Light: Geometric Optics

Introduction to Geometric Optics

Geometric optics is the study of light propagation in terms of rays. This model is highly effective for understanding the behavior of light as it interacts with mirrors, lenses, and various media.

• Ray Model of Light: Light is represented as straight lines (rays) that travel from an object. This is an idealization but is very useful for analyzing optical systems.

• Applications: Used in designing optical instruments, understanding vision, and explaining everyday phenomena like reflection and refraction.

Reflection and Image Formation

Law of Reflection

Reflection occurs when light bounces off a surface. The law of reflection governs this process:

• Law of Reflection: The angle of reflection equals the angle of incidence, both measured from the normal (a line perpendicular to the surface).

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

• Diffuse Reflection: Occurs on rough surfaces, scattering light in many directions, but the law of reflection still applies locally.

Plane Mirrors and Virtual Images

• Plane Mirror: A flat mirror forms images that appear to be behind the mirror.

• Virtual Image: The image formed does not involve actual light passing through the image location; it cannot be projected onto a screen.

• Image Distance: The distance from the mirror to the image equals the distance from the mirror to the object.

Formation of Images by Spherical Mirrors

Types of Spherical Mirrors

• Concave Mirror: Reflective on the inside of a sphere; can form real or virtual images.

• Convex Mirror: Reflective on the outside; always forms virtual, upright, and smaller images.

Focal Point and Focal Length

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

• Focal Length (f): For a spherical mirror, , where is the radius of curvature.

Spherical Aberration

• Occurs when rays far from the principal axis do not converge at the same point, causing a blurred image. Reduced by using mirrors with small curvature or parabolic reflectors.

Ray Diagrams for Mirrors

• Three principal rays are used:

  • Ray parallel to axis reflects through the focal point.

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

  • Ray perpendicular to the mirror reflects back on itself.

• The intersection of these rays locates the image.

Mirror Equation and Magnification

• Mirror Equation:

  where is focal length, is object distance, is image distance.

• Magnification (m):

  where is image height, is object height. Negative indicates an inverted image.

Image Characteristics by Object Location

• Object inside focal point (concave): Image is upright, larger, and virtual.

• Object outside center of curvature (concave): Image is inverted, smaller, and real.

• Convex mirror: Image is always virtual, upright, and smaller.

Sign Conventions

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

• Magnification is positive for upright images, negative for inverted images.

Refraction and Index of Refraction

Index of Refraction

When light enters a medium, its speed changes. The index of refraction quantifies this effect:

• Definition: , where is the speed of light in vacuum, is the speed in the medium.

Refraction and Snell's Law

• Refraction: The bending of light as it passes from one medium to another due to a change in speed.

• Snell's Law:

  where , are indices of refraction, , are angles with respect to the normal.

• Explains phenomena such as the apparent bending of objects in water.

Total Internal Reflection and Fiber Optics

Total Internal Reflection

• Occurs when light attempts to move from a medium with higher to lower at an angle greater than the critical angle.

• Critical Angle ():

  for .

• For angles of incidence greater than , all light is reflected internally.

Applications

• Binoculars: Use total internal reflection for efficient light transmission.

• Fiber Optics: Light is guided along fibers by repeated total internal reflection, enabling data transmission and medical imaging.

Thin Lenses and Ray Tracing

Types of Thin Lenses

• Converging (Convex) Lens: Thicker at the center; focuses parallel rays to a point (focal point).

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

Lens Power

• Definition: , where is in meters, in diopters (D).

• Converging lenses have positive power; diverging lenses have negative power.

Ray Tracing for Lenses

• Three principal rays:

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

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

  • Ray through the center of the lens passes straight through, undeflected.

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

The Thin Lens Equation

• Thin Lens Equation:

  where is focal length, is object distance, is image distance.

• Magnification:

  Positive means upright image; negative means inverted image.

• Sign Conventions:

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

  • Object distance is positive if object is on the side where light enters.

  • Image distance is positive if image is on the opposite side from the object.

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

Problem Solving Steps for Lenses and Mirrors

1. Draw a ray diagram to locate the image.

2. Apply the appropriate equation (mirror or lens equation).

3. Use correct sign conventions.

4. Check that the solution matches the ray diagram.

Combinations of Lenses

• In systems with multiple lenses, the image from the first lens serves as the object for the next.

• Object distances may be negative depending on the configuration.

Lensmaker’s Equation

• Relates the focal length of a lens to its radii of curvature and the index of refraction:

• where is the index of refraction, and are the radii of curvature of the two surfaces.

Summary Table: Key Equations in Geometric Optics

Additional info:

• Parabolic mirrors eliminate spherical aberration but are more complex to manufacture.

• Fiber optics revolutionized telecommunications and medical imaging by enabling light transmission over long distances with minimal loss.