Geometrical optics forms the foundation of understanding how light interacts with surfaces and materials. It explains reflection, refraction, and total internal reflection, which are crucial for designing optical systems like mirrors, lenses, and prisms.
Image formation is a key application of geometrical optics. By using ray diagrams and equations, we can predict how lenses and mirrors create images. This knowledge is essential for designing optical devices, from simple magnifying glasses to complex telescopes and cameras.
Geometrical Optics
Laws of reflection and refraction
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Law of reflection states that the incident ray, reflected ray, and normal to the surface all lie in the same plane, and the angle of incidence (θi) equals the angle of reflection (θr)
Applies to both plane and curved surfaces (mirrors)
Law of refraction, also known as Snell's law, relates the angles of incidence and refraction to the refractive indices of the two media: n1sinθ1=n2sinθ2
n1 and n2 represent the refractive indices of the initial and final media, respectively
θ1 and θ2 denote the angles of incidence and refraction, measured from the normal to the surface
Solving problems involving plane surfaces requires applying trigonometry and geometry to determine angles and distances based on the given information and the laws of reflection and refraction
Curved surfaces can be treated as a collection of infinitesimal plane surfaces, and the laws of reflection and refraction are applied at each point on the surface to determine the path of light rays
Total internal reflection concepts
Total internal reflection (TIR) occurs when light travels from a medium with a higher refractive index to one with a lower refractive index, and the incident angle exceeds the critical angle
Under these conditions, all light is reflected back into the higher refractive index medium instead of being refracted
Critical angle (θc) is the minimum angle of incidence at which TIR occurs and is determined by the ratio of the refractive indices of the two media: sinθc=n1n2
Light incident at angles greater than the critical angle will experience TIR
TIR has important applications in various optical systems
Optical fibers rely on TIR to efficiently propagate light through the fiber core for use in telecommunications (internet) and medical imaging (endoscopes)
Prisms utilize TIR to reflect light, finding applications in binoculars, periscopes, and reflectors (road signs)
Image Formation and Optical Systems
Image formation by lenses and mirrors
Ray diagrams are used to graphically determine the location, size, and orientation of images formed by lenses and mirrors
Principal rays, such as those parallel to the optical axis, passing through the center of the lens, or passing through the focal point, are traced to locate the image
Thin lens equation relates the focal length (f) of the lens to the object distance (do) and image distance (di): f1=do1+di1
This equation helps calculate the position and size of the image formed by a thin lens
Mirrors have a similar equation, f1=do1+di1, but the sign conventions for mirrors differ from those for lenses due to the different nature of image formation
Calculations for optical systems
Magnification (M) is the ratio of the image height (hi) to the object height (ho) and can be calculated using M=hohi or M=−dodi
The negative sign in the latter equation accounts for the inversion of the image
Focal length is the distance from the lens or mirror to the focal point and is determined by the curvature and refractive index of the optical element
Thin lenses: f1=(n−1)(R11−R21), where n is the refractive index of the lens material, and R1 and R2 are the radii of curvature of the lens surfaces
Mirrors: f=2R, where R is the radius of curvature of the mirror
Image distance is the distance from the lens or mirror to the image and can be calculated using the thin lens equation or mirror equation, depending on the optical element
Simple optical systems can consist of a single lens or mirror, or combinations of lenses and mirrors in sequence
Matrix methods (ABCD matrices) or sequential ray tracing can be used to analyze the behavior of light in these systems and determine the final image properties