) \times (3.00 \times 10^{8}\,m/s)}{400 \times 10^{-9}\,m} \approx 5 \times 10^{-19}\,J$ . * Correct Choice: B ( $5 \times 10^{-19}\,J$ ).
Planck Constant in eV $\cdot$ s: * Value in Joule-seconds: $h = 6.626 \times 10^{-34}\,J \cdot s$ . * Conversion factor: $1\,eV = 1.602 \times 10^{-19}\,J$ . * Calculation: $\frac{6.626 \times 10^{-34}\,J \cdot s}{1.602 \times 10^{-19}\,J} = 4.136 \times 10^{-15}\,eV \cdot s$ .

Law of Reflection

The states of the law: 1. The incident ray and the reflected ray are in the same plane normal to the surface. 2. The angle of reflection equals the angle of incidence: $\theta_{r} = \theta_{i}$ .
Fermat’s Principle of Least Time: Light travels between two points along the path that requires the smallest transit time. Since speed is constant in a single medium, the minimum time path is the minimum distance path.
Types of Reflection: * Specular Reflection: Reflection from a smooth surface (like a mirror) where the eye must be in a specific position to see the image. * Diffuse Reflection: Reflection from a rough surface; the law of reflection holds at every point, but the angle of incidence varies across the surface, causing the eye to see reflected light at all angles.

Image Formation by a Plane Mirror

Virtual Image: The image appears to be behind the mirror. It is called virtual because light does not actually go through it.
Geometry: The distance of the image from the mirror ( $d_{i}$ ) is equal to the distance of the object from the mirror ( $d_{o}$ ). Additionally, object height ( $h$ ) equals image height ( $h'$ ).
Magnification ( $M$ ): For a flat mirror, $M = \frac{h'}{h} = 1$ .

Stop to Think! (Mirror Applications)

The word "ME": When printed on a transparent sheet and held to a mirror, the image will appear laterally inverted.
Right-Angle Mirrors: How many images of a ball can you see in two mirrors perpendicular to each other? * Answer: 3 images.
Perpendicular Reflection: A ray reflected from two perpendicular mirrors will end up traveling in the exact opposite direction ( $180^{\circ}$ ) relative to its original direction.
Minimum Mirror Height: To see one's full length, a person of height $h_{o}$ needs a mirror with a length of at least $\frac{1}{2}h_{o}$ .
Example Problem: A person eyes are $H = 1.57\,m$ above the floor, stands $2.20\,m$ in front of a mirror with a bottom edge $38\,cm$ above the floor. Find the horizontal distance $x$ to the nearest point on the floor seen in the reflection. * Answer: $70.3\,cm$ .

Refraction and the Index of Refraction

Definition: Refraction is the bending of light as it crosses the interface between two different transparent media.
Speed Change: When light travels from a vacuum into a transparent material, its speed decreases, but its frequency ( $f$ ) remains constant.
Index of Refraction ( $n$ ): The ratio of the speed of light in a vacuum ( $c$ ) to the speed of light in the medium ( $v$ ): * $n = \frac{c}{v}$ * Note: Since $c$ is the maximum speed, $n$ is always $\geq 1$ .
Wavelength Change: Since $v = f\lambda$ and frequency is constant, the wavelength decreases in a medium: * $\lambda_{n} = \frac{\lambda}{n}$
Example Calculation: A light pulse takes $2.43\,ns$ to travel $0.500\,m$ . Find $n$ . * $v = \frac{0.500\,m}{2.43 \times 10^{-9}\,s} \approx 2.057 \times 10^{8}\,m/s$ . * $n = \frac{3.00 \times 10^{8}}{2.057 \times 10^{8}} = 1.46$ .

Snell's Law

Formula: $n_{1} \sin(\theta_{1}) = n_{2} \sin(\theta_{2})$ .
Problem: Ray from water ( $n = 1.33$ ) to glass ( $n = 1.52$ ) at an incident angle of $60^{\circ}$ . Find the refracted angle. * $1.33 \sin(60^{\circ}) = 1.52 \sin(\theta_{b})$ * $1.33(0.866) = 1.52 \sin(\theta_{b})$ * $\theta_{b} = 49.3^{\circ}$ .
Plane Slab Property: A light ray striking a glass slab with parallel faces submerged in air emerges propagating parallel to its original direction ( $\theta_{1} = \theta_{4}$ ).

