Aberrations

FIELDS OF STUDY: Optics; Relativity; Quantum Physics

ABSTRACT: Aberrations in the motion of light and other wave phenomena produce distortions in the perception of those phenomena. Waves moving at low velocities are described by Newton’s laws of motion. Waves moving at relativistic velocities are described using relativity theory and the Lorentz transformation.

PRINCIPAL TERMS

• chromatic aberration: a distortion created by the separation of white light into its component wavelengths when passing from one medium into another, such as through a lens.
• coma: the extended geometric image formed along the optical axis of a lens by light entering the lens obliquely.
• inertial reference frame: a means of describing relative motions through space according to Newtonian mechanics.
• Lorentz transformation: a means of describing relative motions through space according to the mathematics of relativity.
• refraction: a change in the direction of light due to the different speeds of light when passing through various media.
• relativistic beaming: the effect in which a luminous beam appears brightest when pointing directly at an observer.
• special relativity: the theory that physical laws are constant for matter with a uniform motion.
• spherical aberration: the blurring or distortion of an image produced by a spherical lens or mirror due to differences in the refraction of light that enters the optical system along the optical axis and light that enters closer to the edge of the lens.

Aberrations

The image from a lens is produced by refraction of light rays passing from one medium, such as air, into another medium, such as glass. A convex lens refracts light rays in such a way that they converge at the focal length of the lens. The focal length is determined by the radius of curvature of the lens and the lens material’s index of refraction. The focal length describes how strongly a lens converges or diverges light. Errors in images produced by lenses and other optical systems are called aberrations. Some common lens aberrations include coma, chromatic aberration, and spherical aberration.

Coma results when light rays enter a lens at an oblique angle. Coma creates an elongated image rather than a clear image at the focal length of the lens. A simple example of coma can be seen using an ordinary magnifying glass and holding it at an angle to a point light source such as the sun or a light bulb. A coma produces several images of various sizes. Rather than a clear image of the light source, the image will appear to resemble a comet, with a distinct head and a blurred tail spreading out behind it. This aberration occurs because the magnification of a lens is different for light that enters the lens near its center and light that enters near the edge.

Chromatic aberration results when light rays pass through two different media, such as air and the lens. Each medium has a specific index of refraction. The index of refraction is the ratio of the speed of light in a vacuum to its speed in a material. Light is diffracted by the medium in relation to its wavelength. As light passes through a lens, each wavelength of light changes direction by a slightly different amount. For example, sunlight that passes through a prism and becomes separated into bands of colored light. Chromatic aberration occurs when light entering a lens is diffracted, producing colored outlines, halos, and rings around the image. This is often seen in low-cost optical devices of poor quality, but it is also a major concern for even advanced optical telescopes and microscopes.

Spherical aberration is a problem that occurs in optical systems with spherical lenses or mirrors. For a curved surface, the reflected or refracted rays from each radial location on the surface converge along a central axis. However, the spherical geometry requires them to converge at different foci along that axis. The clearest image from a spherical lens is found at a point that is effectively the average value along the central axis. This point is termed the circle of least confusion.

Distortions

Light and sound depend on the movement of waves. Sound waves require a physical medium such as air to propagate. On the other hand, light propagates as electromagnetic waves that do not require a physical medium. They can thus travel easily through a vacuum. Each light wave has a specific wavelength and corresponding frequency that defines its essential properties. Any aberration, or change in the motion of the light waves, causes a distortion of the light as perceived by an observer. Aberrations and distortions are most commonly associated with lenses and mirrors, such as those used in powerful telescopes and similar optical devices.

Light coming through space from distant stars undergoes aberration due to the relative motion of Earth as it moves through its orbit around the sun. The sun itself is also moving through space relative to the other stars and galaxies in the universe. This relative motion can cause observers to see the stars as though they are ahead of their actual positions. The effect is similar to how a person sees raindrops through the window of a car. When the car is stationary, the raindrops are seen to fall straight down, but when the car is moving, the raindrops appear to be falling on an angle. The faster the car moves in relation to the raindrops, the steeper is the angle at which the raindrops appear to be falling.

Frames of Reference

The stationary car in the example above provides the view from an inertial reference frame. An inertial reference frame must be moving at a constant speed relative to the observed phenomena. The direction of the raindrops seen when the car is stationary can be described by the laws of motion English physicist Isaac Newton (1642–1727) formulated. However, when the car is moving, any mathematical description of the raindrops must account for the effect of the relative motion of the car and the raindrops. Similarly, the motion of light through space relative to Earth must be accounted for mathematically. This is something that Newton’s laws of motion are not capable of describing. Instead, this is a job for the mathematics of relativity.

Relativity

Newtonian mechanics, which describes motion and energy relationships, can account for the motion of objects with nonrelativistic velocities. The simple equations describe force as the product of mass and acceleration, and momentum as the product of mass and velocity. These equations are quite sufficient for objects that are moving very slowly relative to the speed of light. However, as velocity increases, the ability of Newtonian mechanics to accurately describe motion decreases. Therefore, one must use the Lorentz transformation to accurately describe its properties. The Lorentz transformation accounts for the differences in observations of the same phenomenon from two inertial reference frames that are moving at constant velocities with respect to one another. Any phenomenon that two observers want to describe can be converted from one reference frame to the other using the Lorentz transformation. In the theory of special relativity, formulated by German-born scientist Albert Einstein (1879–1955) in 1905, two assumptions are made. For one, it is held that the laws of physics are constant in all inertial reference frames. The second and most important assumption is that the velocity of light in a vacuum is constant regardless of the velocity of the light source. Because of this, an observer always sees light traveling at the same speed regardless of their reference frame.

This is a difficult concept to understand, because it is counterintuitive to the perception of human-scale motion governed by Newtonian mechanics. In everyday life, the motions of wave phenomena such as sound and water are seen to be affected by the motion of the source of the waves. Sound increases in pitch when the sound source is moving toward the observer and decreases when it is moving away. This is called the Doppler effect. What is observed for light sources in motion, such as stars, is the red shift and blue shift of the wavelength that they emit. A light source that is red-shifted appears to have a longer wavelength because it is moving away from the observer. A blue-shifted light source appears to have a shorter wavelength because it is moving toward the observer.

This perception masks the fact that the velocity of sound through some conductive medium is not affected by the motion of the medium. Similarly, the velocity of light through space or some other transparent medium is not affected by the relative motion of the medium and is thus constant. If the velocity of light is not affected by the motion of the medium that it travels through, then the velocity of light from a source moving through space is not affected by the motion of the source. For luminous beams traveling at relativistic velocities, such as those emitted by certain stars, an effect called relativistic beaming can be observed. The beam and its apparent source will appear brightest when the beam is oriented directly toward the observer. This is sometimes referred to as the headlight effect because the headlights of a vehicle moving toward an observer appear brightest when pointed directly at the observer but are barely visible from other angles.

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