Chapter 23: Light – Geometric Optics

Chapter 23: Light – Geometric Optics

Overview

This chapter introduces the fundamental principles of geometric optics, focusing on the behavior of light as rays. It covers the formation of images by mirrors and lenses, the laws of reflection and refraction, and applications such as fiber optics.

The Ray Model of Light

Introduction to the Ray Model

Light can often be modeled as traveling in straight lines, called rays. This idealization is the foundation of geometric optics and is useful for analyzing how light interacts with mirrors, lenses, and other optical devices.

• Ray: A straight line that represents the path along which light energy travels.

• Bundle of Rays: Multiple rays can be used to represent the spread of light from a point source.

• Application: The ray model is used to predict the formation and properties of images in optical systems.

Reflection; Image Formation by a Plane Mirror

Law of Reflection

When a narrow beam of light strikes a flat (plane) surface, it is reflected according to the law of reflection.

• Angle of Incidence (\(\theta_i\)): The angle between the incident ray and the normal (perpendicular) to the surface.

• Angle of Reflection (\(\theta_r\)): The angle between the reflected ray and the normal.

• Law of Reflection: The angle of reflection equals the angle of incidence.

$\theta_r = \theta_i$

• The incident ray, the reflected ray, and the normal all lie in the same plane.

Types of Reflection

• Specular Reflection: Occurs on smooth surfaces (like mirrors), where parallel incident rays remain parallel after reflection.

• Diffuse Reflection: Occurs on rough surfaces, where the law of reflection still holds locally, but the reflected rays scatter in many directions due to varying surface angles.

Image Formation by a Plane Mirror

• A plane mirror forms a virtual image that appears to be behind the mirror.

• The image is upright and the same size as the object.

• Every point on the object appears to be the same distance behind the mirror as it is in front.

• To see your full reflection, a mirror only needs to be half your height.

Formation of Images by Spherical Mirrors

Types of Spherical Mirrors

• Concave Mirror: Reflective surface is on the inner side of a sphere.

• Convex Mirror: Reflective surface is on the outer side of a sphere.

Focal Length and Center of Curvature

• Center of Curvature (C): The center of the sphere from which the mirror segment is taken.

• Focal Point (F): The point where parallel rays converge (concave) or appear to diverge from (convex).

• Focal Length (f): The distance from the mirror to the focal point.

$f = \frac{r}{2}$

• Spherical Aberration: When the curvature is large, not all rays converge at the same point, causing a blurred image. Parabolic mirrors can correct this.

Ray Diagrams for Spherical Mirrors

• Three principal rays are used:

  1. Ray parallel to the axis reflects through the focal point.

  2. Ray through the focal point reflects parallel to the axis.

  3. Ray through the center of curvature reflects back on itself.

• The intersection of reflected rays locates the image.

Mirror Equation and Magnification

• The relationship between object distance (\(d_o\)), image distance (\(d_i\)), and focal length (\(f\)) is:

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

• Magnification (m): The ratio of image height (\(h_i\)) to object height (\(h_o\)):

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

• Negative magnification indicates an inverted image.

Image Properties for Spherical Mirrors

• Concave Mirror:

  • Object outside center of curvature: image is real, inverted, and smaller.

  • Object between center and focal point: image is real, inverted, and larger.

  • Object inside focal point: image is virtual, upright, and larger.

• Convex Mirror: Image is always virtual, upright, and smaller than the object.

Sign Conventions

• Distances are positive if measured from the mirror along the direction of incoming light; negative otherwise.

• Image height is positive if upright, negative if inverted.

Index of Refraction

Definition

The index of refraction (n) of a medium is the ratio of the speed of light in vacuum (c) to the speed of light in the medium (v):

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

• Light slows down when entering a medium with a higher index of refraction.

Table: Indices of Refraction for Common Materials

Material	Index of Refraction (n)
Air (at STP)	1.00
Water	1.33
Ethyl alcohol	1.36
Glass (typical)	1.5
Diamond	2.42
Plastic (Acrylic)	1.49
"High index" glass	1.7
Sodium chloride	1.54

Refraction: Snell’s Law

Law of Refraction

When light passes from one medium to another, it changes direction—a phenomenon called refraction. The relationship between the angles and indices of refraction is given by Snell’s Law:

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

• \(\theta_1\): Angle of incidence (in medium 1)

• \(\theta_2\): Angle of refraction (in medium 2)

• Light bends toward the normal when entering a medium with higher n; away from the normal when entering a medium with lower n.

Applications and Examples

• Refraction explains why objects partially submerged in water appear bent or displaced.

• When light passes through parallel-sided glass, the emerging ray is parallel to the incident ray but displaced sideways.

Total Internal Reflection; Fiber Optics

Critical Angle and Total Internal Reflection

• When light passes from a medium with higher n to lower n, there is a critical angle (\(\theta_c\)) above which all light is reflected back into the medium.

$\sin \theta_c = \frac{n_2}{n_1}$

• Total Internal Reflection: Occurs when the angle of incidence exceeds the critical angle; no transmission occurs.

• Used in fiber optics to transmit light over long distances with minimal loss.

Thin Lenses; Ray Tracing

Types of Thin Lenses

• Converging (Convex) Lens: Thicker at the center; brings parallel rays to a focus.

• Diverging (Concave) Lens: Thicker at the edges; causes parallel rays to diverge as if from a focal point.

Lens Power

• Power (P): The inverse of focal length (in meters):

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

• Measured in diopters (D), where 1 D = 1 m-1.

Ray Tracing for Lenses

• Three principal rays:

  1. Ray parallel to axis passes through (or appears to come from) the focal point after refraction.

  2. Ray through the focal point emerges parallel to the axis.

  3. Ray through the center of the lens continues undeviated.

• For diverging lenses, the image is always upright and virtual.

The Thin Lens Equation

Equation and Sign Conventions

• The thin lens equation relates object distance (\(d_o\)), image distance (\(d_i\)), and focal length (\(f\)):

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

• Focal length is positive for converging lenses, negative for diverging lenses.

• Object distance is positive if the object is on the side from which light enters the lens.

• Image distance is positive if the image is on the opposite side from the object (real image), negative if on the same side (virtual image).

• Image height is positive if upright, negative if inverted.

Magnification for Lenses

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

• Positive magnification: image is upright.

• Negative magnification: image is inverted.

Combinations of Lenses

Multiple Lens Systems

• The image formed by the first lens acts as the object for the second lens.

• Object distances for subsequent lenses may be negative, depending on the system's geometry.

Lensmaker’s Equation

Relating Lens Shape and Material to Focal Length

The lensmaker’s equation relates the focal length of a lens to the radii of curvature of its surfaces and the index of refraction of the lens material:

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

• \(n\): Index of refraction of the lens material

• \(R_1, R_2\): Radii of curvature of the two lens surfaces (signs depend on convention)

Summary Table: Key Equations in Geometric Optics

Concept	Equation
Law of Reflection	$\theta_r = \theta_i$
Index of Refraction	$n = \frac{c}{v}$
Snell’s Law	$n_1 \sin \theta_1 = n_2 \sin \theta_2$
Critical Angle	$\sin \theta_c = \frac{n_2}{n_1}$
Mirror/Lens Equation	$\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}$
Magnification	$m = -\frac{d_i}{d_o}$
Lens Power	$P = \frac{1}{f}$
Lensmaker’s Equation	$\frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)$

Additional info: Some context and definitions have been expanded for clarity and completeness, and tables have been reconstructed for study purposes.