1. Reflection of Light and Spherical Mirrors

Ray optics, or geometrical optics, treats light as rays travelling in straight lines. The reflection of light occurs when light bounces off a surface. The law of reflection states that the angle of incidence equals the angle of reflection ($\theta_i = \theta_r$). Spherical mirrors, which are sections of a sphere, form images through reflection. Concave mirrors can form both real and virtual images, depending on the object's position, and are used in applications like headlights and telescopes. Convex mirrors always form virtual, erect, and diminished images, used as rearview mirrors for a wider field of view.

2. Refraction of Light and Refractive Index

Refraction is the bending of light as it passes from one medium to another, caused by a change in the speed of light. Snell's Law describes this phenomenon: $n_1 \sin \theta_1 = n_2 \sin \theta_2$, where $n_1$ and $n_2$ are the refractive indices of the two media, and $\theta_1$ and $\theta_2$ are the angles of incidence and refraction, respectively. The refractive index ($n$) of a medium is the ratio of the speed of light in a vacuum ($c$) to the speed of light in that medium ($v$), $n = c/v$. A higher refractive index means light travels slower in that medium.

3. Refraction by Lenses and Spherical Surfaces

Lenses are optical devices that refract light to form images. Convex lenses converge parallel light rays to a focal point, while concave lenses diverge them. The behavior of lenses can be predicted using ray diagrams and the lens formula: $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$, where $f$ is the focal length, $v$ is the image distance, and $u$ is the object distance. Refraction also occurs at spherical surfaces, described by the formula $\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$, where $R$ is the radius of curvature.

4. Total Internal Reflection and Prisms

Total internal reflection (TIR) occurs when light travels from a denser medium to a rarer medium at an angle of incidence greater than the critical angle. At this angle, all light is reflected back into the denser medium. TIR is the principle behind optical fibers, used in telecommunications and medical endoscopes, which transmit light signals over long distances with minimal loss. Prisms utilize refraction and sometimes TIR to deviate, disperse, or reflect light. For example, right-angled prisms are often used to turn light by 90$\degree$ or 180$\degree$ via TIR.

5. Dispersion and Scattering

Dispersion is the phenomenon where white light splits into its constituent colors when passed through a prism or diffraction grating. This happens because the refractive index of the medium depends slightly on the wavelength of light, causing different colors to refract at slightly different angles. Scattering, such as Rayleigh scattering, explains why the sky appears blue; shorter blue wavelengths of sunlight are scattered more effectively by atmospheric molecules than longer red wavelengths. This is also responsible for the reddish appearance of the sun at sunrise and sunset.

6. The Human Eye and Optical Instruments

The human eye functions much like a camera, using a lens to focus light onto the retina, where photoreceptor cells convert light into electrical signals. The eye's lens adjusts its focal length (accommodation) to focus on objects at different distances. Optical instruments like cameras, microscopes, and telescopes utilize lenses and mirrors to magnify distant or small objects, or to capture images. Understanding the principles of refraction and reflection is key to designing and utilizing these instruments effectively for observation and scientific research.

7. Additional: Lensmaker's Formula and Aberrations

The Lensmaker's Formula relates the focal length ($f$) of a lens to its refractive index ($n_l$), the refractive index of the surrounding medium ($n_m$), and the radii of curvature of its surfaces ($R_1$ and $R_2$): $\frac{1}{f} = (n_l - n_m) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)$. Aberrations are defects in image formation by lenses and mirrors. Common types include chromatic aberration (different colors focus at different points due to dispersion) and spherical aberration (rays hitting the edges of a spherical mirror or lens focus at a different point than those hitting the center), which optical designers work to minimize.