Optics

D

Aberration

Geometrical optics predicts that rays of light emanating from a point are imaged by spherical optical elements as a small blur. The outer parts of a spherical surface have a focal length different from that of the central area, and this defect causes a point to be imaged as a small circle. The difference in focal length for the various parts of the spherical section is called spherical aberration. If, instead of being a portion of a sphere, a concave mirror is a section of a paraboloid (see Parabola) of revolution, parallel rays incident on all areas of the surface are reflected to a point without spherical aberration. Combinations of convex and concave lenses can help to correct spherical aberration, but this defect cannot be eliminated from a single spherical lens for a real object and image.

The result of differences in lateral magnification for rays coming from an object point not on the optic axis is an effect called coma. If coma is present, light from a point is spread out into a family of circles that fit into a cone, and in a plane perpendicular to the optic axis the image pattern is comet-shaped. Coma may be eliminated for a single object-image point pair, but not for all such points, by a suitable choice of surfaces. Corresponding, or conjugate, object and image points, free from both spherical aberration and coma, are known as aplanatic points, and a lens having such a pair of points is called an aplanatic lens.

Astigmatism is the defect in which the light coming from an off-axis object point is spread along the direction of the optic axis. If the object is a vertical line, the cross section of the refracted beam at successively greater distances from the lens is an ellipse that collapses first into a horizontal line, spreads out again, and later becomes a vertical line. If, for a flat object, the surface of best focus is curved, the situation is described as curvature of field.Distortion arises from a variation of magnification with axial distance and is not caused by a lack of sharpness in the image.

Because the index of refraction varies with wavelength, the focal length of a lens also varies and causes longitudinal or axial chromatic aberration. Each wavelength forms an image of a slightly different size, giving rise to what is known as lateral chromatic aberration. Combinations of converging and diverging lenses, and of components made of glasses with different dispersions, help to minimize chromatic aberration. Mirrors are free of this defect. In general, achromatic lens combinations are corrected for chromatic aberration for two or three colours.

IV

Physical Optics

This branch of optical science is concerned with such aspects of the behaviour of light as its emission, composition, and absorption, and with polarization, interference, and diffraction.

A

Polarization of Light

The atoms in an ordinary light source emit pulses of radiation of extremely short duration. Each pulse from a single atom is a nearly monochromatic (single-wavelength) wave train. The electric vector corresponding to the wave does not rotate about the wave’s direction of travel, but keeps the same angle, or azimuth, with respect to it. The initial azimuth can have any value. When a large number of atoms are emitting light, these azimuths are randomly distributed, the properties of the light beam are the same in all directions, and the light is said to be unpolarized. If the electric vectors for each wave all have the same azimuth angle (that is, all the transverse waves lie in the same plane), the light is plane, or linearly, polarized.

The equations that describe the behaviour of electromagnetic waves involve two sets of waves, one in which the electric vector vibrates perpendicular to the plane of incidence and the other in which it vibrates parallel to that plane. All light can be considered as having a component of its electric vector vibrating in each of these planes. There may be a constant or continually varying phase difference between the two vibrations of the component.If light is linearly polarized, for example, this phase difference becomes zero or 180°. If the phase relationship is random, but more of one component is present, the light is partially polarized. When light is scattered by dust particles, for instance, the light scattered at 90° to the original path of the beam is plane-polarized, explaining why skylight from the zenith (directly overhead) is markedly polarized.

At angles other than zero or 90° of incidence, the amount of reflection at the boundary between two media is not the same for the two components of the light. Less of the component that vibrates parallel to the plane of incidence is reflected. If light is incident on a non-absorbing medium at the so-called Brewster angle, named after the 19th-century British physicist David Brewster, the component vibrating parallel to the plane of incidence is not reflected. At this angle of incidence, the reflected ray is perpendicular to the refracted ray, and the tangent of this angle of incidence is equal to the ratio of the refractive index of the second medium to that of the first.

Certain substances are anisotropic, or display properties with different values when measured along axes in different directions. The speed of light in these materials depends on the direction in which the light travels through them. Some crystals are birefringent, or exhibit double refraction. Unless light is travelling parallel to one of the crystal’s axes of symmetry (an optic axis of the crystal), it is separated into two parts that travel with different speeds. A uniaxial crystal has one axis. The component with the electric vector vibrating in a plane containing the optic axis is the ordinary ray; its speed is the same in all directions through the crystal, and Snell’s law of refraction holds. The component vibrating perpendicular to the plane containing the optic axis forms the extraordinary ray, and the speed of this ray depends on its direction through the crystal. If the ordinary ray travels faster than the extraordinary ray, the birefringence is positive; otherwise the birefringence is negative.

