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1.3: The Law of Reflection

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    Learning Objectives

    By the end of this section, you will be able to:

    • Explain the reflection of light from polished and rough surfaces
    • Describe the principle and applications of corner reflectors

    Whenever we look into a mirror, or squint at sunlight glinting from a lake, we are seeing a reflection. When you look at a piece of white paper, you are seeing light scattered from it. Large telescopes use reflection to form an image of stars and other astronomical objects.

    The law of reflection states that the angle of reflection equals the angle of incidence:

    \[θ_r=θ_i \label{law of reflection} \]

    The law of reflection is illustrated in Figure \(\PageIndex{1}\), which also shows how the angle of incidence and angle of reflection are measured relative to the perpendicular to the surface at the point where the light ray strikes.

    A light ray is incident on a smooth surface and is making an angle theta i relative to a line drawn perpendicular to the surface at the point where the incident ray strikes it. The reflected light ray makes an angle theta r with the same perpendicular drawn to the surface. Both incident and reflected ray are on the same side of the surface but opposite sides of the perpendicular line.
    Figure \(\PageIndex{1}\): The law of reflection states that the angle of reflection equals the angle of incidence—θri. The angles are measured relative to the perpendicular to the surface at the point where the ray strikes the surface.

    We expect to see reflections from smooth surfaces, but Figure \(\PageIndex{2}\) illustrates how a rough surface reflects light. Since the light strikes different parts of the surface at different angles, it is reflected in many different directions, or diffused. Diffused light is what allows us to see a sheet of paper from any angle, as shown in Figure \(\PageIndex{1a}\).

    The figure shown parallel light rays falling on a rough surface. The rays hit the surface at different angles to the perpendicular lines to the surface at the points of incidence, and the reflected rays get scattered in different directions.
    Figure \(\PageIndex{2}\): Light is diffused when it reflects from a rough surface. Here, many parallel rays are incident, but they are reflected at many different angles, because the surface is rough.

    People, clothing, leaves, and walls all have rough surfaces and can be seen from all sides. A mirror, on the other hand, has a smooth surface (compared with the wavelength of light) and reflects light at specific angles, as illustrated in Figure \(\PageIndex{3b}\). When the Moon reflects from a lake, as shown in Figure \(\PageIndex{1c}\), a combination of these effects takes place.

    Figure a shows the rays of light from a flashlight falling on a page of paper. The light gets reflected at many angles as the surface is rough. Reflected light reaches eyes placed at many location. Figure b shows the rays of light from a flashlight falling on mirror. All of the light gets reflected at the same angle since the surface is smooth. Reflected light only reaches an eye placed so that the reflected beam hits it. An observer not at the angle of the reflected light does not see it. Figure c shows a photograph of moonlight falling on a lake. The lake’s shiny surface reflects it. A bright , slightly rippled strip of moonlight is seen reflecting from the lake on a dark background.
    Figure \(\PageIndex{3}\): (a) When a sheet of paper is illuminated with many parallel incident rays, it can be seen at many different angles, because its surface is rough and diffuses the light. (b) A mirror illuminated by many parallel rays reflects them in only one direction, because its surface is very smooth. Only the observer at a particular angle sees the reflected light. (c) Moonlight is spread out when it is reflected by the lake, because the surface is shiny but uneven. (credit c: modification of work by Diego Torres Silvestre)

    When you see yourself in a mirror, it appears that the image is actually behind the mirror (Figure \(\PageIndex{4}\)). We see the light coming from a direction determined by the law of reflection. The angles are such that the image is exactly the same distance behind the mirror as you stand in front of the mirror. If the mirror is on the wall of a room, the images in it are all behind the mirror, which can make the room seem bigger. Although these mirror images make objects appear to be where they cannot be (like behind a solid wall), the images are not figments of your imagination. Mirror images can be photographed and videotaped by instruments and look just as they do with our eyes (which are optical instruments themselves). The precise manner in which images are formed by mirrors and lenses is discussed in an upcoming chapter on Geometric Optics and Image Formation.

    Figure a is a drawing of a  girl standing in front of a mirror and looking at her image. The mirror is about half as tall as the girl, with the top of the mirror above her eyes but below the top of her head.  The light rays from her feet reach the bottom of the mirror and reflect to her eyes following the law of reflection: the angle of incidence theta is equal to the angle of reflection theta. The rays from the top of her head reach the top of the mirror and reflect to her eyes. Figure b is a drawing of the same girl looking at her twin. The twin is facing her and is at the same location, relative to her, that her image is in figure a. The rays from the twin’s feet and head travel directly to the girl’s eyes, reaching them in the same direction as the reflected rays in figure a.
    Figure \(\PageIndex{4}\): (a) Your image in a mirror is behind the mirror. The two rays shown are those that strike the mirror at just the correct angles to be reflected into the eyes of the person. The image appears to be behind the mirror at the same distance away as (b) if you were looking at your twin directly, with no mirror.

    Corner Reflectors (Retroreflectors)

    A light ray that strikes an object consisting of two mutually perpendicular reflecting surfaces is reflected back exactly parallel to the direction from which it came (Figure \(\PageIndex{5}\)). This is true whenever the reflecting surfaces are perpendicular, and it is independent of the angle of incidence. Such an object is called a corner reflector, since the light bounces from its inside corner. Corner reflectors are a subclass of retroreflectors, which all reflect rays back in the directions from which they came. Although the geometry of the proof is much more complex, corner reflectors can also be built with three mutually perpendicular reflecting surfaces and are useful in three-dimensional applications.

    Two mirrors meet each other at a right angle. An incoming ray of light is reflected by one mirror and then the other, such that the outgoing ray is parallel to the incoming ray.
    Figure \(\PageIndex{5}\): A light ray that strikes two mutually perpendicular reflecting surfaces is reflected back exactly parallel to the direction from which it came.

    Many inexpensive reflector buttons on bicycles, cars, and warning signs have corner reflectors designed to return light in the direction from which it originated. Rather than simply reflecting light over a wide angle, retroreflection ensures high visibility if the observer and the light source are located together, such as a car’s driver and headlights. The Apollo astronauts placed a true corner reflector on the Moon (Figure \(\PageIndex{6}\)). Laser signals from Earth can be bounced from that corner reflector to measure the gradually increasing distance to the Moon of a few centimeters per year.

    Figure a is a photograph of an astronaut placing a corner reflector on the moon. Figure b is a photograph of two bicycle safety reflectors.
    Figure \(\PageIndex{6}\): (a) Astronauts placed a corner reflector on the Moon to measure its gradually increasing orbital distance. (b) The bright spots on these bicycle safety reflectors are reflections of the flash of the camera that took this picture on a dark night. (credit a: modification of work by NASA; credit b: modification of work by “Julo”/Wikimedia Commons)

    Working on the same principle as these optical reflectors, corner reflectors are routinely used as radar reflectors (Figure \(\PageIndex{7}\)) for radio-frequency applications. Under most circumstances, small boats made of fiberglass or wood do not strongly reflect radio waves emitted by radar systems. To make these boats visible to radar (to avoid collisions, for example), radar reflectors are attached to boats, usually in high places.

    A photograph of a radar reflector on the rigging of a sailboat.
    Figure \(\PageIndex{7}\): A radar reflector hoisted on a sailboat is a type of corner reflector. (credit: Tim Sheerman-Chase)

    As a counterexample, if you are interested in building a stealth airplane, radar reflections should be minimized to evade detection. One of the design considerations would then be to avoid building 90°90° corners into the airframe.

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