13 Optical Properties of Materials 13.1.Interaction of Light with Matter The most apparent properties of metals,their luster and their color,have been known to mankind since materials were known. Because of these properties,metals were already used in antiq- uity for mirrors and jewelry.The color was utilized 4000 years ago by the ancient Chinese as a guide to determine the compo- sition of the melt of copper alloys:the hue of a preliminary cast indicated whether the melt,from which bells or mirrors were to be made,already had the right tin content. The German poet Goethe was probably the first one who ex- plicitly spelled out 200 years ago in his Treatise on Color that color is not an absolute property of matter(such as the resistiv- ity),but requires a living being for its perception and descrip- tion.Goethe realized that the perceived color of a region in the visual field depends not only on the properties of light coming from that region,but also on the light coming from the rest of the visual field.Applying Goethe's findings,it was possible to ex- plain qualitatively the color of,say,gold in simple terms.Goethe wrote:"If the color blue is removed from the spectrum,then blue, violet,and green are missing and red and yellow remain."Thin gold films are bluish-green when viewed in transmission.These colors are missing in reflection.Consequently,gold appears reddish-yellow.On the other hand,Newton stated quite correctly in his "Opticks"that light rays are not colored.The nature of color remained,however,unclear. This chapter treats the optical properties from a completely different point of view.Measurable quantities such as the index of refraction or the reflectivity and their spectral variations are
13 The most apparent properties of metals, their luster and their color, have been known to mankind since materials were known. Because of these properties, metals were already used in antiquity for mirrors and jewelry. The color was utilized 4000 years ago by the ancient Chinese as a guide to determine the composition of the melt of copper alloys: the hue of a preliminary cast indicated whether the melt, from which bells or mirrors were to be made, already had the right tin content. The German poet Goethe was probably the first one who explicitly spelled out 200 years ago in his Treatise on Color that color is not an absolute property of matter (such as the resistivity), but requires a living being for its perception and description. Goethe realized that the perceived color of a region in the visual field depends not only on the properties of light coming from that region, but also on the light coming from the rest of the visual field. Applying Goethe’s findings, it was possible to explain qualitatively the color of, say, gold in simple terms. Goethe wrote: “If the color blue is removed from the spectrum, then blue, violet, and green are missing and red and yellow remain.” Thin gold films are bluish-green when viewed in transmission. These colors are missing in reflection. Consequently, gold appears reddish-yellow. On the other hand, Newton stated quite correctly in his “Opticks” that light rays are not colored. The nature of color remained, however, unclear. This chapter treats the optical properties from a completely different point of view. Measurable quantities such as the index of refraction or the reflectivity and their spectral variations are Optical Properties of Materials 13.1 • Interaction of Light with Matter
246 13.Optical Properties of Materials used to characterize materials.In doing so,the term "color"will almost completely disappear from our vocabulary.Instead,it will be postulated that the interactions of light with the electrons of a material are responsible for the optical properties. At the beginning of the 20th century,the study of the interac- tions of light with matter (black-body radiation,etc.)laid the foundations for quantum theory.Today,optical methods are among the most important tools for elucidating the electron structure of matter.Most recently,a number of optical devices such as lasers,photodetectors,waveguides,etc.,have gained con- siderable technological importance.They are used in telecom- munication,fiber optics,CD players,laser printers,medical di- agnostics,night viewing,solar applications,optical computing, and for optoelectronic purposes.Traditional utilizations of opti- cal materials for windows,antireflection coatings,lenses,mir- rors,etc.,should be likewise mentioned. We perceive light intuitively as a wave(specifically,an elec- tromagnetic wave)that travels in undulations from a given source to a point of observation.The color of the light is related to its wavelength.Many crucial experiments,such as diffraction,in- terference,and dispersion,clearly confirm the wavelike nature of light.Nevertheless,at least since the discovery of the photo- electric effect in 1887 by Hertz,and its interpretation in 1905 by Einstein,do we know that light also has a particle nature.