Optics

Electromagnetic Waves

Electromagnetic Waves

Text Box: Introduction Optics, branch of physical science dealing with the propagation and behaviour of light. In a general sense, light is that part of the spectrum of electromagnetic radiation that extends from X-rays to microwaves and includes the radiant energy that produces the sensation of vision. The study of optics is divided into geometrical optics and physical optics, and these branches are discussed below. NATURE OF LIGHT Radiant energy has a dual nature and obeys laws that may be explained either in terms of a stream of particles, or packets of energy, called photons, or in terms of a train of transverse waves (see Wave Motion). The concept of photons is used to explain the interactions of light and matter that result in a change in the form of energy, as in the case of the photoelectric effect or luminescence. The concept of waves is usually used to explain the propagation of light and some of the phenomena of image formation. In light waves as in other types of electromagnetic wave, there are rapidly fluctuating electric and magnetic fields at each point in space. Since they have both direction and magnitude, the fields are vector quantities. The electric and magnetic fields are at right angles to each other and to the direction of movement of the wave. The simplest sort of light wave is a pure sine wave, so called because a graph of the electric or magnetic field intensity drawn along the direction of travel at any moment would trace out a sine curve. The number of complete oscillations, or vibrations, per second of a point on the light wave is known as the frequency. The wavelength is the distance parallel to the axis between two points of the same phase—that is, points occupying equivalent positions on the wave. For example, the wavelength equals the distance from maximum to maximum or from minimum to minimum of the sine wave. In the visible spectrum differences in wavelength manifest themselves as differences in colour. The visible range extends from about 350 nanometres (violet) to 750 nanometres (red), a nanometre being equal to a billionth of a metre, or 4 × 10-8 in. White light is a mixture of the visible wavelengths. No sharp boundaries exist between wavelength regions, but 10 nanometres may be taken as the low-wavelength limit for ultraviolet radiation. Infrared radiation, which includes radiant heat energy, spans the wavelengths from about 700 nanometres to approximately 1 millimetre. The speed of an electromagnetic wave is the product of the frequency and the wavelength. In a vacuum this speed is the same for all wavelengths. The speed of light in material substances is less than in a vacuum, and is different for different wavelengths, an effect called dispersion. The ratio of the speed of light in vacuum to the speed of a particular wavelength of light in a substance is known as the index of refraction of that substance for the given wavelength. The index of refraction of air for all wavelengths is 1.00029, but for most applications it is sufficiently accurate to take it to be 1. The laws of reflection and refraction of light are usually derived using the wave theory of light introduced by the 17th-century Dutch mathematician, astronomer, and physical scientist Christiaan Huygens. Huygens’ principle states that every point on an initial wave front may be considered as the source of small, secondary spherical wavelets that spread out in all directions from their centres with the same speed, frequency, and wavelength as the parent wave front. A new wave front can be defined, encompassing the wavelets. Since the light progresses at right angles to this wave front, changes in the direction of the light can be worked out using Huygens’ principle. When the wavelets encounter another medium or object, each point on the boundary becomes a source of two new sets of waves. The reflected set travels back into the first medium, and the refracted set enters the second medium. The behaviour of the reflected and refracted rays can be explained by Huygens’ principle. It is simpler and sometimes sufficient to represent the propagation of light by rays rather than by waves. The ray is the flow line, or direction of travel, of radiant energy. In geometrical optics the wave theory of light is ignored and the assumption is made that light does not bend round corners. This approximation is valid when lenses, apertures, and so on are large in comparison with the wavelength of the light. Rays are traced through an optical system by applying the laws of reflection and refraction. Geometric Optics This area of optical science concerns the application of laws of reflection and refraction of light in the design of lenses (see Lenses below) and other optical components of instruments. Reflection and Refraction If a light ray that is travelling through a homogeneous medium is incident on the surface of a second homogeneous medium, part of the light is reflected and part may enter the second medium as the refracted ray, and may or may not undergo absorption there. The amount of light reflected depends on the ratio of the refractive indexes for the two media. The plane of incidence is defined as the plane containing the incident ray and the normal (that is, the line perpendicular to the surface) at the point of incidence. The angle of incidence is the angle between the incident ray and this normal. The angles of reflection and refraction are defined correspondingly. The laws of reflection state that the angle of incidence is equal to the angle of reflection and that the incident ray, the reflected ray, and the normal at the point