Geometric Optics: Reflection, Refraction, and Lenses

Geometric Optics

Introduction

Geometric optics is the study of light propagation in terms of rays. It explains how light interacts with surfaces, including reflection, refraction, and the formation of images by mirrors and lenses. This chapter focuses on the behavior of light at plane and spherical surfaces, as well as thin lenses.

Reflections at a Plane Surface

Concept of Image Formation

• Image: The reproduction of an object formed by light rays coming from the object and interacting with a surface (mirror or lens).

• The distance to the object is denoted by s, and the distance to the image is denoted by s'.

Types of Images

• Real Image:

  • Formed when actual light rays converge and meet at a point.

  • Can be projected onto a screen (light physically exists at the image location).

  • Typically inverted (upside down).

  • Example: Image formed on a movie screen by a projector.

• Virtual Image:

  • Formed when light rays appear to diverge from a point, but never actually meet.

  • Cannot be projected because no light exists at the apparent image location.

  • Usually upright.

  • Example: Your reflection in a flat mirror.

Classification Table: Image Types

Situation	Image Type
Flat mirror	Virtual
Camera lens	Real
Projector	Real
Magnifying mirror (close)	Virtual
Concave mirror (far)	Real

Magnification

Definition and Formula

• Magnification (m): The ratio of the height of the image (y') to the height of the object (y), or equivalently, the ratio of image distance to object distance (with a negative sign):

$m = \frac{y'}{y} = -\frac{s'}{s}$

• If m > 0, the image is upright.

• If m < 0, the image is inverted.

Spherical Mirrors

Reflections from Spherical Mirrors

• Depend on the radius of curvature (R) of the mirror.

• For a spherical mirror, the focal length f is related to the radius of curvature by:

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

• The mirror equation relates object distance (s), image distance (s'), and focal length (f):

$\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$

Example Problem

• A 3.0 cm tall statue is 24 cm in front of a concave mirror with a radius of curvature of 20 cm. Determine if the image is real or virtual, and if it is upright or inverted.

Solution steps:

• Calculate focal length: $f = 10$ cm.

• Use the mirror equation to solve for $s'$ and determine image properties.

Thin Lenses

Definition and Types

• A thin lens is a system with two refracting surfaces, typically spherical, close enough that the thickness can be ignored.

• Converging (Convex) Lens: Brings parallel rays to a focus.

• Diverging (Concave) Lens: Causes parallel rays to spread out as if from a focal point.

Focal Points

• First focal point (F1): Point from which rays emerge parallel to the axis after passing through the lens.

• Second focal point (F2): Point at which rays parallel to the axis converge after passing through the lens.

• For a converging lens, $f$ is positive; for a diverging lens, $f$ is negative.

The Thin Lens Equation

• Relates object distance (s), image distance (s'), and focal length (f):

$\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$

• The lens maker's equation relates the focal length to the radii of curvature and the index of refraction (n):

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

• Where $R_1$ and $R_2$ are the radii of curvature of the lens surfaces.

Examples of Thin Lens Analysis

• Object outside focal point: Real, inverted image.

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

Optical Fibers and Total Internal Reflection

Principle of Operation

• Light is trapped inside the core of an optical fiber by total internal reflection at the core/cladding boundary.

• The critical angle for total internal reflection is given by:

$\theta_c = \arcsin\left(\frac{n_b}{n_a}\right)$

• Where $n_a$ is the index of refraction of the core, and $n_b$ is that of the cladding ($n_a > n_b$).

• Light rays entering at angles greater than the critical angle are totally internally reflected and guided along the fiber.

Example Calculation

• Given $n_a = 1.46$ (core), $n_b = 1.44$ (cladding):

$\theta_c = \arcsin\left(\frac{1.44}{1.46}\right) \approx 80.6^\circ$

• This is the largest angle with respect to the normal for which total internal reflection occurs.

Summary Table: Key Equations

Concept	Equation
Magnification	$m = \frac{y'}{y} = -\frac{s'}{s}$
Mirror Equation	$\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$
Focal Length (Mirror)	$f = \frac{R}{2}$
Thin Lens Equation	$\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$
Lens Maker's Equation	$\frac{1}{f} = (n-1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)$
Critical Angle (TIR)	$\theta_c = \arcsin\left(\frac{n_b}{n_a}\right)$

Important Concepts

• Concave spherical mirrors: Image formation, focal length, and magnification.

• Thin lenses: Converging and diverging lenses, graphical analysis, and equations.

• Total internal reflection: Application in optical fibers.

Additional info: These notes are based on lecture slides for Chapter 24 (Geometric Optics) from a College Physics course, covering the fundamentals of image formation, mirrors, lenses, and optical fibers.