9.6 Mirrors

Types of Mirrors

Mirrors reflect light to form images, and the type of mirror determines what kind of image you get. Different mirror shapes redirect light in different ways, which is why a bathroom mirror behaves nothing like the curved mirror on a car's passenger side.

Plane Mirrors

A plane mirror is a flat, smooth reflective surface. It produces a virtual image that appears to be the same distance behind the mirror as the object is in front. The image is the same size as the object (magnification = 1), upright, and laterally inverted.

Plane mirrors follow the law of reflection, and because the surface is flat, every reflected ray obeys that law at the same angle relative to the surface. You see these everywhere: bathroom mirrors, dressing mirrors, rearview mirrors.

Curved Mirrors

Curved mirrors have a reflective surface that curves inward or outward. They come in two main types:

• Concave mirrors curve inward (like the inside of a spoon). They can converge light to a focal point.
• Convex mirrors curve outward (like the back of a spoon). They spread light out, making it appear to diverge from a focal point behind the mirror.

Because of this curvature, curved mirrors can produce images that are magnified, reduced, or inverted depending on where the object is placed.

Spherical vs. Parabolic Mirrors

Both are types of curved mirrors, but their geometry differs in an important way.

• Spherical mirrors have a surface shaped like a section of a sphere. They're easier to manufacture, but they suffer from spherical aberration because rays far from the center don't all converge at the same point.
• Parabolic mirrors have a surface shaped like a paraboloid. Their geometry focuses all parallel rays to a single focal point, eliminating spherical aberration. That's why they're used in high-precision instruments like astronomical telescopes and satellite dishes.

Both types have a defined focal point where reflected light converges (concave) or appears to diverge from (convex).

Reflection of Light

Reflection describes what happens when light hits a surface and bounces back. Every mirror, and every image you see in a mirror, depends on the principles covered here.

Law of Reflection

The law of reflection states:

Both angles are measured from the normal, which is an imaginary line perpendicular to the surface at the point where the light hits. This law holds for every reflective surface, whether flat or curved.

Regular vs. Diffuse Reflection

• Regular (specular) reflection occurs on smooth surfaces like mirrors or still water. Parallel incoming rays stay parallel after reflecting, which is why you see a clear image.
• Diffuse reflection occurs on rough surfaces like paper or unpolished wood. Incoming rays scatter in many directions because the surface normal varies from point to point. You don't see a clear image, but diffuse reflection is the reason you can see most objects from any angle.

Properties of Plane Mirrors

Plane mirrors are the simplest mirrors to analyze, and they illustrate the core ideas of image formation that carry over to curved mirrors.

Image Formation

A plane mirror creates an image with these properties:

• The image is virtual (it appears behind the mirror where light doesn't actually reach).
• The image distance equals the object distance from the mirror surface.
• The image is the same size as the object (magnification = 1).
• The image is upright (not inverted).

Virtual vs. Real Images

This distinction matters for every type of mirror and lens you'll encounter:

• A virtual image is formed where reflected rays appear to intersect when extended behind the mirror. It cannot be projected onto a screen. Plane mirrors always produce virtual images.
• A real image is formed where reflected rays actually intersect in front of the mirror. It can be projected onto a screen. Plane mirrors never produce these, but concave mirrors can.

Lateral Inversion

When you look in a plane mirror, your left hand appears to be on the right side of the image, and vice versa. This is lateral inversion. It happens because the mirror reverses the axis perpendicular to its surface (front-to-back), not truly left-to-right. That's why text appears backwards in a mirror.

Curved Mirror Characteristics

These terms and relationships apply to both concave and convex mirrors. You'll need them for ray diagrams and calculations.

Focal Point and Focal Length

The focal point (F) is the point where rays parallel to the principal axis converge after reflecting from a concave mirror, or appear to diverge from behind a convex mirror.

The focal length (f) is the distance from the mirror's vertex (its center surface point) to the focal point. It's related to the radius of curvature by:

$f = \frac{R}{2}$

A shorter focal length means a more strongly curved mirror that bends light more sharply.

Center of Curvature

The center of curvature (C) is the center of the sphere that the mirror's surface is a part of. It sits at a distance $R$ from the vertex, which is twice the focal length. Any light ray that passes through C and hits the mirror reflects straight back along the same path. This property makes it one of the key rays in ray diagrams.

Principal Axis

The principal axis is the imaginary straight line that passes through both the center of curvature and the vertex of the mirror. It's perpendicular to the mirror surface at the vertex. All distances in mirror equations (object distance, image distance, focal length) are measured along this axis.

Spherical Mirrors

Spherical mirrors are the most common curved mirrors you'll work with in this course. Their behavior depends on whether they're concave or convex.

Concave Mirrors

Concave mirrors reflect light inward. The type of image they produce depends on where the object is placed relative to the focal point:

• Object beyond C: real, inverted, diminished image (between F and C)
• Object at C: real, inverted, same-size image (at C)
• Object between C and F: real, inverted, magnified image (beyond C)
• Object at F: no image forms (reflected rays are parallel)
• Object between F and the mirror: virtual, upright, magnified image (behind the mirror)

This is why concave mirrors are used in makeup mirrors (object close to the mirror gives a magnified view) and in headlights (light source at F produces a parallel beam).

Convex Mirrors

Convex mirrors reflect light outward. They always produce images that are:

• Virtual (behind the mirror)
• Upright
• Diminished (smaller than the object)

Because of this, convex mirrors provide a wider field of view than plane mirrors of the same size. That's why they're used as vehicle side mirrors ("objects in mirror are closer than they appear") and as security mirrors in stores.

