光学宝典

1 General Principles 1.1 The Electromagnetic Spectrum This book deals with certain phenomena associated with a relatively narrow slice of the electromagnetic spectrum. Optics is often defined as being concerned with radiation visible to the human eye; however, in view of the recent expansion of optical applications in the regions of the spectrum on either side of the visible region, it seems not only pru- dent, but necessary, to include certain aspects of the infrared and ultraviolet regions in our discussions. The known electromagnetic spectrum is diagramed in Fig. 1.1 and ranges from cosmic rays to radio waves. All the electromagnetic radi- ations transport energy and all have a common velocity in vacuum of c � 2.998 � 1010 cm/s. In other respects, however, the nature of the radiation varies widely, as might be expected from the tremendous range of wavelengths represented. At the short end of the spectrum we find gamma radiation with wavelengths extending below a billionth of a micron (one micron or micrometer � 1 �m � 10�6 m) and at the long end, radio waves with wavelengths measurable in miles. At the short end of the spectrum, electromagnetic radiation tends to be quite parti- clelike in its behavior, whereas toward the long wavelength end the behavior is mostly wavelike. Since the optical portion of the spectrum occupies an intermediate position, it is not surprising that optical radi- ation exhibits both wave and particle behavior. The visible portion of this spectrum (Fig. 1.2) takes up less than one octave, ranging from violet light with a wavelength of 0.4 �m to red light with a wavelength of 0.76 �m. Beyond the red end of the spec- trum lies the infrared region, which blends into the microwave region Chapter 1 at a wavelength of about one millimeter. The ultraviolet region extends from the lower end of the visible spectrum to a wavelength of about 0.01 �m at the beginning of the x-ray region. The wavelengths associated with the colors seen by the eye are indicated in Fig. 1.2. The ordinary units of wavelength measure in the optical region are the angstrom (Å); the millimicron (m�), or nanometer (nm); and the micrometer (�m), or micron (�). One micron is a millionth of a meter, a millimicron is a thousandth of a micron, and an angstrom is one ten- thousandth of a micron (see Table 1.1). Thus, 1.0 Å � 0.1 nm � 10�4 �m. The frequency equals the velocity c divided by the wavelength, and the wavenumber is the reciprocal of the wavelength, with the usual dimen- sion of cm�1. 1.2 Light Wave Propagation If we consider light waves radiating from a point source in a vacuum as shown in Fig. 1.3, it is apparent that at a given instant each wave front is spherical in shape, with the curvature (reciprocal of the radius) decreasing as the wave front travels away from the point source. At a sufficient distance from the source the radius of the wave front may be regarded as infinite. Such a wave front is called a plane wave. 2 Chapter One Figure 1.1 The electromagnetic spectrum. The distance between successive waves is of course the wavelength of the radiation. The velocity of propagation of light waves in vacuum is approximately 3 � 1010 cm/s. In other media the velocity is less than in vacuum. In ordinary glass, for example, the velocity is about two- thirds of the velocity in free space. The ratio of the velocity in vacuum to the velocity in a medium is called the index of refraction of that medium, denoted by the letter n. Index of refraction n � (1.1) Both wavelength and velocity are reduced by a factor of the index; the frequency remains constant. velocity in vacuum ��� velocity in medium General Principles 3 Figure 1.2 The “optical” portion of the electromagnetic spectrum. TABLE 1.1 Commonly Used Wavelength Units Centimeter � 10�2 meter Millimeter � 10�3 meter Micrometer � 10�6 meter � 10�3 millimeter Micron � 10�6 meter � 10�3 millimeter Millimicron � 10�3 micron � 1.0 nanometer � 10�6 millimeter � 10�9 meter Nanometer � 10�9 meter � 1.0 millimicron Angstrom � 10�10 meter � 0.1 nanometer Ordinary air has an index of refraction of about 1.0003, and since almost all optical work (including measurement of the index of refrac- tion) is carried out in a normal atmosphere, it is a highly convenient convention to express the index of a material relative to that of air (rather than vacuum), which is then assumed to have an index of exactly 1.0. The actual index of refraction for air at 15°C is given by (n�1) �108 � 8342.1 � � where � � 1/ ( � wavelength, in �m). At other temperatures the index may be calculated from (nt�1) � The change in index with pressure is 0.0003 per 15 lb/in2, or 0.00002/psi. If we trace the path of a hypothetical point on the surface of a wave front as it moves through space, we see that the point progresses as a straight line. The path of the point is thus what is called a ray of light. Such a light ray is an extremely convenient fiction, of great utility