250 13.Optical Properties of Materials X ←-Vacuum Material ep(e】 FIGURE 13.2.Exponential decrease of the amplitude of electromagnetic radiation in optically dense materials such as metals. We define a characteristic penetration depth,W,as that dis- tance at which the intensity of the light wave,which travels through a material,has decreased to 1/e or 37%of its original value,i.e.,when: L=1=e1 (13.14) Io e This definition yields,in conjunction with Eq.(13.13), &=w=k (13.15) Table 13.2 presents experimental values for k and W for some materials obtained by using sodium vapor light (A=589.3 nm). The inverse of W is sometimes called the (exponential)atten- uation or the absorbance,a,which is,by making use of Eq. (13.15)and(13.12), a=Amk=Amk =2me2 (13.16) A c An It is measured,for example,in cm-1.The energy loss per unit length(given,for example,in decibels,dB,per centimeter)is ob- tained by multiplying the absorbance,a,with 4.34,see Problem 13.7.(1dB=101ogI1o). TABLE 13.2.Characteristic penetration depth,W,and damping con- stant,k,for some materials (A=589.3 nm) Material Water Flint glass Graphite Gold W(cm) 32 29 6×10-6 1.5×10-6 k 1.4×10-7 1.5×10-7 0.8 3.2
We define a characteristic penetration depth, W, as that distance at which the intensity of the light wave, which travels through a material, has decreased to 1/e or 37% of its original value, i.e., when: I I 0 1 e e1. (13.14) This definition yields, in conjunction with Eq. (13.13), z W 4 c k 4 k . (13.15) Table 13.2 presents experimental values for k and W for some materials obtained by using sodium vapor light ( 589.3 nm). The inverse of W is sometimes called the (exponential) attenuation or the absorbance, , which is, by making use of Eq. (13.15) and (13.12), 4 k 4 c k 2 n 2 . (13.16) It is measured, for example, in cm1. The energy loss per unit length (given, for example, in decibels, dB, per centimeter) is obtained by multiplying the absorbance, , with 4.34, see Problem 13.7. (1 dB 10 log I/I0). FIGURE 13.2. Exponential decrease of the amplitude of electromagnetic radiation in optically dense materials such as metals. 250 13 • Optical Properties of Materials x Vacuum Material exp – &kz c z TABLE 13.2. Characteristic penetration depth, W, and damping constant, k, for some materials ( 589.3 nm) Material Water Flint glass Graphite Gold W (cm) 32 29 6 106 1.5 106 k 1.4 107 1.5 107 0.8 3.2
13.2.The Optical Constants 。 251 The ratio between the reflected intensity IR and the incom- ing intensity lo of the light is the reflectivity: R=会 (13.17) Quite similarly,one defines the ratio between the transmitted in- tensity,Ir,and the impinging light intensity as the transmissiv- ity: (13.18) The reflectivity is connected with n and k(assuming normal in- cidence)through: R=n-12+k2 (n+1)2+k2 (13.19) (Beer equation).The reflectivity is a unitless material constant and is often given in percent of the incoming light (see Table 13.1).R is,like the index of refraction,a function of the wave- length of the light.For insulators (k=0)one finds that R de- pends solely on the index of refraction: R=-1)2 (13.20) (n+1)2 Metals are characterized by a large reflectivity.This stems from the fact that light penetrates metals only a short distance,as shown in Figure 13.2 and Table 13.2.Thus,only a small part of the impinging energy is converted into heat.The major part of the energy is reflected (in some cases as much as 99%,see Table 13.1).In contrast to this,visible light penetrates into glass (and many other dielectrics)much farther than into metals,that is, approximately seven orders of magnitude more;see Table 13.2. As a consequence,very little light is reflected by glass.Never- theless,a piece of glass about 1 or 2 m thick eventually dissipates a substantial part of the impinging light into heat.(In practical applications,one does not observe this large reduction in light intensity because windows are,as a rule,only a few millimeters thick.)It should be noted that window panes,lenses,etc.,reflect the light on the front as well as on the back side. An energy conservation law requires that the intensity of the light impinging on a material,Io,must be equal to the reflected inten- sity,IR,plus the transmitted intensity,I7,plus that intensity which has been extinct,IE,for example,transferred into heat,that is, Io IR IT IE. (13.21)
The ratio between the reflected intensity IR and the incoming intensity I0 of the light is the reflectivity: R I I R 0 . (13.17) Quite similarly, one defines the ratio between the transmitted intensity, IT, and the impinging light intensity as the transmissivity: T I I T 0 . (13.18) The reflectivity is connected with n and k (assuming normal incidence) through: R (13.19) (Beer equation). The reflectivity is a unitless material constant and is often given in percent of the incoming light (see Table 13.1). R is, like the index of refraction, a function of the wavelength of the light. For insulators (k 0) one finds that R depends solely on the index of refraction: R . (13.20) Metals are characterized by a large reflectivity. This stems from the fact that light penetrates metals only a short distance, as shown in Figure 13.2 and Table 13.2. Thus, only a small part of the impinging energy