The Digital Camera: Reflection

According to Greek mythology, the goddess Nemesis tricked the handsome hunter Narcissus into falling in love with the beauty of his own reflection. Narcissus did not recognize that his reflection was simply an image, and so he gazed at his own likeness until he died.

Carvaggio’s depiction of Narcissus’s fatal gaze at his own reflection.

Although not as handsome as Narcissus, the Greek inventor Archimedes of Syracuse did understand that light reflects from shiny surfaces, and that — under the right circumstances — those reflections can fool us by projecting realistic virtual images.  He also understood the focusing power of mirrors, and, according to ancient legends, Archimedes designed a parabolic arrangement of mirrors that focused several reflections from the sun onto one location, which, in the process, created a ray of heat capable of setting a ship on fire. The Archimedes heat ray legend may or may not be true, but the principle of reflection on which the legend is based is valid.

The basic rule for the reflection of light is simple: when a light ray falls upon a shiny surface like a mirror it reflects so that the angle of reflection is equal to the angle of incidence. To keep track of the angles that are involved we typically use the Greek letter  (pronounced “theta”) with a subscript i to denote the angle of incidence, and the Greek letter θ with a subscript r to denote the angle of reflectance. We say that

\theta_r = \theta_i,

and this means, for example, that a light ray that strikes the mirror at an angle of 10 degrees will reflect at an angle of 10 degrees.

The law of reflection: the angle of reflection is equal to the angle of incidence.

Every time we use a mirror as an aid in combing our hair or brushing our teeth, we react to — and subconsciously adjust for — the consequences of this rule. But this important law of optics is also used in more deliberate ways by optical engineers when they design sophisticated mirrors for state-of-the-art telescopes that enable astronomers to observe and study distant planets, stars, and galaxies.

As they did with Narcissus, the reflections from a mirror can fool us into believing that an object is behind a mirror when it is actually in front. This deception is due to the fact that the reflected rays are identical to those that would originate from a source that is located behind the mirror. Optical scientists call this apparent source of light behind the mirror a virtual image, alluding to the fact that the image does not really exist, but the rays we see are virtually the same as those that would come from this image if it did. Microscopes, telescopes, and binoculars are other optical instruments that form virtual images.

The reflections from a mirror bend the light rays in a way that can make us think an object is behind a mirror when it is actually in front. The image we see behind the mirror is called a virtual image.

The law of reflection holds for curved mirrors exactly as it does for flat mirrors. For curved mirrors, though, we must be more careful when we specify the angles of incidence and reflection. To do this, we define a direction that is normal to the mirror at each position on its surface. The normal direction is perpendicular to the surface at the particular position, and extends from the surface in a way that makes the shape of a ’T’. Any place that a light ray strikes the mirror the reflected ray will deviate from the normal direction by exactly the same angle as the incident ray. With curved mirrors, then, it is possible to make the virtual image seem farther from or closer to the mirror’s surface, and it is possible to distort the image like the ’fun house’ mirrors at carnivals and circuses.

The normal directions on the surface of a mirror area perpendicular to the surface at each particular position.

Even when the surface of a mirror has an irregular shape, each reflected ray obeys the law of reflection.

The light rays from an object that is very far away – like the sun or a distance star or planet – will appear to be parallel when observed over a limited region. If a mirror is constructed with an appropriately curved surface, it is possible to bend these parallel rays in a way that brings them all to focus in one spot. Because a mirror like this brings the light into focus and creates an image that can be recorded by film or detectors, optical scientists refer to the focused light as a real image, alluding to the fact that the image can be recorded. If the rays originate from a spot in the scene that is displaced to the right or left of center, then the rays from that spot will remain parallel, but they will slope to the right or left corresponding to the displacement of the spot. This causes the corresponding point of focus for that spot to move to the left or right, and, as a result, we can obtain faithful images of complicated scenes.

The rays from a distant source will seem to be parallel over a limited region.

A parabolic mirror bends all of the rays from a distance source so that they all cross at the same focal point.

The shape for the curvature that forms images in this way is known as a parabola, and a mirror with this curvature is know as a parabolic mirror. Nearly all large astronomical telescopes use parabolic mirrors to focus light and form images. This same principle is also used to focus radio waves and form images with a 305 meter wide parabolic antenna at the Arecibo Radio Observatory in Puerto Rico.

Reflection from mirrors is relatively simple: the angle of reflection is equal to the angle of incidence. This simple fact, though, can be used to make sophisticated imaging instruments with reflective mirrors. Digital cameras, however, use transmissive lenses to form their images, so we need to understand the more complicated situation that arises when light rays pass from air into glass, and from glass into air.

© 2011 Timothy Schulz

The Digital Camera
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David BradyJuly 15, 2011 - 2:31 pm

Beautiful start! 800 pages to go.

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