Dispersion of Light

Definition: Dispersion is the slight variation of the index of refraction ( $n$ ) with wavelength (color).
Spectrum: White light passing through a prism disperses into a rainbow spectrum.
Wavelength Relationship: $n$ is larger for shorter wavelengths. Consequently, violet light ( $400\,nm$ ) refracts more than red light ( $700\,nm$ ).
Visible Spectrum Table (General Ranges): * Violet: $\approx 400\,nm$ * Red: $\approx 700\,nm$
Rainbows: Formed by sunlight dispersed by water droplets. No two people see the same rainbow because it depends on the relative positions of the Sun, droplet, and observer.

Scattering of Light (Rayleigh Criteria)

Why is the Sky Blue? Scattering by air molecules depends on the wavelength. Blue light has a wavelength closer to the size of air molecules and is scattered most strongly.
Proportionality: Molecular scattering is proportional to $\frac{1}{\lambda^{4}}$ .
Why are Sunsets Red? When the sun is near the horizon, light travels through more atmosphere. Blue light is scattered away, leaving only the red light to reach the observer.

Huygens’ Principle

Geometric Method (1678): Used to deduce the laws of reflection and refraction based on wave motion.
Construction: All points on a wavefront act as point sources for production of spherical secondary waves called wavelets. The new position of the wavefront is the surface tangent to these wavelets.

Total Internal Reflection (TIR)

Condition: Occurs when light travels from a medium with a higher index of refraction ( $n_{1}$ ) to one with a lower index ( $n_{2}$ ).
Critical Angle ( $\theta_{C}$ ): The incident angle at which the refracted ray moves at $90^{\circ}$ (parallel to the boundary). * $\sin(\theta_{C}) = \frac{n_{2}}{n_{1}}$
Occurrence: If the angle of incidence > \theta_{C}, no transmission occurs, and all light is reflected back into the first medium.
Examples: * Glass-to-air ( $n=1.50$ ): $\theta_{C} = \sin^{-1}(\frac{1.00}{1.50}) = 41.8^{\circ}$ . * Water-to-air ( $n=1.33$ ): $\theta_{C} = \sin^{-1}(\frac{1.00}{1.33}) = 48.8^{\circ}$ .
Applications: * Fiber Optics: Light is transmitted along a fiber even if curved. * Binoculars: Use prisms for $100\%$ reflection. * Diamonds: Cut specifically to maximize TIR on back surfaces to create brilliance.

Chapter 23: Mirrors and Lenses

Spherical Mirrors

Construction: Shaped like a section of a sphere. * Concave: Reflective on the inside. * Convex: Reflective on the outside.
Paraxial Approximation: Rays from faraway objects are effectively parallel to the mirror axis.
Focal Point ( $F$ ) and Focal Length ( $f$ ): * For a spherical mirror: $f = \frac{R}{2}$ . * Concave Mirror: Real focus (same side as incident light). * Convex Mirror: Virtual focus (opposite side of the mirror).
Spherical Aberration: Occurs when parallel rays striking a large-curvature spherical mirror do not converge at a single point. * Solution: Use a parabolic reflector to ensure all parallel rays focus at one point (used in research telescopes and flashlights).

Ray Tracing for Mirrors

Ray 1: Parallel to the axis; reflects through the focal point.
Ray 2: Through the focal point; reflects parallel to the axis.
Ray 3: Incident on the center of the mirror; reflects symmetrically ( $\theta_{i} = \theta_{r}$ ).
Ray 4: Perpendicular to the mirror; reflects back on itself.

Mirror Image Characteristics

Concave Mirrors: * Object outside Center of Curvature ( $C$ ): Image is inverted, smaller, and real. * Object at $C$ : Image is at $C$ , same size, inverted, and real. * Object inside $F$ : Image is upright, larger, and virtual.
Convex Mirrors: Image is always virtual, upright, and smaller, regardless of object distance.

The Mirror Equation and Magnification

Mirror Equation: $\frac{1}{d_{o}} + \frac{1}{d_{i}} = \frac{1}{f}$
Magnification ( $M$ ): $M = \frac{h'}{h} = -\frac{d_{i}}{d_{o}}$
Sign Conventions for Mirrors: * $d_{i}, d_{o}$ are positive if in front of the mirror (reflective side). * $d_{i}, d_{o}$ are negative if behind the mirror. * $f$ is positive for concave, negative for convex. * $M$ is positive for upright images, negative for inverted.