of incidence all lie in the same plane. If the surface of the second medium is smooth it may act as a mirror and produce a reflected image. A light ray from an object striking a flat, or plane, mirror will be reflected away from the surface. To an observer in front of the mirror, the reflected ray appears to have come from a point behind the mirror that is a continuation of that reflected ray. The image of the object appears to lie as far behind the mirror as the object lies in front of it. If the surface of the second medium is rough, then normal’s to various points of the surface lie in random directions. In that case, rays that may lie in the same plane when they emerge from a point source nevertheless lie in random planes of incidence, and therefore of reflection, so are scattered and cannot form an image. Snell’s Law This important law, named after the Dutch mathematician Willebrord van Roijen Snell, states that the product of the refractive index and the sine of the angle of incidence of a ray in one medium is equal to the product of the refractive index and the sine of the angle of refraction in a successive medium. Algebraically, this can be written n1 sinθ1 = n2 sinθ2, where n1, n2 are the two values of refractive index, and θ1, θ2 are the angles of incidence and refraction. The incident ray, the refracted ray, and the normal to the boundary at the point of incidence all lie in the same plane. Generally, the refractive index of a denser transparent substance is higher than that of a less dense material; that is, the speed of light is lower in the denser substance. So if a ray is incident obliquely, then a ray entering a medium with a higher refractive index is bent towards the normal, and a ray entering a medium of lower refractive index is deviated away from the normal. Rays incident along the normal are reflected and refracted along the normal. To an observer in a less dense medium such as air, an object in a denser medium appears to lie closer to the boundary than is the actual case. A common example is that of an object lying underwater and observed from above the water. An oblique ray from the object is bent away from the normal towards the position of the observer. The object, therefore, appears to lie slightly away from its true position and at a point where a straight from the observer intersects a line normal to the surface of the water and passing through the object. In the case of light passing through more than two media with parallel boundaries, another effect occurs. If the refractive index of the first and last medium is the same, but different from the that of the intermediate medium, the ray emerges parallel to the incident ray, but is displaced laterally. If light passes through a prism, a transparent object with flat, surfaces and a uniform cross-section, the exit ray is no longer parallel to the incident ray. Because the refractive index of a substance varies for the different wavelengths, a prism can spread out the various wavelengths of light contained in an incident beam and form a spectrum. In this, the angle between the path of the incident ray and the path of the emergent ray is the angle of deviation. It can be shown that when the angle of incidence is such that it is equal to the angle made by the emergent ray, the deviation is at a minimum. The refractive index of the prism can be calculated by measuring the angle of minimum deviation and the angle between the faces of the prism. Critical Angle Given that a ray is bent away from the normal when it enters a less dense medium, and that the deviation from the normal increases as the angle of incidence increases, an angle of incidence exists, known as the critical angle, such that the refracted ray makes an angle of 90° with the normal and travels along the boundary between the two media. If the angle of incidence is increased beyond the critical angle, the light rays will be totally reflected. Total reflection cannot occur if light is travelling from a less dense medium to a denser one. In recent years, a new, practical application of total reflection has been found in fiber optics. If light enters a solid glass or plastic tube at one end, it can be totally reflected at the boundary of the tube and, after a number of successive total reflections, emerge from the other end. Glass fibers can be drawn to a very small diameter, coated with a material of lower refractive index, and then assembled into flexible bundles or fused into plates of fibers that are used to transmit images. The flexible bundles, which can be used to provide illumination as well as to transmit images, are valuable in medical examination, as they can be passed along narrow passages or even blood vessels. Spherical and Non-spherical Surfaces Most of the traditional terminology of geometrical optics was developed with reference to spherical reflecting and refracting surfaces. Non-spherical surfaces, however, are sometimes involved. The optic axis is a reference line that is an axis of symmetry. The optic axis passes through the centre of a spherical lens or mirror and through its centre of curvature. If a narrow beam of rays travelling along the optic axis is incident on the spherical surface of a mirror or a thin lens, the rays are reflected or refracted so that they intersect or appear to intersect at a point on the optic axis. The distance between this point and the mirror or lens is the focal length. If a lens is thick, calculations are made with reference to planes called principal planes, rather than to the surface of the lens. A lens may have two focal lengths, if the surfaces are not alike, depending on which surface