(The photoelectric effect describes the emission of electrons from a metallic surface after it has been illuminated by light of appro- priately high energy,e.g.,by blue light.)Interestingly enough, Newton,about 300 years ago,was a strong proponent of the par- ticle concept of light.His original ideas,however,were in need of some refinement,which was eventually provided in 1901 by quantum theory.We know today (based on Planck's famous hy- pothesis)that a certain minimal energy of light,that is,at least one light quantum,called a photon,with the energy: E=vh oh (13.1) needs to impinge on a metal in order that a negatively charged electron may overcome its binding energy to its positively charged nucleus,and can escape into free space.(This is true regardless of the intensity of the light.)In Eq.(13.1),h is the Planck con- stant whose numerical value is given in Appendix II and v is the frequency of light given as the number of vibrations(cycles)per second or hertz (Hz).Frequently,the reduced Planck constant: 方 2T (13.2)
used to characterize materials. In doing so, the term “color” will almost completely disappear from our vocabulary. Instead, it will be postulated that the interactions of light with the electrons of a material are responsible for the optical properties. At the beginning of the 20th century, the study of the interactions of light with matter (black-body radiation, etc.) laid the foundations for quantum theory. Today, optical methods are among the most important tools for elucidating the electron structure of matter. Most recently, a number of optical devices such as lasers, photodetectors, waveguides, etc., have gained considerable technological importance. They are used in telecommunication, fiber optics, CD players, laser printers, medical diagnostics, night viewing, solar applications, optical computing, and for optoelectronic purposes. Traditional utilizations of optical materials for windows, antireflection coatings, lenses, mirrors, etc., should be likewise mentioned. We perceive light intuitively as a wave (specifically, an electromagnetic wave) that travels in undulations from a given source to a point of observation. The color of the light is related to its wavelength. Many crucial experiments, such as diffraction, interference, and dispersion, clearly confirm the wavelike nature of light. Nevertheless, at least since the discovery of the photoelectric effect in 1887 by Hertz, and its interpretation in 1905 by Einstein, do we know that light also has a particle nature. (The photoelectric effect describes the emission of electrons from a metallic surface after it has been illuminated by light of appropriately high energy, e.g., by blue light.) Interestingly enough, Newton, about 300 years ago, was a strong proponent of the particle concept of light. His original ideas, however, were in need of some refinement, which was eventually provided in 1901 by quantum theory. We know today (based on Planck’s famous hypothesis) that a certain minimal energy of light, that is, at least one light quantum, called a photon, with the energy: E h &' (13.1) needs to impinge on a metal in order that a negatively charged electron may overcome its binding energy to its positively charged nucleus, and can escape into free space. (This is true regardless of the intensity of the light.) In Eq. (13.1), h is the Planck constant whose numerical value is given in Appendix II and is the frequency of light given as the number of vibrations (cycles) per second or hertz (Hz). Frequently, the reduced Planck constant: ' 2 h (13.2) 246 13 • Optical Properties of Materials