is converted into heat. The major part of the energy is reflected (in some cases as much as 99%, see Table 13.1). In contrast to this, visible light penetrates into glass (and many other dielectrics) much farther than into metals, that is, approximately seven orders of magnitude more; see Table 13.2. As a consequence, very little light is reflected by glass. Nevertheless, a piece of glass about 1 or 2 m thick eventually dissipates a substantial part of the impinging light into heat. (In practical applications, one does not observe this large reduction in light intensity because windows are, as a rule, only a few millimeters thick.) It should be noted that window panes, lenses, etc., reflect the light on the front as well as on the back side. An energy conservation law requires that the intensity of the light impinging on a material, I0, must be equal to the reflected intensity, IR, plus the transmitted intensity, IT, plus that intensity which has been extinct, IE, for example, transferred into heat, that is, I0 IR IT IE. (13.21) (n 1) 2 (n 1)2 (n 1)2 k 2 (n 1)2 k2 13.2 • The Optical Constants 251
252 13.Optical Properties of Materials Dividing Eq.(13.21)by Io and making use of Eq.(13.17)and (13.18)yields: R+T+E=1. (13.22) (It has been assumed for these considerations that the light which has been scattered inside the material may be transmitted through the sides and is therefore contained in Ir and IE.) The reflection losses encountered in optical instruments such as lenses can be significantly reduced by coating the surfaces with a thin layer of a dielectric material such as magnesium flu- oride.This results in the well-known blue hue on lenses for cam- eras. Metals are generally opaque in the visible spectral region be- cause of their comparatively high damping constant and thus high reflectivity.Still,very thin metal films (up to about 50 nm thickness)may allow some light to be transmitted.Dielectric ma- terials,on the other hand,are often transparent.Occasionally, however,some opacifiers are inherently or artificially added to dielectrics which cause the light to be internally deflected by mul- tiple scattering.Finally,if the diffuse scattering is not very se- vere,dielectrics might appear translucent,that is,objects viewed through them are vaguely seen,but not clearly distinguishable. Scattering of light may occur,for example,due to residual poros- ity in ceramic materials,or on grain boundaries(which have a small variation in refractive index compared to the matrix),or on finely dispersed particles,or on boundaries between crys- talline and amorphous regions in polymers,to mention only a few mechanisms. There exists an important equation which relates the reflec- tivity of light at low frequencies (infrared spectral region)with the direct-current conductivity,o: R=1-4,T0- (13.23) This relation,which was experimentally found at the end of the 19th century by Hagen and Rubens,states that materials having a large electrical conductivity (such as metals)also possess es- sentially a large reflectivity (and vice versa). 13.3.Absorption of Light If light impinges on a material,it is either re-emitted in one form or another (reflection,transmission)or its energy is extinct,for example,transformed into heat.In any of these cases,some in-
Dividing Eq. (13.21) by I0 and making use of Eq. (13.17) and (13.18) yields: R T E 1. (13.22) (It has been assumed for these considerations that the light which has been scattered inside the material may be transmitted through the sides and is therefore contained in IT and IE.) The reflection losses encountered in optical instruments such as lenses can be significantly reduced by coating the surfaces with a thin layer of a dielectric material such as magnesium fluoride. This results in the well-known blue hue on lenses for cameras. Metals are generally opaque in the visible spectral region because of their comparatively high damping constant and thus high reflectivity. Still, very thin metal films (up to about 50 nm thickness) may allow some light to be transmitted. Dielectric materials, on the other hand, are often transparent. Occasionally, however, some opacifiers are inherently or artificially added to dielectrics which cause the light to be internally deflected by multiple scattering. Finally, if the diffuse scattering is not very severe, dielectrics might appear translucent, that is, objects viewed through them are vaguely seen, but not clearly distinguishable. Scattering of light may occur, for example, due to residual porosity in ceramic materials, or on grain boundaries (which have a small variation in refractive index compared to the matrix), or on finely dispersed particles, or on boundaries between crystalline and amorphous regions in polymers, to mention only a few mechanisms. There exists an important equation which relates the reflectivity of light at low frequencies (infrared spectral region) with the direct-current conductivity, : R 1 4 0 . (13.23) This relation, which was experimentally found at the end of the 19th century by Hagen and Rubens, states that materials having a large electrical conductivity (such as metals) also possess essentially a large reflectivity (and vice versa). If light impinges on a material, it is either re-emitted in one form or another (reflection, transmission) or its energy is extinct, for example, transformed into heat. In any of these cases, some in- 252 13 • Optical Properties of Materials 13.3 • Absorption of Light