in understanding and analyzing the action of optical systems, and we shall devote the greater portion of this volume to the study of light rays. Note that the ray is normal to the wave front, and vice versa. The preceding discussion of wave fronts has assumed that the light waves were in a vacuum, and of course that the vacuum was isotropic, i.e., of uniform index in all directions. Several optical crystals are anisotropic; in such media wave fronts as sketched in Fig. 1.3 are not spherical. The waves travel at different velocities in different direc- tions, and thus at a given instant a wave in one direction will be fur- ther from the source than will a wave traveling in a direction for which the media has a larger index of refraction. 1.0549 (n15° � 1) ��� (1 � 0.00366t) 15,996 �� (38.9��2) 2,406,030 �� (130��2) 4 Chapter One Figure 1.3 Light waves radiat- ing from a point source in an isotropic medium take a spheri- cal form; the radius of curvature of the wave front is equal to the distance from the point source. The path of a point on the wave front is called a light ray, and in an isotropic medium is a straight line. Note also that the ray is normal to the wave front. Although most optical materials may be assumed to be isotropic, with a completely homogeneous index of refraction, there are some sig- nificant exceptions. The earth’s atmosphere at any given elevation is quite uniform in index, but when considered over a large range of alti- tudes, the index varies from about 1.0003 at sea level to 1.0 at extreme altitudes. Therefore, light rays passing through the atmosphere do not travel in exactly straight lines; they are refracted to curve toward the earth, i.e., toward the higher index. Gradient index optical glasses are deliberately fabricated to bend light rays in controlled curved paths. We shall assume homogeneous media unless specifically stated otherwise. 1.3 Snell’s Law of Refraction Let us now consider a plane wave front incident upon a plane surface separating two media, as shown in Fig. 1.4. The light is progressing from the top of the figure downward and approaches the boundary sur- face at an angle. The parallel lines represent the positions of a wave front at regular intervals of time. The index of the upper medium we shall call n1 and that of the lower n2. From Eq. 1.1, we find that the velocity in the upper medium is given by v1 � c/n1 (where c is the veloc- ity in vacuum ≈ 3 � 1010 cm/s) and in the lower by v2 � c/n2. Thus, the velocity in the upper medium is n2/n1 times the velocity in the lower, and the distance which the wave front travels in a given interval of time in the upper medium will also be n2/n1 times that in the lower. In Fig. 1.4 the index of the lower medium is assumed to be larger so that the velocity in the lower medium is less than that in the upper medium. At time t0 our wave front intersects the boundary at point A; at time t1 � t0 � t it intersects the boundary at B. During this time it has moved a distance d1 � v1 t � t (1.2a) in the upper medium, and a distance c � n1 General Principles 5 Figure 1.4 A plane wave front passing through the boundary between two media of differing indices of refraction (n2 � n1). d2 � v2 t � t (1.2b) in the lower medium. In Fig. 1.5 we have added a ray to the wave diagram; this ray is the path of the point on the wave front which passes through point B on the surface and is normal to the wave front. If the lines represent the positions of the wave at equal intervals of time, AB and BC, the dis- tances between intersections, must be equal. The angle between the wave front and the surface (I1 or I2) is equal to the angle between the ray (which is normal to the wave) and the normal to the surface XX′. Thus we have from Fig. 1.5 AB � � BC � and if we substitute the values of d1 and d2 from Eq. 1.2, we get � which, after canceling and rearranging, yields n1 sin I1 � n2 sin I2 (1.3) This expression is the basic relationship by which the passage of light rays is traced through optical systems. It is called Snell’s law after one of its discoverers. Since Snell’s law relates the sines of the angles between a light ray and the normal to the surface, it is readily applicable to surfaces other c t � n2 sin I2 c t � n1 sin I1 d2� sin I2 d1� sin I1 c � n2 6 Chapter One Figure 1.5 than the plane which we used in the example above; the path of a light ray may be calculated through any surface for which we can determine the point of intersection of the ray and the normal to the surface at that point. The angle I1 between the incident ray and surface normal is cus- tomarily referred to as the angle of incidence; the angle I2 is called the angle of refraction. For all optical media the index of refraction varies with the wave- length of light. In general the index is higher for short wavelengths than for long wavelengths. In the preceding discussion it has been assumed that the light incident on the refracting