Lenses and curved mirrors

Curved optical elements

Plane mirrors change ray direction but do not focus light. Curved mirrors and lenses can converge or diverge rays, forming real or virtual images. They are the basis of cameras, eyes, microscopes, telescopes, projectors, and many scientific instruments.

In the thin-lens and paraxial approximations, rays stay close to the optical axis and make small angles. These approximations allow simple equations to describe image formation.

Spherical mirrors

A concave mirror curves inward like the inside of a sphere and can converge parallel rays to a focal point. A convex mirror curves outward and diverges rays, producing virtual images.

For a spherical mirror with radius of curvature $R$, the focal length is approximately

f=rac{R}{2}

using the paraxial approximation.

Mirror and lens equation

For both thin lenses and spherical mirrors under standard sign conventions, object distance $d_o$, image distance $d_i$, and focal length $f$ are related by

rac{1}{f}=rac{1}{d_o}+rac{1}{d_i}

This equation is one of the central tools of geometric optics.

A real image forms where light rays actually converge. A virtual image forms where rays appear to originate when traced backward.

Magnification

The lateral magnification is

m=rac{h_i}{h_o}=-rac{d_i}{d_o}

where $h_i$ is image height and $h_o$ is object height. A negative magnification indicates an inverted image; a positive magnification indicates an upright image.

The magnitude $|m|$ tells how much larger or smaller the image is than the object.

Converging lenses

A converging lens is thicker in the middle than at the edges and has positive focal length in air. Parallel incoming rays converge to the far focal point.

For an object outside the focal length, a converging lens can form a real inverted image. If the object is inside the focal length, it forms a virtual upright magnified image. This is the principle of a simple magnifying glass.

Diverging lenses

A diverging lens is thinner in the middle and has negative focal length. It causes parallel incoming rays to spread out as if they came from the near focal point. A single diverging lens with a real object forms a virtual upright reduced image.

Diverging lenses are used in eyeglasses for nearsightedness and in optical systems to expand beams or control focus.

Ray diagrams

Three principal rays are especially useful for lenses. A ray parallel to the axis refracts through or away from the focal point. A ray through the center of a thin lens continues nearly straight. A ray through the near focal point exits parallel to the axis.

For mirrors, a parallel ray reflects through the focal point, a ray through the focal point reflects parallel, and a ray through the center of curvature reflects back on itself.

Lensmaker's equation

For a thin lens in air, focal length depends on refractive index and surface curvatures. The lensmaker's equation is

ight)$$ with sign conventions for radii. This shows that lens power comes from refraction at curved surfaces. ## The big idea Lenses and curved mirrors form images by focusing or diverging rays. The equation $1/f=1/d_o+1/d_i$ and magnification formula $m=-d_i/d_o$ connect object location, image location, and size. Ray diagrams provide the geometric understanding behind the algebra.