Geometrical Optics

Introduction

Geometrical optics is the branch of physics that deals with the behavior of light as it interacts with mirrors, lenses, and other optical devices. This chapter covers the fundamental principles of reflection, refraction, image formation, and dispersion, providing a foundation for understanding optical phenomena.

The Reflection of Light

Wave Fronts and Rays

- Wave fronts are surfaces along which the phase of the wave is constant. - Rays are lines perpendicular to the wave fronts, indicating the direction of wave propagation. - As distance from a point source increases, wave fronts become flatter, resembling plane waves.

Law of Reflection

- The law of reflection states that the angle of incidence equals the angle of reflection. - - Specular reflection occurs on smooth surfaces, producing clear images. - Diffuse reflection occurs on rough surfaces, scattering light in many directions.

Forming Images with a Plane Mirror

Image Formation

- Light reflected from an object obeys the law of reflection and enters the observer's eye. - The eye perceives the image as being behind the mirror, along the straight-line path of the reflected ray.

Properties of Plane Mirror Images

- Mirror images are upright but reversed left to right. - The image appears the same distance behind the mirror as the object is in front. - The image is the same size as the object.

Corner Reflectors

- A corner reflector reflects light parallel to the incident ray, regardless of the angle of incidence.

Spherical Mirrors

Types of Spherical Mirrors

- Concave mirrors are curved inward (like the inside of a sphere). - Convex mirrors are curved outward (like the outside of a sphere).

Key Features

- Spherical mirrors have a central axis and a center of curvature. - Parallel rays striking a concave mirror converge at the focal point; for convex mirrors, rays appear to diverge from the focal point.

Focal Length

- For a convex mirror, the focal length is negative; for a concave mirror, it is positive. - (R = radius of curvature) - Spherical aberration occurs when rays far from the principal axis blur the image; parabolic mirrors can correct this.

Ray Tracing and the Mirror Equation

Principal Rays for Mirrors

- Parallel ray (P ray): Reflects through the focal point. - Focal ray (F ray): Reflects parallel to the axis. - Center-of-curvature ray (C ray): Reflects back along its incoming path.

Mirror Equation

- The mirror equation relates object distance (), image distance (), and focal length (): -

Magnification

- Magnification () describes the ratio of image height to object height: -

Sign Conventions

The Refraction of Light

Refraction and Index of Refraction

- Light changes speed when moving between media, causing it to bend (refract). - The index of refraction () is defined as: - - = speed of light in vacuum, = speed of light in the medium.

Snell's Law

- Snell's Law relates the angles and indices of refraction: -

Properties of Refraction

- Light entering a medium with higher bends toward the normal. - Light entering a medium with lower $n$ bends away from the normal. - No bending occurs if $n$ does not change. - If light travels along the normal, its direction is unchanged.

Total Internal Reflection and Critical Angle

- When light moves from higher to lower , it can be totally reflected if the angle of incidence exceeds the critical angle. - Total internal reflection is used in optical fibers and binoculars.

Brewster's Angle

- At Brewster's angle, reflected light is completely polarized. - Brewster's angle () is given by: -

Ray Tracing for Lenses

Types of Lenses

- Double convex (converging) lenses focus light to a point. - Double concave (diverging) lenses spread light apart.

Principal Rays for Lenses

- Parallel ray (P ray): Approaches the lens parallel to its axis. - Focal ray (F ray): Drawn toward or through the focal point. - Midpoint ray (M ray): Passes through the center of the lens without deflection (thin-lens approximation).

The Thin-Lens Equation

Thin-Lens Equation and Magnification

- The thin-lens equation is: - - Magnification: -

Sign Conventions for Lenses

Dispersion and the Rainbow

Dispersion

- The index of refraction varies with the frequency of light; higher frequency light has a higher index. - Dispersion is the spreading of light into its component colors due to variation in refractive index.

Rainbows

- Rainbows are formed by the dispersion of light as it refracts and reflects inside raindrops. - All colors arrive at the observer's eye, sometimes producing a secondary arc.

Summary Table: Key Equations and Concepts

Additional info:

- Spherical aberration is a common issue in spherical mirrors and lenses, corrected by using parabolic shapes. - Total internal reflection is critical in fiber optics and advanced optical devices. - Dispersion explains the formation of rainbows and other color phenomena in nature.