Geometric Optics: Mirrors and Lenses

Geometric Optics

Introduction to Geometric Optics

Geometric optics is the study of light propagation in terms of rays. It explains how devices such as mirrors and lenses form images by bending or reflecting light rays. Applications of geometric optics are found in daily life (mirrors, lenses, fiber optics) and in medicine (endoscopes, laser-based surgery).

• Reflection occurs when light bounces off a surface.

• Refraction is the bending of light as it passes from one medium to another.

• Mirrors and lenses are the primary tools for manipulating light in geometric optics.

This chapter covers:

• Reflection from plane and spherical surfaces

• Ray tracing and graphical methods for mirrors

• Refraction at spherical surfaces and thin lenses

• Graphical methods for lenses

Reflection at a Plane Surface

Image Formation by Plane Mirrors

When light rays diverge from an object point and strike a plane mirror, they reflect according to the law of reflection. The reflected rays appear to originate from a point behind the mirror, forming a virtual image.

• Object distance (s): Distance from the object to the mirror.

• Image distance (s'): Distance from the image to the mirror (behind the mirror for a plane mirror).

• Law of Reflection: Angle of incidence equals angle of reflection.

• Virtual Image: The image appears behind the mirror; rays do not actually pass through this point.

• Magnification (m): $m = \frac{y'}{y}$, where $y$ is object height and $y'$ is image height.

• For a plane mirror: $s = -s'$, $m = 1$ (image is same size and upright).

Key Properties:

• Image is always upright and laterally inverted (left-right reversal).

• Image distance equals object distance (behind the mirror).

Reflection at a Spherical Surface

Concave and Convex Mirrors

Spherical mirrors are segments of a sphere and can be either concave (reflective surface curves inward) or convex (reflective surface curves outward).

• Center of Curvature (C): Center of the sphere of which the mirror is a part.

• Vertex (V): Center of the mirror surface.

• Optical Axis: Line passing through C and V.

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

• Focal Length (f): $f = \frac{R}{2}$, where $R$ is the radius of curvature.

• Spherical Mirror Equation: $\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$ $f = \frac{R}{2}$

• Sign Conventions:

  • $R$ is positive for concave, negative for convex mirrors.

  • $s$ is positive for real objects, $s'$ is positive for real images (same side as object).

• Magnification: $m = \frac{y'}{y} = -\frac{s'}{s}$

Example: Calculating image position and magnification for a lamp in front of a concave mirror.

Convex Mirrors

Convex mirrors have their reflective surface bulging outward. The center of curvature is on the opposite side from the outgoing rays, so $R$ is negative. Images formed are always virtual, upright, and reduced in size.

Example: Calculating the image of Santa in a shiny Christmas ornament (convex mirror).

Graphical Methods for Mirrors

Principal Rays and Image Construction

Ray diagrams are used to locate images formed by mirrors. Four principal rays are commonly used, but two are sufficient for image construction:

• Ray parallel to axis reflects through focal point.

• Ray through focal point reflects parallel to axis.

• Ray through center of curvature reflects back on itself.

• Ray to vertex reflects symmetrically about the axis.

Image Properties:

• Concave mirrors can form real or virtual images depending on object position.

• Convex mirrors always form virtual images.

Example: Graphical construction of images for various object distances using a concave mirror.

Thin Lenses

Types of Lenses and Image Formation

A lens is an optical system with two refracting surfaces. Lenses are classified as converging (thicker at the center) or diverging (thicker at the edges).

• Converging Lens: Parallel rays converge to a real focal point after passing through the lens. Focal length is positive.

• Diverging Lens: Parallel rays diverge as if coming from a virtual focal point. Focal length is negative.

Thin Lens Equation:

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

• $n$ = index of refraction of lens material

• $R_1$, $R_2$ = radii of curvature of lens surfaces

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

Sign Conventions:

• $R_1$ is negative if center of curvature is toward incoming ray.

• $R_2$ is positive if center of curvature is toward outgoing ray.

Image Properties:

• Concave (diverging) lens: Image is always virtual.

• Convex (converging) lens: Image can be real or virtual depending on object position.

Graphical Methods: Three principal rays are used for image construction, but two are sufficient.

Applications: Lenses are used in eyeglasses, cameras, microscopes, and contact lenses.

Examples: Problems involving calculation of image position and magnification for various lens types.

Additional info: For a complete understanding, students should practice ray diagrams and apply sign conventions consistently when solving problems involving mirrors and lenses.