What do solar cookers, satellite dishes, reflector telescopes and flashlights have in common? It might seem like an outlandish question, but the truth is that they all work based on the same thing: parabolic reflectors.
These reflectors essentially exploits the benefits of a parabolic shape, in particular its ability to focus light onto a single point, in order to concentrate either a radio wave signal (in the case of satellite dishes) or visible light (in the case of flashlights and reflector telescopes) to allow us to detect it or use the energy. Learning about the basics of the parabolic mirror helps you understand these pieces of technology and a lot more.
Before getting into the details, you need to understand how a parabolic mirror reflects light rays, and there is some important terminology you’ll need to understand.
First, the focal point is a point where parallel rays converge after reflecting off the surface, and the focal length of a parabolic mirror is the distance from the center of the mirror to the focal point. In some cases (e.g., a convex parabolic mirror) the focal point isn’t where parallel rays actually meet after reflecting, but where they appear to have emanated from after being reflected.
The optical axis of a parabolic mirror or a spherical mirror is the line of symmetry of the reflector, which is essentially a horizontal line through the center if you imagine the reflective surface of the mirror stood up vertically.
A light ray is a straight-line approximation for the path of travel of light. This is a huge oversimplification in most cases, because any object will have light traveling away from it in all directions, but by focusing on a few specific lines, the main features of the effect of a surface on light can be determined.
For example, an extended object in front of a mirror will have light rays emerging from it vertically and in the opposite direction to the mirror, which will never make contact with the mirror’s surface, but you can understand how the mirror works by looking solely at some of the rays traveling in its direction.
The geometry of a parabola makes it a particularly good choice for applications where you need to focus light waves on a single location. The parabolic shape is such that incident parallel rays will converge at a single focal point no matter where on the surface of the mirror they actually strike. This is why the parabolic mirror is the key component of a reflecting telescope along with many other devices designed to focus light.
The light rays do have to be incident parallel to the optical axis of the mirror for this to work perfectly, but it’s important to remember that if an object is very far away from the surface of the mirror, all of the light rays coming from it are approximately parallel by the time they reach it. This means that in many cases, you can treat the rays as parallel even if they technically wouldn’t be. As well as simplifying calculations, this means you don’t have to go through the process of ray tracing for a parabolic reflector in some cases.
Ray tracing is an invaluable technique in cases where the rays aren’t parallel and so can’t be assumed to all reflect towards the focal point. The technique essentially involves drawing individual light rays coming off the object and using the law of reflection (along with some useful tips for ray tracing specifically) to determine where the reflective surface will focus light to. In other words, using the position of the object and the position of the mirror, along with some simple reasoning, you can find where the image of the object will be located using ray tracing.
The image for a concave mirror (one where the inside of the bowl faces the object) will be a “real image,” which is one where light rays physically converge to form an image. It helps to think about what would happen if you placed a projector screen at this location: For a real image, the image would be displayed on the screen, in focus.
For a convex paraboloid or spherical mirror, the image will be “virtual,” so light rays don’t physically converge at its location. If you placed a screen at this location, there would be no image. The way the mirror affects the light simply makes it look like that’s where the image is. If you look at yourself in a regular plane mirror you can see this effect: It looks like the image is behind the mirror, but of course there is no light and no image actually behind the mirror.
A concave mirror has a curve such that the “bowl” of the mirror faces the object – you can think of the interior as a little “cave” to remember the difference between concave and convex. The focal point for a concave mirror is on the same side as the object, and it is assigned a positive focal length. The images created in this way are real images.
To do ray tracing for a concave mirror, there are a few key rules you can apply as needed. First, any ray coming from the object that is parallel to the optical axis of the mirror will pass through the focal point after reflection. The opposite of this is also true: Any light ray coming from the object that passes through the focal point on its journey to the mirror will reflect so it is parallel to the optical axis. Finally, the law of reflection applies to any ray that strikes the vertex of the surface of the mirror, so the angle of incidence matches the angle of reflection.
By drawing two or three of these rays in a ray diagram for a single point on the object, you can pinpoint the location of the image of that point.
A convex mirror has a curve opposite that of a concave mirror, so the outside of the “bowl” of the mirror faces the object. The focal point for a convex spherical or parabolic mirror is on the opposite side to the object, and they are assigned a negative focal length to reflect this and the fact that the images produced are virtual.