surface was mono- chromatic, i.e., composed of only one wavelength of light. Figure 1.6 shows a ray of white light broken into its various component wave- lengths by refraction at a surface. Notice that the blue light ray is bent, or refracted, through a greater angle than is the ray of red light. This is because n2 for blue light is larger than n2 for red. Since n2 sin I2 � n1 sin I1 � a constant in this case, it is apparent that if n2 is larg- er for blue light than red, then I2 must be smaller for blue than red. This variation in index with wavelength is called dispersion; when used as a differential it is written dn, otherwise dispersion is given by n � n 1 � n 2, where 1 and 2 are the wavelengths of the two colors of light for which the dispersion is given. Relative dispersion is given by n/(n � 1) and, in effect, expresses the “spread” of the colors of light as a fraction of the amount that light of a median wavelength is bent. General Principles 7 Figure 1.6 Showing the disper- sion of white light into its con- stituent colors by refraction (exaggerated for clarity). All of the light incident upon a boundary surface is not transmitted through the surface; some portion is reflected back into the incident medium. A construction similar to that used in Fig. 1.5 can be used to demonstrate that the angle between the surface normal and the reflected ray (the angle of reflection) is equal to the angle of incidence, and that the reflected ray is on the opposite side of the normal from the incident ray (as is the refracted ray). Thus, for reflection, Snell’s law takes on the form Iincident � �Ireflected (1.4) Figure 1.7 shows the relationship between a ray incident on a plane surface and the reflected and refracted rays which result. At this point it should be emphasized that the incident ray, the nor- mal, the reflected ray, and the refracted ray all lie in a common plane, called the plane of incidence, which in Fig. 1.7 is the plane of the paper. 1.4 The Action of Simple Lenses and Prisms on Wave Fronts In Fig. 1.8 a point source P is emitting light; as before, the arcs cen- tered about P represent the successive positions of a wave front at reg- ular intervals of time. The wave front is incident on a biconvex lens consisting of two surfaces of rotation bounding a medium of (in this instance) higher index of refraction than the medium in which the 8 Chapter One Figure 1.7 Relationship between a ray incident on a plane surface and the reflected and refracted rays which result. source is located. In each interval of time the wave front may be assumed to travel a distance d1 in the medium of the source; it will travel a lesser distance d2 in the medium of the lens. (As in the pre- ceding discussion, these distances are related by n1d1 � n2d2.) At some instant, the vertex of the wave front will just contact the vertex of the lens surface at point A. In the succeeding interval, the portion of the wave front inside the lens will move a distance d2, while the portion of the same wave front still outside the lens will have moved d1. As the wave front passes through the lens, this effect is repeated in reverse at the second surface. It can be seen that the wave front has been retard- ed by the medium of the lens and that this retardation has been greater in the thicker central portion of the lens, causing the curvature of the wave front to be reversed. At the left of the lens the light from P was diverging, and to the right of the lens the light is now converging in the general direction of point P′. If a screen or sheet of paper were placed at P′, a concentration of light could be observed at this point. The lens is said to have formed an image of P at P′. A lens of this type is called a converging, or positive, lens. The object and image are said to be conjugates. Figure 1.8 diagrams the action of a convex lens—that is, a lens which is thicker at its center than at its edges. A convex lens with an index higher than that of the surrounding medium is a converging lens, in that it will increase the convergence (or reduce the divergence) of a wave front passing through it. In Fig. 1.9 the action of a concave lens is sketched. In this case the lens is thicker at the edge and thus retards the wave front more at the edge than at the center and increases the divergence. After passing through the lens, the wave front appears to have originated from the neighborhood of point P′, which is the image of point P formed by the lens. In this case, however, it would be futile to place a screen at P′ and General Principles 9 Figure 1.8 The passage of a wave front through a converging, or posi- tive, lens element. expect to find a concentration of light; all that would be observed would be the general illumination produced by the light emanating from P. This type of image is called a virtual image to distinguish it from the type of image diagramed in Fig. 1.8, which is called a real image. Thus a virtual