7 Geometric Optics

Learn how ray models predict reflection, refraction, and image formation in mirrors, lenses, and optical instruments.

Rays, reflection, and refraction

Geometric optics models light as rays: straight-line paths that indicate the direction light travels. This approximation works well when the objects and openings involved are much larger than light’s wavelength. By tracing rays, we can predict where images form and whether they are real or virtual.

When a ray meets a surface, measure its angle from the normal, an imaginary line perpendicular to that surface. For reflection, the angle of reflection equals the angle of incidence:

\theta_r=\theta_i

A smooth surface gives orderly, specular reflection. A rough surface scatters reflected light in many directions, producing diffuse reflection.

When light crosses into a different material, its speed changes and its path may bend. The is the ratio of the speed of light in vacuum to its speed in the material:

n=\frac{c}{v}

The direction of bending is described by :

n_1\sin\theta_1=n_2\sin\theta_2

The angles are measured from the normal. A ray entering a material with a higher bends toward the normal; entering a lower-index material, it bends away. For example, light from a fish bends as it leaves water for air, making the fish appear closer to the surface than it really is.

Takeaway: Reflection changes a ray’s direction at a surface; refraction changes its direction as it crosses between materials.

Images and image calculations

An image forms where rays from an object point meet, or where they appear to meet. A forms at the actual convergence of rays and can be projected onto a screen. A forms where rays only appear to originate, so it cannot be projected onto a screen.

A gives a visual way to locate an image. Start at the top of the object and trace at least two rays with predictable paths. The image tip is where the rays intersect. If the rays spread apart, extend them backward; their apparent intersection locates a .

For spherical mirrors and thin lenses, distances and focal length can also be related algebraically:

\frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{f}

Here, $d_o$ is the object distance, $d_i$ is the image distance, and $f$ is the focal length. relates the image height to the object height:

m=\frac{h_i}{h_o}=-\frac{d_i}{d_o}

In the sign convention used here, a has positive $d_i$ , while a has negative $d_i$ . Converging lenses and concave mirrors have positive $f$ ; diverging lenses and convex mirrors have negative $f$ . Positive $m$ indicates an upright image, and negative $m$ indicates an inverted one. Use consistent units and apply one sign convention throughout.

Takeaway: Ray diagrams show where an image forms; the image and equations quantify its location, size, and orientation.

Mirrors and lenses

A plane mirror forms an upright, that is the same size as the object. The image appears as far behind the mirror as the object is in front of it.

A spherical mirror has a reflecting surface that curves inward or outward. A concave mirror curves inward and brings parallel rays to a real focus. A convex mirror curves outward and reflects rays so that they diverge as if from a virtual focus. For a spherical mirror in the small-angle, or paraxial, approximation, focal length is half the radius of curvature:

f=\frac{R}{2}

A concave mirror’s image depends on the object’s position. If the object is beyond the focal point, the image is real and inverted. If the object is between the mirror and its focal point, the image is virtual, upright, and enlarged. A convex mirror forms a virtual, upright, reduced image and provides a wide field of view, making it useful in security and vehicle mirrors.

A , usually thicker at its center, bends parallel rays toward a real focal point. A diverging lens, usually thinner at its center, spreads parallel rays so they appear to come from a virtual focal point. Their focal lengths are positive and negative, respectively.

For a thin lens, three principal rays help locate images: a ray parallel to the optical axis refracts through the far focal point (or appears to come from the near focal point for a diverging lens); a ray through the lens center continues approximately straight; and a ray aimed through the near focal point emerges parallel to the axis.

Example: A has a focal length of $10\,\text{cm}$ , and an object is $30\,\text{cm}$ away. Applying the thin-lens equation gives:

\frac{1}{d_i}=\frac{1}{10\,\text{cm}}-\frac{1}{30\,\text{cm}},\qquad d_i=15\,\text{cm}

The positive image distance indicates a . Its is:

m=-\frac{15}{30}=-0.50

So the image is inverted and half as tall as the object.

Takeaway: The mirror or lens type and the object’s position determine whether the image is real or virtual, upright or inverted, and enlarged or reduced.

Optical instruments

Optical instruments combine reflection and refraction to form useful images.

Eye: The cornea and lens focus a real, inverted image on the retina. The eye changes the lens shape to focus on objects at different distances; this adjustment is called .
Camera: A focuses a onto film or an electronic sensor. Moving the lens brings objects at different distances into focus.
Simple magnifier: A held with the object inside its focal length forms a virtual, upright, enlarged image. When the final image is at infinity, its angular is approximately $\frac{25\,\text{cm}}{f}$ , using the conventional near point of $25\,\text{cm}$ .
Compound microscope: A short-focal-length objective first creates a real, enlarged, inverted intermediate image. The eyepiece then acts as a magnifier to produce a larger virtual final image.
Refracting telescope: The objective lens forms an image of a distant object, and the eyepiece magnifies it for viewing. A reflecting telescope instead uses a concave mirror to collect and focus light, avoiding the color fringing, or chromatic aberration, caused by lenses.

Takeaway: Optical instruments use the same image-forming principles as individual mirrors and lenses, arranged to meet different viewing needs.