Optics: Refraction, Lenses, and Diffraction

The refractive index (n) of a medium is a measure of how much the speed of light is reduced when it enters that medium. It is defined as the ratio of the speed of light in a vacuum (c) to the speed of light in the medium (v), i.e., n = c/v. When light passes from one medium to another, it changes direction, a phenomenon called refraction. Snell's Law describes this change in direction: when light travels from air into a medium, the refractive index of the medium (n) is given by the ratio of the sine of the angle of incidence (i) to the sine of the angle of refraction (r). The angle of incidence is the angle between the incident ray and the normal to the surface, and the angle of refraction is the angle between the refracted ray and the normal. The general form of Snell's Law for light passing from medium 1 (refractive index n₁) to medium 2 (refractive index n₂) is n₁ sin θ₁ = n₂ sin θ₂.

Total internal reflection occurs when light travelling from a denser optical medium to a less dense optical medium strikes the boundary at an angle of incidence greater than the critical angle. For TIR to occur, two conditions must be met: 1. Light must be travelling from an optically denser medium to an optically less dense medium. 2. The angle of incidence (i) in the denser medium must be greater than the critical angle (C). The critical angle is the angle of incidence in the denser medium for which the angle of refraction in the less dense medium is 90°. It can be calculated using Snell's Law: sin C = 1/n, where n is the refractive index of the denser medium relative to the less dense medium (assuming the less dense medium is air or vacuum).

Lenses are optical devices that refract light to form images. Converging (convex) lenses bring parallel rays of light to a focus, while diverging (concave) lenses spread parallel rays of light outwards, appearing to originate from a virtual focus. The focal length (f) is the distance from the optical centre of the lens to its principal focus. The lens formula relates the object distance (u), image distance (v), and focal length (f) of a lens. It is crucial to use the correct sign conventions: - Object distance (u) is positive for real objects. - Image distance (v) is positive for real images (formed on the opposite side of the lens from the object) and negative for virtual images (formed on the same side as the object). - Focal length (f) is positive for converging lenses and negative for diverging lenses. - Magnification (M) = v/u = image height / object height. A positive M indicates an upright image, a negative M indicates an inverted image.

A diffraction grating is an optical component with a periodic structure that diffracts light into several beams travelling in different directions. The directions of these beams depend on the spacing of the grating and the wavelength of the light. When monochromatic light passes through a diffraction grating, a pattern of bright lines (maxima) and dark lines (minima) is observed. The grating constant (d) is the distance between adjacent slits on the grating. The formula for the maxima produced by a diffraction grating relates the order of the maximum (n), the wavelength of light (λ), the grating constant (d), and the angle of diffraction (θ) from the central maximum.