the light strikes first. If an object is at the focal point, the rays emerging from it are parallel to the optic axis after reflection or refraction. If rays are converged by a lens or mirror so that they intersect in front of it, the image is real and inverted (upside down). If the rays diverge after reflection or refraction so that they only appear to come from a point through which they have not actually passed, the image is erect and is described as virtual. The ratio of the height of the image to the height of the object is the lateral magnification. If it is understood that distances measured from the surface of a lens or mirror to objects or to real images are positive and distances measured to virtual images are negative, then, if u is the object distance, v is the image distance, and f is the focal length of a mirror or of a thin lens, the equation 1/v + 1/u = 1/f applies to spherical mirrors and spherical lenses. If a simple lens has surfaces with radii r1 and r2, and the ratio of its refractive index to that of the medium surrounding it is n, then 1/f = (n - 1) (1/r1 + 1/r2) The radii r1, r2 are taken to be positive or negative, depending on whether the surfaces are convex or concave, respectively. The focal length of a spherical mirror is equal to half the radius of curvature. A narrow beam of rays travelling along the optic axis and incident on a concave mirror is reflected so that it intersects the radius at the focal point, or principal focus, halfway between the pole, or centre, of the mirror's surface and the mirror's centre of curvature. If the object distance is greater than the distance between the pole and the centre of curvature, the image is real, inverted, and diminished. If the object lies between the centre of curvature and the focal point, the image is real, inverted, and enlarged. If the object is located between the surface of the mirror and the focus, the image is virtual, upright, and enlarged. A convex mirror forms only virtual, erect, and diminished images, unless the mirror is used in conjunction with other optical components. Lenses Magnifying Glass A magnifying glass is a large convex lens commonly used to examine small objects. The lens bends incoming light so that an enlarged, virtual image of the object (in this case a mushroom) appears beyond it. The image is called virtual because the light rays that appear to come from it do not actually pass through it. A virtual image cannot be projected on a screen. Lenses with surfaces of small radii have short focal lengths. A lens with two convex surfaces will always refract rays that are originally parallel to the optic axis so that they converge to a focus on the side of the lens opposite to the object. A concave lens surface will deviate incident rays that are originally parallel to the axis away from the axis. Unless the second surface of the lens is convex and more strongly curved than the first surface, the rays diverge and appear to come from a point on the same side of the lens as the object. Such lenses form only virtual, erect, and diminished images. If the object distance is greater than the focal length, a converging lens forms a real and inverted image. If the object is sufficiently far away, the image is smaller than the object. If the object distance is smaller than the focal length of this lens, the image is virtual, upright, and larger than the object. The observer is then using the lens as a magnifier or simple microscope. The angle subtended at the eye by this virtual enlarged image (that is, its apparent angular size) is greater than would be the angle subtended by the object if it were at the normal viewing distance. The ratio of these two angles is the magnifying power of the lens. A lens with a shorter focal length would form a virtual image subtending a greater angle and would therefore have a greater magnifying power. The magnifying power of an instrument is a measure of its ability to make the object seem closer to the eye. This is distinct from the lateral magnification of a camera (see Photographic Techniques) or telescope, for example, where the ratio of the actual dimensions of a real image to those of the object increases as the focal length increases. The amount of light a lens can admit increases with its diameter. Because the area occupied by an image is proportional to the square of the focal length of the lens, the light intensity over the image area is directly proportional to the diameter of the lens and inversely proportional to the square of the focal length. The image produced by a lens of diameter 3 cm and focal length 20 cm would be one-quarter as bright as the image formed by a lens of the same diameter and focal length 10 cm. The ratio of the focal length to the effective diameter of a lens is its focal ratio, the so-called f-number. The reciprocal of this ratio is called the relative aperture. Lenses having the same relative aperture have the same light-gathering power, regardless of the actual diameters and focal lengths. 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 parabolic 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. 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. 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 unpolarised. 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 analyze 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 unpolarised, 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 analyzer may be physically the same as a polarizer. If a polarizer and analyzer are crossed, the analyzer 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. 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).