Thin Lenses

Definition: Lenses whose thickness is small compared to their radii of curvature.
Converging Lenses: Thicker in the center; bring parallel rays to a real focus.
Diverging Lenses: Thicker at the edges; parallel rays seem to diverge from a virtual focus.
Lens Power ( $P$ ): The inverse of the focal length, measured in diopters ( $D$ ). * $P = \frac{1}{f}$ (where $f$ is in meters). * $1\,D = 1\,m^{-1}$ .

Ray Tracing for Lenses

Ray 1: Comes in parallel to the axis; exits through/from the focal point.
Ray 2: Goes through the center of the lens; remains undeflected.
Ray 3: Comes in through the focal point; exits parallel to the axis.

Thin Lens Equation and Magnification

The formula is identical to mirrors: $\frac{1}{d_{o}} + \frac{1}{d_{i}} = \frac{1}{f}$ and $M = -\frac{d_{i}}{d_{o}}$ .
Sign Conventions for Lenses: * $f$ is positive for converging, negative for diverging. * $d_{o}$ is positive on the incident light side. * $d_{i}$ is positive on the opposite side of the incident light (real image). * $h_{i}$ is positive if upright, negative if inverted.

Refraction at Curved Surfaces and Lensmaker's Equation

Lensmaker's Equation: Relates radii of curvature ( $R$ ) and index of refraction ( $n$ ) to focal length: * $\frac{1}{f} = (n - 1) \left( \frac{1}{R_{1}} - \frac{1}{R_{2}} \right)$
Sign for Radius: Convex refracting surface faces object: $R$ is positive. Concave surface faces object: $R$ is negative.

Chapter 24: Wave Optics

Superposition and Coherence

Superposition Principle: When two or more waves meet, the resulting displacement is the sum of individual displacements.
Coherence: Two waves are coherent if their phase relationship remains fixed over time. They must have the same frequency and wavelength.
Interference Types: * Constructive: Waves are in phase; waveforms reinforce ( $c = a + b$ ). * Destructive: Waves are $180^{\circ}$ out of phase; waveforms cancel.

Young’s Double-Slit Experiment

Purpose: Proves the wave nature of light via interference patterns.
Path Difference: $\Delta L = |r_{1} - r_{2}| \approx d \sin(\theta)$ .
Bright Fringes (Maxima): $d \sin(\theta) = m\lambda$ for $m = 0, 1, 2, \dots$
Dark Fringes (Minima): $d \sin(\theta) = (m + \frac{1}{2})\lambda$ for $m = 0, 1, 2, \dots$
Small Angle Approximation: For $y \ll L$ , $\sin(\theta) \approx \tan(\theta) = \frac{y}{L}$ .
Fringe Spacing ( $\Delta y$ ): $\Delta y = \frac{\lambda L}{d}$ . * Spacing is uniform and independent of order $m$ .

Change of Phase Due to Reflection

Rule 1: A $180^{\circ}$ phase shift occurs when light reflects from a medium with a higher index of refraction (n_{2} > n_{1}).
Rule 2: No phase shift occurs when light reflects from a medium with a lower index of refraction (n_{2} < n_{1}).
Transmitted Waves: Undergo no phase change when crossing a boundary.

Interference in Thin Films

Thin film interference causes the colors seen in soap bubbles or oil slicks.
Condition for Anti-Reflective Coating ( $n_{air} < n_{film} < n_{glass}$ ): * Both reflections (top and bottom) have a $180^{\circ}$ shift. * Constructive: $2t = m\lambda_{f}$ . * Destructive: $2t = (m + \frac{1}{2})\lambda_{f}$ .
Condition for Soap Bubbles/Oil on Water ( $n_{air} < n_{film} > n_{water}$ ): * Only top reflection has a $180^{\circ}$ shift. * Constructive: $2t = (m + \frac{1}{2})\lambda_{f}$ . * Destructive: $2t = m\lambda_{f}$ .

Newton’s Rings and Air-Wedges

Newton's Rings: Concentric circles formed by a curved glass on a flat glass surface.
Air Wedge: A wedge-shaped gap of air between two glass slides produces equally spaced parallel fringes. * Destructive (Dark): $2t = m\lambda$ . * Constructive (Bright): $2t = (m + \frac{1}{2})\lambda$ .