Ray tracing for a convex mirror follows the same general pattern as for a concave mirror, but it requires a little more abstraction to get the result. A ray traveling parallel to the optical axis of the mirror will reflect at an angle that makes it look like it originated from the focal point of the mirror. Any ray from the object that travels towards the focal point will reflect parallel to the optical axis of the mirror. Finally, rays that reflect from the surface at the vertex will reflect at an angle equal to their angle of incidence, just on the opposite side of the optical axis.
For both convex and concave spherical mirrors, if you draw a ray that passes through the center of curvature (if you imagine extending the mirror surface into a sphere) or that would pass through it, the ray would reflect back along exactly the same path. Drawing two or three rays on a diagram will help you find the image location for a single point on an object, noting that on a convex mirror this will be a virtual image on the opposite side of the mirror.
Spherical mirrors affect light in a very similar way to parabolic mirrors, except the curved surface forms part of a sphere rather than being a generic paraboloid. In many cases, light will reflect from a spherical mirror just like it would from a parabolic mirror, but if the angle of incidence of the light is farther from the optical axis of the mirror, the deviation of the reflected ray is increased.
This means spherical mirrors are less dependable than parabolic mirrors, because they’re prone to what’s known as spherical aberration, as well as comatic aberration. Spherical aberration occurs when light rays parallel to the optical axis are incident on a spherical mirror, because the rays farther from the optical axis are reflected at larger angles, so there isn’t a clearly defined focal point. In fact, there are effectively multiple focal lengths, depending on how far the incident ray is from the optical axis.
For comatic aberration, parallel rays farther from the optical axis respond in a similar way, but their focal points vary in height as well as focal length. This produces a “tail” effect, similar to the appearance of a comet, which is where the phenomenon gets its name.
Focal Length Equations for Curved Mirrors
The focal length of a mirror or lens is one of the most important characteristics to define it, but the expression isn’t as simple for a parabolic mirror as it is for a lens. For a light ray incident on the mirror at a height y (where y = 0 at the deepest part of the curve) and making an angle of θ to the tangent to the curve of the mirror, the focal length is:
For spherical mirrors, things are a little simpler, and the mirror equation takes a similar form to the lens equation. For the distance to the object do, the distance to the image di and the radius of the curvature of the mirror (i.e., if the curve were extended into a circle or sphere, the radius of that shape) R, the expression is:
Where do is the distance to the object and di is the distance to the image, measured from the mirror’s surface on the optical axis. For very small angles of incidence, you can replace 2/R with 1/f, to obtain an explicit expression for the focal length.
Applications of Parabolic Mirrors
The dependable behavior of parabolic mirrors allows them to be used for many different purposes. One of the most “everyday” items is the simple flashlight; by having a source of light at the focal point of a parabolic mirror surrounding it, the light emitted reflects off the mirror and emerges from the other side parallel to the optical axis. This design means that essentially no light produced by the bulb is “wasted” and all of it emerges from the end of the flashlight.
Solar cookers work in a very similar way, except they concentrate parallel rays from the sun towards the focal point of the parabolic mirror. This is a very efficient (and eco-friendly) way to generate heat, and if you place a cooking pot directly at the focal point, then it absorbs the reflected energy from the whole parabola. Some solar cookers use other shapes for the reflective surface, but as you’ve learned, the parabola is really the best choice in terms of efficiency.
Satellite dishes and radio telescopes essentially work in the same way as solar cookers, except they are designed to reflect radio wavelength light instead of visible light. The parabolic shapes of both of these are designed to reflect light onto a receiver, which is positioned at the focal point of the dish. Both radio telescopes and satellite dishes do this for the same reason: to maximize the amount of waves they detect.
- Australian Telescope National Facility: How Does a Radio Telescope Work?
- Science ABC: Why Are TV Dish Antennas Concave?
- Albert: Parabola as a Flashlight
- Solar Cooker at Cantina West: How Does Solar Cooking Work?
- Isaac Physics: Parabolic Mirror
- University of California, Santa Cruz: Focusing Properties of Spherical and Parabolic Mirrors
- Physics LibreTexts: Spherical Mirrors
- Lumen: Image Formation by Mirrors