image may be observed directly or may serve as a source to be reimaged by a subsequent lens system, but it cannot be produced on a screen. The terms “real” and “virtual” also may be applied to rays, where “virtual” applies to the extended part of a real ray. The path of a ray of light through the lenses of Figs. 1.8 and 1.9 is the path traced by a point on the wave front. In Fig. 1.10 several ray paths have been drawn for the case of a converging lens. Note that the rays originate at point P and proceed in straight lines (since the media involved are isotropic) to the surface of the lens where they are refracted according to Snell’s law (Eq. 1.3.) After refraction at the second surface the rays converge at the image P′. (In practice the rays will converge exactly at P′ only if the lens surfaces are suitably chosen surfaces of rotation, usually nonspherical, whose axes are coincident and pass through P.) This would lead one to expect that the concentration of light at P′ would be a perfect point. However, the wave nature of light caus- es it to be diffracted in passing through the limiting aperture of the lens so that the image, even for a “perfect” lens, is spread out into a small disc of light surrounded by faint rings as discussed in Chap. 6. In Fig. 1.11 a wave front from a source so far distant that the cur- vature of the wave front is negligible is shown approaching a prism, which has two flat polished faces. As it passes through each face of the prism, the light is refracted downward so that the direction of propa- gation is deviated. The angle of deviation of the prism is the angle between the incident ray and the emergent ray. Note that the wave front remains plane as it passes through the prism. If the radiation incident on the prism consisted of more than one wavelength, the shorter-wavelength radiation would be slowed down more by the medium composing the prism and thus deviated through a greater angle. This is one of the methods used to separate different wavelengths of light and is, of course, the basis for Isaac Newton’s clas- sic demonstration of the spectrum. 10 Chapter One Figure 1.9 The passage of a wave front through a diverging, or negative, lens element. 1.5 Interference and Diffraction If a stone is dropped into still water, a series of concentric ripples, or waves, is generated and spreads outward over the surface of the water. If two stones are dropped some distance apart, a careful observer will notice that where the waves from the two sources meet there are areas with waves twice as large as the original waves and also areas which are almost free of waves. This is because the waves can reinforce or cancel out the action of each other. Thus if the crests (or troughs) of two waves arrive simultaneously at the same point, the crest (or trough) generated is the sum of the two wave actions. However, if the crest of one wave arrives at the same instant as the trough of the oth- er, the result is a cancellation. A more spectacular display of wave rein- forcement can often be seen along a sea wall where an ocean wave which has struck the wall and been reflected back out to sea will com- bine with the next incoming wave to produce an eruption where they meet. Similar phenomena occur when light waves are made to interfere. In general, light from the same point on the source must be made to trav- el two separate paths and then be recombined, in order to produce optical interference. The familiar colors seen in soap bubbles or in oil films on wet pavements are produced by interference. General Principles 11 Figure 1.10 Showing the relationship between light rays and the wave front in passing through a positive lens element. Figure 1.11 The passage of a plane wave front through a re- fracting prism. Young’s experiment, which is diagramed schematically in Fig. 1.12, illustrates both diffraction and interference. Light from a source to the left of the figure is caused to pass through a slit or pinhole s in an opaque screen. According to Huygens’ principle, the propagation of a wave front can be constructed by considering each point on the wave front as a source of new spherical wavelets; the envelope of these new wavelets indicates the new position of the wave front. Thus s may be considered as the center of a new spherical or cylindrical wave (depending on whether s is a pinhole or a slit), provided that the size of s is sufficiently small. These diffracted wave fronts from s travel to a second opaque screen which has two slits (or pinholes), A and B, from which new wave fronts originate. The wave fronts again spread out by diffraction and fall on an observing screen some distance away. Now, considering a specific point P on the screen, if the wave fronts arrive simultaneously (or in phase), they will reinforce each other and P will be illuminated. However, if the distances AP and BP are such that the waves arrive exactly out of phase, destructive interference will o