13.3.Absorption of Light 253 teraction between light and matter will take place,as was ex- plained in the preceding section.One of the major mechanisms by which this interaction occurs is called absorption of light. The classical description of absorption and reemission of light was developed at the turn of the 20th century by P.Drude,a Ger- man physicist.His concepts were described in Chapter 11.1 when we discussed electrical conduction in metals.As explained there, Drude postulated that some electrons in a metal (essentially the valence electrons)can be considered to be free,that is,they can be separated from their respective nuclei.He further assumed that the free electrons within the crystal can be accelerated by an external electric field.This preliminary Drude model was re- fined by considering that the moving electrons on their path col- lide with certain metal atoms in a nonideal lattice.If an alter- nating electric field (as through interaction with light)is envolved,then the free electrons are thought to perform oscil- lating motions.These vibrations are restrained by the above-men- tioned interactions of the electrons with the atoms of a nonideal lattice.Thus,a friction force is introduced which takes this in- teraction into consideration.The calculation of the frequency de- pendence of the optical constants is accomplished by using the classical equations for vibrations whereby the interactions of electrons with atoms are taken into account by a damping term which is assumed to be proportional to the velocity of the elec- trons.The Newtonian-type equation (Force mass times accel- eration)is essentially identical to that of Eq.(11.9)except that the direct-current excitation force e is now replaced by a peri- odic (i.e.,sinusoidal)excitation force: F=e 80 sin(2Tvt), (13.24) where v is the frequency of the light,t is the time,and o is the maximal field strength of the light wave.In short,the equation describing the motion of free electrons which are excited to per- form forced,periodic vibrations under the influence of light can be written as: m- dv+yv=e &o sin (2mvt) (13.25) t where y is the damping strength which takes the damping of the electron motion into account. The solution of this equation,which shall not be attempted here,yields the frequency dependence (or dispersion)of the op- tical constants. The free electron theory describes,to a certain degree,the dis- persion of the optical constants quite well.This is schematically
teraction between light and matter will take place, as was explained in the preceding section. One of the major mechanisms by which this interaction occurs is called absorption of light. The classical description of absorption and reemission of light was developed at the turn of the 20th century by P. Drude, a German physicist. His concepts were described in Chapter 11.1 when we discussed electrical conduction in metals. As explained there, Drude postulated that some electrons in a metal (essentially the valence electrons) can be considered to be free, that is, they can be separated from their respective nuclei. He further assumed that the free electrons within the crystal can be accelerated by an external electric field. This preliminary Drude model was refined by considering that the moving electrons on their path collide with certain metal atoms in a nonideal lattice. If an alternating electric field (as through interaction with light) is envolved, then the free electrons are thought to perform oscillating motions. These vibrations are restrained by the above-mentioned interactions of the electrons with the atoms of a nonideal lattice. Thus, a friction force is introduced which takes this interaction into consideration. The calculation of the frequency dependence of the optical constants is accomplished by using the classical equations for vibrations whereby the interactions of electrons with atoms are taken into account by a damping term which is assumed to be proportional to the velocity of the electrons. The Newtonian-type equation (Force mass times acceleration) is essentially identical to that of Eq. (11.9) except that the direct-current excitation force e is now replaced by a periodic (i.e., sinusoidal) excitation force: F e 0 sin (2t), (13.24) where is the frequency of the light, t is the time, and 0 is the maximal field strength of the light wave. In short, the equation describing the motion of free electrons which are excited to perform forced, periodic vibrations under the influence of light can be written as: m d d v t v e 0 sin (2t), (13.25) where is the damping strength which takes the damping of the electron motion into account. The solution of this equation, which shall not be attempted here, yields the frequency dependence (or dispersion) of the optical constants. The free electron theory describes, to a certain degree, the dispersion of the optical constants quite well. This is schematically 13.3 • Absorption of Light 253