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Reflectance Methods and Applications
Reflectance Methods and Applications Bruce Hapke, University of Pittsburgh, Pittsburgh, PA, USA ã 2017 Elsevier Ltd. All rights reserved.
Introduction The amount of light reflected or scattered by a solid or particulate medium changes with wavelength. These spectral variations are caused by wavelength-dependent absorption and not only give rise to the sensation of color, but can also be used to infer such properties of the material as composition, complex refractive index, and electronic or molecular structure. The relative amount of light scattered into different directions depends on the physical structure, such as particle size and spacing, of the material. Both the spectral and directional variations of the reflectance can be measured without physically contacting the medium, and hence, they form a major tool in the study of planetary surfaces by remote sensing. Virtually all spacecraft that fly by or orbit the earth or other planets carry imaging spectrometer array detectors for this purpose. Absorption spectra are commonly measured by transmittance, for which a thin slab of material is sliced off and both sides polished. One advantage of measuring spectra by diffuse reflectance is convenience: the only preparation required is that the sample be in powder form. Another is that if the absorbance is too strong, it may be difficult to prepare and handle a section thin enough to pass light through, whereas the powder can simply be ground to a finer size. On the negative side, the process of diffuse reflectance involves multiple scattering of photons from one particle to another and is highly nonlinear. Thus, fairly complex reflectance models are required to extract quantitative spectral absorption coefficients. As with gases, absorption by solids usually occurs in discrete bands, the exception being metals, which absorb over a broad range of wavelengths. However, most solid state bands are much broader than their gaseous counterparts, with Dl/l 10% being typical, where l is the wavelength. For this reason the bands often overlap, complicating their recognition and sometimes producing an apparent absorption continuum. Solid state bands arise from a variety of causes. Absorption in metals and semiconductors is due to photon-induced motion of conduction band electrons, which then collide with impurities or irregularities and transfer this energy to the solid state lattice. In insulators, ultraviolet, visible, and near-infrared photons are absorbed by electronic transitions, while at mid-infrared and radio frequencies absorption occurs by changes in the vibrational and rotational states of the ions of the lattice. The electronic transitions are of three types: valence to conduction band jumps, electronic excitation, in which an electron remains localized on one ion while changing its orbital state, and charge transfer, in which its localization state moves from one ion to another nearby.
Specular Reflection: Kramers-Kronig Spectroscopy If the medium is continuous and homogeneous and is separated from a vacuum by a smooth plane interface, light
Encyclopedia of Spectroscopy and Spectrometry, Third Edition
incident on the surface will be reflected in a specular or mirror-like manner, and the amount reflected is governed by the Fresnel reflection coefficients. For vertically incident light of arbitrary polarization, the specular reflectivity of the intensity is rs ðnÞ ¼
ðnr 1Þ2 þ n2i ðnr þ 1Þ2 þ n2i
[1]
corresponding to the reflection coefficient for the amplitude of the electric field, Ereflected 1 nr ini ¼ Eincident 1 þ nr þ ini
[2]
the where n(n) ¼ nr(n)þini(n)pisffiffiffiffiffiffi ffi complex index of refraction, n is the frequency, and i ¼ 1. If ni21, nr can be calculated from a measurement of rs(n) directly using eqn [1]. If the imaginary component is not negligible, both components of n can still be obtained from rs(n) using the Kramers-Kronig method. The reflection coefficient can be written in the form [3] Ereflected =Eincident ¼ exp ði’Þ pffiffiffiffiffiffiffiffi where ðnÞ ¼ rðnÞ and tan ’ðnÞ ¼ 2ni =½n2r 1 þ n2i . Solving for the components of n, nr ¼
1 2 2 sin ’ , ni ¼ 1 2 cos ’ þ 2 1 2 cos ’ þ 2
[4]
Assuming that n is analytic, the quantities and ’ must satisfy the Cauchy relations: ’ðnÞ ¼
2n p
Z1
ln ½ðnÞ ¼
ln ½ðn0 Þ 0 dn , n0 2 n2
0
2 p
Z1 0
n0 ’ðn0 Þ 0 dn n0 2 n2
[5]
The reflectivity rs is measured over as wide a range of wavepffiffiffiffi lengths as possible, from which ¼ rs is found. A suitable theory, such as the Drude or Lorentz model, is used to extrapolate the measured data to zero and infinity, and the result inserted into eqn [5] for ’; then nr and ni as functions of frequency or wavelength are found from eqns [4].
