Chapter 4 Reflected Light Optics

l CHAPTER 4 REFLECTED LIGHT OPTICS 4.1 INTRODUCTION Light is a form ofelectromagnetic radiation. which may be emitted by matter that is in a suitably "energized" (excited) state (e.g.,the tungsten filament ofa microscope lamp emits light when "energized" by the passage of an electric current). One ofthe interesting consequences ofthe developments in physics in the early part of the twentieth century was the realization that light and other forms of electromagnetic radiation can be described both as waves and as a stream ofparticles (photons). These are not conflicting theories but rather complementary ways of describing light; in different circumstances. either one may be the more appropriate. For most aspects ofmicroscope optics. the "classical" approach ofdescribing light as waves is more applicable. However, particularly (as outlined in Chapter 5) when the relationship between the reflecting process and the structure and composition ofa solid is considered. it is useful to regard light as photons. The electromagnetic radiation detected by the human eye is actually only a very small part of the complete electromagnetic spectrum, which can be re garded as a continuum from the very low energies and long wavelengths characteristic ofradio waves to the very high energies (and shortwavelengths) of gamma rays and cosmic rays. As shown in Figure 4.1, the more familiar regions ofthe infrared, visible light, ultraviolet. and X-rays fall between these extremes of energy and wavelength. Points in the electromagnetic spectrum can be specified using a variety ofenergy or wavelength units. The most com mon energy unit employed by physicists is the electron volt' (eV). whereas 'Oneelectron volt is the energy acquired by an electron accelerated through a potential difference of one volt. 55 58 REFLECTED LIGHT OPTICS nm em-I eV 400 25000 3.11 500 20000 2.5 600 16667 2.07 700 . 14286 1.78 Broadcast band Long radio waves FIGURE 4.1 Theelectromagnetic spectrum between 10-5and 1015 A. Energies within this range are also shown as wave numbers (cm") and electron volts (eV). Energies of the visible light range are shown on an expanded scale. (Reproduced from Mineral Chemistry ofMetal Sulfides by D. 1. Vaughan and J. R. Craig, copyright e 1978, Cam bridge University Press, with the publisher's permission). wavelengths maybe expressed in terms ofAngstrom units (A.. where 1A= 10-8 em) or nanometers (nm, where 1nm = 10-7 cm = lOA). Anotherunit employed in the literature of physics and chemistry is the wavenumber, or reciprocal centimeter (ern-I), which, unlike the wavelength units, varies linearly with energy. The nanometer is most commonly used in mineralogical literature and will be used in this book. In Figure 4.1, the relationships between these units are indicated for different regions ofthe spectrum. Shown in detail is the visible light region, which extends between approximately 390 and 770 nrn; particularwavelength regions within this range are characterized by the eye as different colors. Seven primary colors were recognized by Sir Isaac Newton (the "colors ofthe rainbow"), and their wavelength limits are shown in Figure 4.1. White light from the sun or an artificial light source is comprised of con tributions from all of these wavelengths. Light of a very limited range of wavelengths, such as the characteristic yellow light from a sodium vaporlamp that consists chiefly ofwavelengths 589.0and 589.6nm, is described as mono chromatic. In further developing the wave description of light, transmission can be considered to occur by a transverse wave motion in which the vibrations are perpendicular to the direction of travel of the energy. A wave resulting from this type of (simple harmonic) motion is shown in Figure 4.2a and has the shape ofa sine curve. For such a wave, c = VA (4.1) INTRODUCTION 57 90 · ~-IOI ' ____ __J »2 unp~;~:i -:e~ ; I polarized FIGURE 4.2 The wave motion of light showing (a) unpolarized light with the wave motion represented along the propagation direction and normal to it (on the left); (b) unpolarized light in contrast to light that is linearly polarized. where c = velocity of the wave, v = frequency of the wave, A. = wavelength. As Figure 4.2a shows, the ray's vibration directions within any plane perpen dicular to its path may be represented by a semicircle of radius equal to the vi bration of the wave within that plane, except at points a, b, or c, where the vibration is nil. In addition to this, a single wave of this type has a kind of spiraling motion that can best be explained by describing the behavior of a single point on the wave that on progressing from a - b cha nges its direction of vibration with reference to the circular section normal to the line ahc. At point a , it can be considered as being at point I on the circular section in Figure 4.2; by pointX, it will be at 2, point Yat4,and soon. Th e position ofthis single pointon the wave is termed itsphase. Finally,the amplitudeofthe wave is the maximum vibrational displacement (i.e.. the radius of the semicircular section at point y in Figure 4.2). It is possible to restrict the vibration of the light wave illustrated in Figure 4.2a to a single plane, in which case the light is said to be plane polarized, although the more correct term linearly polarizedis less confusing in a discus sion of reflected-light optics and will be used in this text. The plane ofvibra tion ofthis polarized light is that plane parallel to both the path ofthe ray and the vibration direction. 