Mirror Equation

The mirror equation relates object distance ($u$), image distance ($v$), and focal length ($f$):

$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$

You'll use this alongside the magnification equation:

$m = -\frac{v}{u}$

Together, these two equations let you find the position, size, and orientation of any image formed by a spherical mirror.

Image Formation in Curved Mirrors

Ray Diagrams

Ray diagrams are a graphical way to find where an image forms and what it looks like. You draw at least two of these three principal rays from the top of the object:

1. Parallel ray: Travels parallel to the principal axis, then reflects through the focal point (concave) or appears to come from the focal point (convex).
2. Focal ray: Travels through the focal point (or toward it for convex), then reflects parallel to the principal axis.
3. Center ray: Travels through the center of curvature, then reflects back along the same path.

Where two of these rays intersect (or appear to intersect) is where the top of the image is located. Draw the image from the principal axis to that intersection point.

Magnification

Magnification ($m$) tells you how the image size compares to the object size:

$m = -\frac{v}{u} = \frac{h_i}{h_o}$

where $h_i$ is image height and $h_o$ is object height.

• $|m| > 1$: image is larger than the object
• $|m| < 1$: image is smaller than the object
• $m > 0$: image is upright
• $m < 0$: image is inverted

Sign Conventions

A consistent sign convention keeps your calculations from going wrong. The most common convention for mirrors:

Always check which sign convention your course uses. Some textbooks flip the signs for $u$, so confirm with your instructor.

Applications of Mirrors

Optical Instruments

• Reflecting telescopes use large concave mirrors to collect light from distant stars and galaxies. The larger the mirror, the more light it gathers.
• Microscopes use mirrors to redirect illumination light onto specimens.
• Laser systems use highly reflective mirrors for beam steering, and partially reflective mirrors as output couplers in laser cavities.
• Interferometers split and recombine light using mirrors to make extremely precise measurements of distance or wavelength.

Everyday Uses

• Rearview and side mirrors in vehicles (plane and convex)
• Makeup and shaving mirrors (concave, for magnification)
• Security and surveillance mirrors in stores and parking garages (convex, for wide field of view)
• Decorative mirrors that create illusions of larger spaces

Scientific Applications

• Solar concentrators use large arrays of mirrors to focus sunlight for energy generation.
• Adaptive optics in astronomy correct for atmospheric turbulence in real time, sharpening telescope images.
• Spectroscopy instruments use mirrors to direct and focus light for analyzing spectra of materials.

Aberrations in Mirrors

Aberrations are flaws in image formation that cause blurring or distortion. They matter most in precision instruments where image quality is critical.

Spherical Aberration

This occurs in spherical mirrors because rays hitting the outer edges of the mirror converge at a slightly different point than rays near the center. The result is a blurred image rather than a sharp point. Spherical aberration is more pronounced for larger mirrors and can be eliminated by using a parabolic mirror shape instead.

Coma

Coma affects off-axis points (objects not on the principal axis). It causes point sources of light to appear stretched into a comet-like shape, with a tail pointing away from the axis. It becomes more severe for objects farther from the optical axis and can be reduced through careful mirror design or aspheric surfaces.

Astigmatism

Astigmatism occurs when rays in different planes (say, horizontal vs. vertical) come to focus at different distances from the mirror. A point source ends up looking like a short line rather than a dot. Like coma, it's most noticeable for off-axis objects and can be corrected with additional optical elements.

Mirror Systems

Combining multiple mirrors creates optical systems with capabilities that a single mirror can't achieve.

Multiple Mirror Arrangements

• A Cassegrain telescope uses a large concave primary mirror with a smaller convex secondary mirror. Light reflects off the primary, bounces off the secondary, and passes through a hole in the primary to reach the eyepiece. This folds the optical path into a compact design.
• A Gregorian telescope uses two concave mirrors and produces an upright final image.
• Beam splitters use partially reflective mirrors to divide a single beam into two paths, which is essential in interferometry.
• Laser cavities place two mirrors facing each other to bounce light back and forth, amplifying it with each pass through the gain medium.

Periscopes

A periscope uses two parallel plane mirrors (or prisms) set at 45° angles to redirect light. Light enters the top mirror, reflects down to the bottom mirror, and reflects out toward the viewer. This lets you see over walls or around corners. Submarines use periscopes to observe the surface while remaining submerged.

Kaleidoscopes

Kaleidoscopes typically arrange three mirrors in a triangular tube. Objects at one end undergo multiple reflections, creating repeating symmetrical patterns. They're a simple but vivid demonstration of how multiple reflections compound to produce complex images from simple inputs.

Advanced Mirror Concepts

These topics go beyond standard spherical mirrors and represent technologies used in modern research and engineering.

Adaptive Optics

Adaptive optics systems correct for distortions in real time. In astronomy, Earth's atmosphere bends and distorts incoming starlight. An adaptive optics system measures these distortions (often using a guide star) and adjusts a deformable mirror hundreds of times per second to compensate. The result is much sharper images from ground-based telescopes.

Deformable Mirrors

A deformable mirror has a flexible reflective surface controlled by an array of actuators (tiny mechanical pushers) behind it. Each actuator adjusts a small region of the mirror surface by microscopic amounts. These mirrors are the core component of adaptive optics systems and also find use in laser beam shaping and medical imaging of the eye.

Liquid Mirrors

A liquid mirror is formed by spinning a reflective liquid (such as mercury) in a container. The rotation causes the liquid surface to naturally assume a parabolic shape due to the balance of gravitational and centrifugal forces. This provides a large, smooth parabolic reflector at a fraction of the cost of a polished glass mirror. The tradeoff is that liquid mirrors can only point straight up, limiting their use to zenith-pointing telescopes.