If a crystal is biaxial, no component exists for which the speed is independent of the direction of travel. Birefringent materials can be cut and shaped to introduce specific phase differences between two sets of polarized waves, to separate them, or to analyse the state of polarization of any incident light. A polarizer transmits only one component of vibration, either by reflecting the other away by means of properly cut prism combinations or by absorbing it. A material that preferentially absorbs one component of vibration is said to exhibit dichroism, and Polaroid is an example of this. Polaroid consists of many small dichroic crystals embedded in plastic and identically oriented. If the incident light is unpolarized, Polaroid absorbs approximately half of it. Glare from a large flat surface such as water or a wet road consists of partially polarized light, and properly oriented Polaroid can absorb more than half of it. This explains the effectiveness of Polaroid sunglasses.

The so-called analyser may be physically the same as a polarizer. If a polarizer and analyser are crossed, the analyser is oriented to allow transmission of vibrations lying in a plane perpendicular to those transmitted by the polarizer, and therefore blocks the light passed by the polarizer.

Substances that are optically active rotate the plane of linearly polarized light. A sugar crystal or a solution of sugar, for example, may be optically active. If a solution of sugar is placed between a crossed polarizer and analyser, some of the light is able to pass through. The amount of rotation of the analyser required to restore extinction of the light determines the concentration of the solution. The polarimeter is based on this principle.

Some substances, such as glass and plastic, that are not normally doubly refracting may become so if subjected to stress. If such stressed materials are placed between a polarizer and analyser, the bright and dark coloured areas that are seen give information about the strains. The technology of photoelasticity is based on double refraction produced by stresses.

Birefringence can also be introduced in otherwise homogeneous materials by magnetic and electric fields. A strong magnetic field across a liquid may cause it to become doubly refracting, a phenomenon known as the Kerr effect, after the 19th-century Scottish physicist John Kerr. If an appropriate material is placed between a crossed polarizer and analyser, light may be transmitted, depending on whether the electric field is on or off. This can act as a very rapid light switch or modulator.

B

Interference and Diffraction

When two light beams cross, they may interfere in such a way that the resultant intensity pattern is affected (see Interference). The coherence of two beams is the extent to which their waves are in phase. If the phase relationship changes rapidly and randomly, the beams are incoherent. If two wave trains are coherent and if the maximum of one wave coincides with the maximum of another, the two waves combine to produce a greater intensity in that place than if the two beams were present but not coherent. If they are coherent and the maximum of one wave coincides with the minimum of the other, the two waves will cancel each other in part or completely, thus decreasing the intensity. An interference pattern consisting of dark and bright fringes may be formed. To produce a steady interference pattern the two wave trains must be polarized in the same plane. Atoms in an ordinary light source radiate independently, so a large light source usually emits incoherent radiation. To obtain coherent light from such a source, a small portion of the light is selected by means of a pinhole or slit. If this portion is then again split by double slits, double mirrors, or double prisms, and the two parts are made to travel along paths that differ in length (though not by too much) before they are combined again, an interference pattern results. Devices that do this are called interferometers; they are used in measuring small angles such as the apparent diameters of stars, or small distances such as the deviations of an optical surface from the required shape, in terms of numbers of wavelengths of light. Such an interference pattern was first demonstrated by the British physicist Thomas Young in the experiment illustrated in Fig. 1. Light that had passed through one pinhole illuminated an opaque surface that contained two pinholes. The light that passed through the two pinholes formed a pattern of alternately bright and dark circular fringes on a screen. Wavelets are drawn in the illustration to show that at points such as A, C, and E (intersection of solid line with solid line) the waves from the two pinholes arrive in phase and combine to increase the intensity. At other points, such as B and D (intersection of solid line with dashed line), the waves are 180° out of phase and cancel each other.

Light waves reflected from the two surfaces of an extremely thin transparent film on a smooth surface can interfere with each other. The rainbow colours of a film of oil on water are a result of interference, and they demonstrate the importance of the ratio of film thickness to wavelength. A single film or several films of different material can be used to increase or decrease the reflectance of a surface. Dichroic beam splitters are stacks of films of more than one material, controlled in thickness so that one band of wavelengths is reflected and another is transmitted. An interference filter made of such films transmits an extremely narrow band of wavelengths and reflects the remainder. The shape of the surface of an optical element can be checked by touching it to a master lens, or flat, and observing the fringe pattern formed because of the thin layer of air remaining between the two surfaces.

Light incident on the edge of an obstacle is bent or diffracted, and the obstacle does not form a sharp geometric shadow. The points on the edge of the obstacle act as a source of coherent waves, and a pattern of interference fringes, called a diffraction pattern, is formed. The shape of the edge of the obstacle is not exactly reproduced because part of the wave front is cut off.

