Focal length is the distance between the optical center (or the principal focus) of a lens or a mirror and the point where parallel rays of light converge to a single point, or appear to diverge from, after passing through the lens or reflecting off the mirror. The focal length is typically measured in millimeters (mm) and serves as a crucial parameter for lens selection and composition. When a lens is referred to as a "50 mm lens," its focal length is specified.

A lens with a smaller focal length is categorized as a wide-angle lens, as it captures a wider field of view. 

On the other hand, a lens with a larger focal length is classified as a telephoto lens, as it provides a narrower field of view, allowing for closer and magnified shots.

Focal length of a Convex Mirror and Concave Mirror

A convex mirror curves outward, away from the reflecting side. The focal length of a convex mirror is considered to be negative. This is because the focal point for a convex mirror is virtual - the light rays appear to converge at a point behind the mirror.

A concave mirror curves inward, towards the reflecting side. The focal length of a concave mirror can be positive or negative, depending on whether the mirror is a converging or diverging mirror, respectively.

• For a converging concave mirror, the focal length is positive. The mirror focuses incoming parallel rays of light to a real focal point in front of the mirror.
• For a diverging concave mirror, the focal length is negative. The mirror causes incoming parallel rays of light to appear to diverge from a virtual focal point behind the mirror.

The formula for calculating the focal length of a mirror is derived from the mirror equation:

where:

f is the focal length of the mirror

u is the object distance 

v is the image distance 

For mirrors, if the image is formed on the same side as the object (virtual image), then v is considered negative.

Focal length of a Convex Lens and Concave Lens

In the case of a convex lens, refraction causes parallel rays of light to converge at a specific point known as the principal focal point. The distance from the lens to this focal point is referred to as the principal focal length, denoted as 'f.'

For a convex lens, u is negative. So,

On the other hand, for a concave lens, where the rays of light diverge, the principal focal length is determined by the distance at which the back-projected rays would intersect. In this case, the principal focal length is assigned a negative sign.

Focal length of a concave lens is given by,

To quantify the optical properties of a lens, we use the concept of lens strength, measured in diopters. The lens strength is defined as the reciprocal of the focal length, expressed in meters.

When dealing with a thick lens composed of spherical surfaces, the focal distance varies for different rays, resulting in a phenomenon known as spherical aberration. Spherical aberration refers to the change in focal length across the lens.

Also, the focal length for different wavelengths of light will exhibit slight variations, known as chromatic aberration. This means that the focal length will be slightly different for each wavelength, causing a dispersion effect.

Focal length of a Thin Lens and Thick Lens

A thick lens is a large physical size lens, with spherical surfaces that are noticeably spaced apart. In simpler terms, it is a lens where the distance between its curved surfaces is significant. 

For a thick lens, where the thickness of the lens cannot be ignored, the lens formula becomes a bit more complex. It considers the physical thickness of the lens and the refractive indices of the lens material and the surrounding medium. 

Focal length of a thick lens is given by,

where,

f is the focal length of the thin lens

n is the refractive index of the lens

d is the thickness of the lens

R₁ is the radius of curvature of first curved surface

R₂ is the radius of curvature of second curved surface

A thin lens is an optical element with curved surfaces that is designed to bend and manipulate light in order to focus or diverge it. It is referred to as "thin" because its physical thickness is negligible compared to its focal length and the curvature of its surfaces. Hence, the thickness of the lens can be ignored (i.e. d=0). 

Focal length of a thin lens is given by,

where,

f is the focal length of the thin lens

n is the refractive index of the lens

R₁ is the radius of curvature of the first curved surface

R₂ is the radius of curvature of the second curved surface