13.2.The Optical Constants 247 Wavelength Energy Frequency (m) (ev) (H2) 10-14 108 1022 400nm 10-2 106 violet -----Y-Rays 1020 blue X-Rays--1-- 10-10 104 - nm 1018 500nm UVI 10-8 102 1016 green ----7 10-6 m yellow 100 104 600nm orange Infrared (heat) -不--1-- 10-4 10-2 1012 mm red/ Microwaves 10-2 10-4 1010 700nm 100 -m 106 GHz 108 Visible Radio, spectrum TV 102 10-8 106 -MHz km 104 10-0 104 106 10-2 kHz 102 FIGURE 13.1.The spec- is utilized in conjunction with the angular frequency,@=2mv. trum of electromag- In short,the wave-particle duality of light (or,more generally,of netic radiation.Note electromagnetic radiation)had been firmly established at about the small segment of 1924.The speed of light,c,and the frequency are connected by this spectrum that is the equation: visible to human eyes. C=入， (13.3) where A is the wavelength of the light. Light comprises only an extremely small segment of the entire electromagnetic spectrum,which ranges from radio waves via microwaves,infrared,visible,ultraviolet,X-rays,to y rays,as de- picted in Figure 13.1.Many of the considerations which will be advanced in this chapter are therefore also valid for other wave- length ranges,i.e.,for radio waves or X-rays. 13.2.The Optical Constants When light passes from an optically "thin"medium (e.g.,vac- uum,air)into an optically dense medium one observes that in the dense medium,the angle of refraction B(i.e.,the angle be- tween the refracted light beam and a line perpendicular to the surface)is smaller than the angle of incidence,a.This well-known
is utilized in conjunction with the angular frequency, & 2. In short, the wave-particle duality of light (or, more generally, of electromagnetic radiation) had been firmly established at about 1924. The speed of light, c, and the frequency are connected by the equation: c , (13.3) where is the wavelength of the light. Light comprises only an extremely small segment of the entire electromagnetic spectrum, which ranges from radio waves via microwaves, infrared, visible, ultraviolet, X-rays, to rays, as depicted in Figure 13.1. Many of the considerations which will be advanced in this chapter are therefore also valid for other wavelength ranges, i.e., for radio waves or X-rays. When light passes from an optically “thin” medium (e.g., vacuum, air) into an optically dense medium one observes that in the dense medium, the angle of refraction (i.e., the angle between the refracted light beam and a line perpendicular to the surface) is smaller than the angle of incidence, . This well-known FIGURE 13.1. The spectrum of electromagnetic radiation. Note the small segment of this spectrum that is visible to human eyes. 13.2 • The Optical Constants 247 10–14 10–12 10–10 10–8 10–6 10–4 10–2 100 102 104 106 108 106 104 102 100 10–2 10–4 10–6 10–8 10–10 10–12 1022 1020 1018 1016 1014 1012 1010 108 106 104 102 kHz MHz GHz nm mm m km m 400 nm 500 nm 600 nm 700 nm violet blue green yellow orange red Visible spectrum UV Wavelength (m) Energy (eV) Frequency (Hz) -Rays Microwaves Infrared (heat) Radio, TV X-Rays 13.2 • The Optical Constants
248 13.Optical Properties of Materials phenomenon is used for the definition of the refractive power of a material and is called the Snell law: sin a=nmed=n. (13.4) sin B nvac Commonly,the index of refraction for vacuum nvac is arbitrar- ily set to be unity.The refraction is caused by the different ve- locities,c,of the light in the two media: sin a Cvac (13.5) sin B Cmed Thus,if light passes from vacuum into a medium,we find: n=Cvac=c (13.6) Cmed v' where v cmed is the velocity of light in the material.The mag- nitude of the refractive index depends on the wavelength of the incident light.This property is called dispersion.In metals,the index of refraction varies also with the angle of incidence.This is particularly true when n is small. The index of refraction is generally a complex number,desig- nated as n,which is comprised of a real and an imaginary part ni and n2,respectively,i.e., n=ni-in2. (13.7) In the literature,the imaginary part of ni is often denoted by k. Equation (13.7)is then written as: n=n-ik. (13.8) We will call n2 or k the damping constant.(In some books,n2 and k are named absorption constant,attenuation index,or ex- tinction coefficient.We will not follow this practice because of its potential to be misleading.)The square of the (complex)index of refraction is equal to the (complex)dielectric constant(Sec- tion11.8): 2=e=e1-ie2, (13.9) which yields,with Eq.(13.8), n2=n2-k2-2nki=e1-i e2 (13.10) Equating individually the real and imaginary parts in Eq.(13.10) yields: e1=n2-k2 (13.11)