4.1.1 Interaction of Light with Transparent Media The velocity ofa light wave (c) or other electromagnetic wave is constant in a vacuum (299,793 km/sec) but changes if the wave passes into another trans mitting medium, a change expressed by the concept ofthe refractive index of that medium: c n=- (4.2) 58 REFLECTED LIGHT OPTICS where n is the refractive index of the medium and c and Cm represent the velocity of light in a vacuum and in the medium, respectively. Since Cm is always less thanc,n is always greater than 1.0,although for airn = 1.0003 (~I) . Since the refractive index is a ratio of two velocities, it is a dimensionless number. Those materials through which monochromatic light travels at the same speed, regardless ofthe direction oflightvibration relative to the medium, are optically isotropic. A vacuum, all gases, most liquids, glasses, and cubic (iso metric) crystalline substances are isotropic; other materials (chiefly non isometric crystals) are optically anisotropic, and light rays may travel through them at different speeds, depending on the direction oflight vibration within them. The optical indicatrix shows how the refractive index of a transparent material varies with the vibration direction ofthe (monochromatic) light wave in the material. Ifan infinite number ofvectors are imagined radiating in all directions from one point within the substance and if each vector has a length that is proportional to the refractive index for light vibrating parallel to that vector direction, then the indicatrix is the surface connecting the tips ofthese vectors. For an isotropic substance, therefore, the indicatrix is a sphere of radius n (Figure 4.3a). However, it should be noted that the value ofn does still vary as a function ofwavelength oflight. For anisotropic crystals, the refractive index varies with vibration direction in the crystal, even for monochromatic light, so that the indicatrix is not a sphere but an ellipsoid. The optical indicatrix for crystalline substances with hexagonal (used here as including the trigonal system) or tetragonal crystal symmetry is such that one cross section through the ellipsoid is circular and all other sections are elliptical. The indicatrix is termed uniaxial, and the direction normal to this unique circularsection is the optic axis and is parallel to the c axis ofthe crystal. The value ofthe refractive index in this circular section through the ellipsoid (termed the ordinary vibration direction and often symbolized as co or 0) may be either the maximum or the minimum value ofthe refractive index, so that the indicatrix may be either prolate or oblate, as shown in Figures 4.3b and 4.3c. Conversely, the value of the refractive index of the crystal for the vi bration along the direction of the optic axis (crystal C axis) and termed the extraordinary vibration direction (often symbolized cor e) must also be eithera maximum or a minimum value. Materials in which the value (e-o) is positive (a prolate indicatrix) are termed uniaxialpositive; those with (e-o) negative (an oblate indicatrix) are termed uniaxial negative in their optic signs. Theoptical indicatrix for crystallinesubstances with orthorhombic, mono clinic, or triclinic crystal symmetry, although ellipsoidal (being geometrically termed the "general ellipsoid"), is such that there are two cross sections thatare circular, all others being ellipsoidal. Again, the two directions normal to these sections are opticaxes, so that the indicatrix is termed biaxial (Figure4.3e).The biaxial crystal has three principal refractive indices that are commonly sym holized a, ~ , y, with a and y being the smallest and largest, respectively. The vector directions corresponding to the a and y refractive indices are unique directions termed principal vibration axes and are symbolized X and Z, re- INTRODUCTION 59 ®~~ (a) (b) (e) Z Z I I Circulor_ Sections I f) (g) FIGURE 4.3 The optical indicatrix for (a) an isotropic substance, (b. c) a uniaxial positive and negative crystal , (d, e) a biaxial crystal showing principal vibration axes and optic axes, (f. g) crystallographic orientation of the biaxial indicatrix in ortho rhombic and monoclinic crystals.

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