Single-Slit Diffraction

Concept: Bending of waves passing through an opening ( $a$ ).
Formula for Minima (Dark Fringes): $a \sin(\theta) = m\lambda$ for $m = \pm 1, \pm 2, \dots$ (Note: $m=0$ is the central maximum).
Central Maximum: As slit width $a$ decreases, the central maximum widens.

Diffraction Grating

Definition: Device with a large number ( $N$ ) of equally spaced slits.
Formula: $d \sin(\theta) = m\lambda$ ( $d$ is the slit separation, $d = \frac{1}{\text{lines per mm}}$ ).
Effect of Slits: As $N$ increases, maxima become brighter and much narrower.

Polarization

Definition: Light is polarized when electric fields oscillate in a single plane.
Malus’s Law: For polarized light of intensity $I_{o}$ passing through an analyzer at angle $\theta$ : * $I = I_{o} \cos^{2}(\theta)$ .
Polarizing Angle (Brewster’s Angle): The angle of incidence where reflected light is $100\%$ polarized. * $\tan(\theta_{p}) = n$
Scattering: Blue sky light is polarized because gas electrons absorb and reradiate light (Rayleigh scattering).
Optical Activity: Substances that rotate the plane of polarization of transmitted light are termed "optically active."

Chapter 25: Optical Instruments

The Camera

Basic Components: Lens, light-tight box, shutter, and sensor (Film, CCD, or CMOS).
Light Intensity ( $I$ ): Proportional to the square of lens diameter ( $D^{2}$ ) and inversely proportional to the square of focal length ( $f^{2}$ ).
f-number: $f\text{-number} = \frac{f}{D}$ . * Standard sequence: $1.4, 2, 2.8, 4, 5.6, 8, 11$ .

The Human Eye

Mechanism: Cornea does most refraction; lens adjusts focus (Accommodation).
Near Point: Closest clear focus distance ( $\approx 25\,cm$ for normal vision).
Far Point: Farthest clear focus distance (infinity for normal vision).
Vision Defects: * Nearsightedness (Myopia): Focus is in front of the retina. Corrected with a diverging lens. * Farsightedness (Hyperopia): Focus is behind the retina. Corrected with a converging lens. * Astigmatism: Different curvature radii in different directions; common in human eyes.

The Simple Magnifier

Mechanism: A converging lens used to view objects closer than the near point.
Angular Magnification ( $m$ ): $m = \frac{\theta}{\theta_{o}}$ . * Eye relaxed (image at infinity): $m = \frac{25\,cm}{f}$ . * Eye focused at near point: $m = \frac{25\,cm}{f} + 1$ .

Compound Microscope and Telescope

Microscope: Uses an objective lens (short f_{o} < 1\,cm) and an eyepiece ( $f_{e} \approx$ few cm). * Total Magnification: $M = M_{objective} \times m_{eyepiece} = -\frac{L \times 25\,cm}{f_{o} \times f_{e}}$ .
Refracting Telescope: Two lenses where the objective forms a real image at its focal point ( $f_{o}$ ), viewed by the eyepiece. * Angular Magnification: $m = -\frac{f_{o}}{f_{e}}$ . * Length of telescope: $L \approx f_{o} + f_{e}$ .

Resolution and the Rayleigh Criterion

Rayleigh Criterion: Two images are just resolvable when the center of one diffraction peak is over the first minimum of the other.
Circular Aperture Resolution ( $\theta_{min}$ ): * $\theta_{min} = 1.22 \frac{\lambda}{D}$ .

Michelson Interferometer

Function: Splits a light beam into two paths using a beam splitter, then recombines them to form interference fringes.
Application: Used for extremely accurate measurements of wavelengths and small distances ( $d_{1}$ vs $d_{2}$ ).

Questions & Discussion

Why are clouds white? Cloud droplets ( $\approx 10\,microns$ ) are much larger than visible light wavelengths. They scatter all colors equally (Mie scattering), resulting in white light.
Why is vision blurry underwater? The index of refraction of water is close to the cornea; light rays bend much less than in air. Goggles restore the air-cornea interface.
Why use Reflectors over Refractors? 1. No chromatic aberration (reflection is wavelength-independent). 2. Glass in lenses absorbs UV and Infrared; mirrors don't. 3. Lenses can be supported only at edges; mirrors can be supported across the back. 4. Lenses need two precise surfaces; mirrors only need one.