phenomenon is used for the definition of the refractive power of a material and is called the Snell law: s s i i n n n n m va e c d n. (13.4) Commonly, the index of refraction for vacuum nvac is arbitrarily set to be unity. The refraction is caused by the different velocities, c, of the light in the two media: s s i i n n c c m va e c d . (13.5) Thus, if light passes from vacuum into a medium, we find: n c c m va e c d v c , (13.6) where v cmed is the velocity of light in the material. The magnitude of the refractive index depends on the wavelength of the incident light. This property is called dispersion. In metals, the index of refraction varies also with the angle of incidence. This is particularly true when n is small. The index of refraction is generally a complex number, designated as nˆ, which is comprised of a real and an imaginary part n1 and n2, respectively, i.e., nˆ n1 i n2. (13.7) In the literature, the imaginary part of nˆ is often denoted by k. Equation (13.7) is then written as: nˆ n i k. (13.8) We will call n2 or k the damping constant. (In some books, n2 and k are named absorption constant, attenuation index, or extinction coefficient. We will not follow this practice because of its potential to be misleading.) The square of the (complex) index of refraction is equal to the (complex) dielectric constant (Section 11.8): nˆ 2 ˆ 1 i 2, (13.9) which yields, with Eq. (13.8), nˆ 2 n2 k2 2nki 1 i 2. (13.10) Equating individually the real and imaginary parts in Eq. (13.10) yields: 1 n2 k2 (13.11) 248 13 • Optical Properties of Materials
13.2.The Optical Constants 249 and E2=2nk. (13.12) e is called polarization whereas e2 is known by the name ab- sorption.Values for n and k for some materials are given in Table 13.1.For insulators,k is nearly zero,which yields for dielectrics e1≈n2ande2→0. When electromagnetic radiation (e.g.,light)passes from vac- uum (or air)into an optically denser material,then the ampli- tude of the wave decreases exponentially with increasing damp- ing constant k and for increasing distance,z,from the surface, as shown in Figure 13.2.Specifically,the intensity,I,of the light (that is,the square of the electric field strength,)obeys the fol- lowing equation(which can be derived from the Maxwell equa- tions): 1=82=106 (13.13) TABLE 13.1.Optical constants for some materials (A 600 nm) 么 k R%ob Metals Copper 0.14 3.35 95.6 Silver 0.05 4.09 98.9 Gold 0.21 3.24 92.9 Aluminum 0.97 6.0 90.3 Ceramics Silica glass (Vycor) 1.46 3.50 Soda-lime glass 1.51 4.13 Dense flint glass 1.75 a 7.44 Quartz 1.55 4.65 Al203 1.76 7.58 Polymers Polyethylene 1.51 4.13 Polystyrene 1.60 5.32 Polytetrafluoroethylene 1.35 2.22 Semiconductors Silicon 3.94 0.025 35.42 GaAs 3.91 0.228 35.26 "The damping constant for dielectrics is about 10-7;see Table 13.2. bThe reflection is considered to have occurred on one reflecting surface only. See also Table 15.1
and 2 2nk. (13.12) 1 is called polarization whereas 2 is known by the name absorption. Values for n and k for some materials are given in Table 13.1. For insulators, k is nearly zero, which yields for dielectrics 1 n2 and 2 0. When electromagnetic radiation (e.g., light) passes from vacuum (or air) into an optically denser material, then the amplitude of the wave decreases exponentially with increasing damping constant k and for increasing distance, z, from the surface, as shown in Figure 13.2. Specifically, the intensity, I, of the light (that is, the square of the electric field strength, ) obeys the following equation (which can be derived from the Maxwell equations): I 2 I0 exp 4 c k z . (13.13) 13.2 • The Optical Constants 249 TABLE 13.1. Optical constants for some materials ( 600 nm) n kR %b Metals Copper 0.14 3.35 95.6 Silver 0.05 4.09 98.9 Gold 0.21 3.24 92.9 Aluminum 0.97 6.0 90.3 Ceramicsc Silica glass (Vycor) 1.46 a 3.50 Soda-lime glass 1.51 a 4.13 Dense flint glass 1.75 a 7.44 Quartz 1.55 a 4.65 Al2O3 1.76 a 7.58 Polymers Polyethylene 1.51 a 4.13 Polystyrene 1.60 a 5.32 Polytetrafluoroethylene 1.35 a 2.22 Semiconductors Silicon 3.94 0.025 35.42 GaAs 3.91 0.228 35.26 aThe damping constant for dielectrics is about 107; see Table 13.2. bThe reflection is considered to have occurred on one reflecting surface only. cSee also Table 15.1.