Because light passes through a finite aperture when it goes through a lens, a diffraction pattern is formed around the image of an object. If the object is extremely small, the diffraction pattern appears as a series of concentric bright and dark rings around a central disc called the Airy disc, after the 19th-century English astronomer George Biddell Airy. This is true even for an aberration-free lens. If two particles are so close together that the two diffraction patterns overlap and the bright rings of one fall on the dark rings of the second, the two particles cannot be resolved (distinguished). The 19th-century German physicist Ernst Karl Abbe first explained image formation by a microscope with a theory based on the interference of diffraction patterns of various points on the object.

Fourier analysis is a mathematical treatment, named after the French mathematician Jean Fourier, that represents an optical object as a sum of simple sine waves, called components. Optical systems are sometimes evaluated by choosing an object of known Fourier components and evaluating the Fourier components present in the image. Such procedures measure what is called the optical transfer function. Extrapolations of these techniques sometimes allow extraction of information from poor images. Statistical theories have also been included in analyses of the recording of images.

A diffraction grating consists of several thousand slits that are equal in width and equally spaced (formed by ruling lines on glass or metal with a fine diamond point). Each slit gives rise to a diffraction pattern, and the many diffraction patterns interfere. Bright fringes are formed in different places for different wavelengths. If white light is incident, a continuous spectrum is formed. Prisms and gratings are used in instruments such as monochromators, spectrographs, or spectrophotometers to provide nearly monochromatic light or to analyse the wavelengths present in the incident light (see Spectroscopy; Spectroheliograph).

C

Stimulated Emission

The atoms in common light sources, such as the incandescent lamp, fluorescent lamp, and neon lamp, produce light by spontaneous emission, and the radiation is incoherent. If a sufficient number of atoms absorb energy so that they are excited into appropriate states of higher energy, stimulated emission can occur. Light of a certain wavelength can produce additional light that has the same phase and direction as the original wavelength, and it will be coherent. Stimulated emission amplifies the amount of radiation having a given wavelength, and this radiation has a very narrow beam spread. The material that is excited may be a gas, a solid, or a liquid, but it must be contained or shaped to form an interferometer in which the wavelength being amplified is reflected back and forth many times. A small fraction of the excited radiation is transmitted by one of the mirrors of the interferometers. This device is called a laser, an acronym for “light amplification by stimulated emission of radiation”. Energizing a large number of atoms to be in the appropriate upper state is called pumping. Pumping may be optical or electrical. Because lasers can be made to emit pulses of extremely high energy that have a very narrow beam spread, laser light sent to the Moon and reflected back to the Earth can be detected. The intense narrow beam of the laser has found practical application in surgery and in the cutting of metals.

Holography, the technique of producing three-dimensional images, was made possible by a technique pioneered by the Hungarian-born British physicist and electrical engineer Dennis Gabor. He noted that if the diffraction pattern of an object could be recorded and the phase information also retained, the image of the object could be reconstructed by coherent illumination of the recorded diffraction pattern. Illuminating it with a wavelength longer than that used to produce the diffraction pattern would result in magnification. Because the absolute phase of a light wave cannot be directly detected physically, it was necessary to provide a reference beam coherent with the beam illuminating the object to interfere with the diffraction pattern and provide phase information. Before the development of the laser, the Gabor scheme was limited by the lack of sufficiently intense coherent light sources.

A hologram is a photographic record of the interference between a reference beam and the diffraction pattern of the object. Light from a single laser is separated into two beams. The reference beam illuminates the photographic plate, perhaps via a lens and mirror, and the second beam illuminates the object. The reference beam and the light reflected from the object jointly form a diffraction pattern on the photographic plate. If the processed hologram is illuminated by coherent light, not necessarily of the same wavelength that was used to make the hologram, a three-dimensional image of the object can be obtained. Holograms of a theoretical object can be produced by computer, and the images of these objects can then be reconstructed.

Intense, coherent laser beams permit the study of new optical effects that are produced by the interaction of certain substances with electric fields and that depend on the square or the third power of the field strength. This is called non-linear optics, and the interactions being studied affect the refractive index of the substances. The Kerr effect, mentioned earlier, belongs to this group of phenomena.

Harmonic generation of light has been observed. Infrared laser light of wavelength 1.06 micrometres, for example, can be changed to green light with a wavelength of 0.53 micrometres in a crystal of barium sodium niobate. Broadly tunable sources of coherent light in the visible and near infrared ranges can be produced by pumping suitable media with light or shorter wavelengths. A lithium niobate crystal can be made to fluoresce in red, yellow, and green by pumping it with greenish-blue laser light having a wavelength of 488 nanometres (19.2 millionths of an inch). Certain scattering phenomena can be stimulated by a single laser to produce intense, pulsed light at a wide range of monochromatic wavelengths. One of the phenomena observed in high-power optical experiments is a self-focusing effect that produces extremely short-lived filaments as small as 5 micrometres (200 millionths of an inch) in diameter. Non-linear optical effects are applied in developing efficient broadband modulators for communication systems (see Radio: Modulation).