Diffuse Reflectance from a Particulate Medium The reflectances of disordered particulate media, such as powders in the laboratory, soils, or planetary regoliths, are problems of particular theoretical and practical interest. Most reflectance measurements are one of two types: bidirectional reflectance, denoted by rbd, in which the surface is illuminated from one direction by highly collimated light, such as sunlight
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or a laser, and is observed by a detector that subtends a small angle as seen from the surface; and directional-hemispherical reflectance, denoted by rh, in which the surface is illuminated by collimated light, but the total light scattered into all directions is integrated and measured using an integrating sphere. The latter type of reflectance is also known as the hemispherical reflectance, hemispherical albedo, and plane albedo. Other types of reflectance, including hemispherical-directional and bihemispherical, are also possible. In planetary photometry the total fraction of incident sunlight scattered into all directions by a solar system object is called the Bond albedo, and is equal to the bihemispherical reflectance. The most elementary bidirectional reflectance is an empirical expression known as Lambert’s law, 1 rL ði, e, gÞ ¼ cos i p
scattered by an element of the medium from direction O0 to direction O. The first term on the right-hand side of eqn [7] gives the decrease of I caused by scattering and absorption as the wave propagates through the medium, the second term describes the increase of I due to light traveling in direction O0 scattered into direction O, and the third term is the source term and describes how I is increased as incident irradiance is scattered from direction O0 into direction O. Define the scattering coefficient S, single scattering phase function p(y), and single scattering albedo w by G(O0 ,O) ¼ Sp (y), where y is the angle between O0 and O, and p(y) is normalized so that 1 2
where # is the angle between O and the positive z axis, O0 is the direction of the incident irradiance, i, E is the extinction coefficient, and G(O0 ,O) is the average probability of light being
pðyÞ sin ydy ¼ 1
[8]
0
[6]
where i is the angle between the direction to the source of illumination and the normal to the sample surface, e is the angle between the direction to the detector and the normal, and g is the phase angle, the angle between the directions to the source and detector subtended at the surface. This equation is widely used because of its simplicity. Although no material is known to obey Lambert’s Law exactly, several purified powdered forms of high albedo materials, such as titanium dioxide, barium sulfate, and polytetrafluoroethylene, come close. Thus, these materials can provide a convenient empirical reflectance standard. For mechanical reasons many reflectance spectrometers cannot measure the incident irradiance J directly. Instead, the measured radiance I of light scattered by the material of interest is ratioed to the radiance scattered at i ¼ e ¼ 0 by the standard, which is assumed to be a Lambert surface. A quantity F is defined as F ¼ J/p and the ratio of reflectances is called the radiance factor, denoted by I/F. At present the exact solution of Maxwell’s equations for scattering by a plane electromagnetic wave incident on a disordered semi-infinite medium of interacting particles can be found numerically only for systems of a few particles of simple shapes. Hence, approximate methods and models must be used. The most widely used of these models assume that the powder can be treated as a continuous medium containing embedded scattering and absorption centers in which plane waves or photons diffuse through it in a manner that can be described by a form of the Boltzmann transport equation known as the equation of radiative transfer. Let a particulate medium be illuminated from above by collimated light of irradiance J from a distant source of electromagnetic radiation. The surface of the medium is the z ¼ 0 plane and is examined by a distant detector located in the empty space above the surface. Let the radiance within and above the surface be I(z, O), where O is the direction of propagation. Then the equation of radiative transfer is Z @Iðz, OÞ 1 ¼ EIðz, OÞ þ Iðz, O0 ÞGðO0 , OÞdO0 cos # @z 4p O¼4p 1 Ez þ J GðO0 , OÞ exp [7] 4p cos i
Zp
and w ¼ S/E; w is the fraction of light scattered by an average scattering element of the medium. For a particulate medium of identical particles E ¼ N s QE S ¼ Ns QS and w ¼ QS =QE
[9]
where N is the number of particles per unit volume, s is their rotationally averaged cross-sectional area, QE is the extinction efficiency (which includes both scattering and absorption), and QS is the scattering efficiency. If the particles are large compared with the wavelength, the scattering elements of the medium can be identified with the individual particles so that w is the fraction of light scattered by a particle. If the particles are smaller than the wavelength, the nature of the scattering element is rather nebulous, but is probably defined by the clumps and statistical irregularities in the packing of the particles. With these definitions, eqn [7] can be put into the form Z @Iðz, OÞ w cos # Iðz, O0 ÞpðyÞdO0 þ J ¼ Iðz, OÞ þ @t 4p O¼4p w t pðy0 Þ exp [10] 4p cos i where y0 is the angle between O0 and O, and t is the optical depth: Z1 t¼
Edz0
[11]
z
No exact analytic solutions to eqn [10] in closed form have been obtained. However, because this equation is important in theories of stellar and planetary atmospheres, including the earth’s, it has been studied extensively and numerical solutions tabulated for many cases of interest. The simplest case is that of isotropically scattering particles, p(y) ¼ 1, for which the radiance emerging from the surface in the direction toward the detector is given by Iði, e, gÞ ¼ J
w m Hðw, m0 ÞHðw, mÞ 4p m0 þ m
[12]
where m ¼ cose, m0 ¼ cosi, and H(w,x) are the AmbartsumianChandrasekhar H-functions given by the solutions to the integral equation
Reflectance Methods and Applications
Hðw, xÞ ¼ 1 þ
Z1 w Hðw, x0 Þ 0 dx xHðw, xÞ 2 x þ x0
[13]
0
These functions are tabulated in many references. An approximate expression for them, accurate to better than 3%, is Hðw, xÞ
1 þ 2x pffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2x 1 w
[14]
If the particles of the medium do not scatter isotropically, but are not too anisotropic, a useful approximate solution for the bidirectional reflectance is rbd ði, e, gÞ ¼
Iði, e, gÞ J w m0 ¼ ½pðgÞ þ Hðw, m0 ÞHðw, mÞ 1 4p m0 þ m
[15]
where H(w, x) is given by eqn [14], and g ¼ py is the phase angle. The directional-hemispherical reflectance for isotropic particles is rh ¼
1 Jm0
Zp=2 Iði, e, gÞ cos e sin ede ¼ 1
pffiffiffiffiffiffiffiffiffiffiffiffi 1 wHðw, m0 Þ [16]
0
To predict the reflectance of a medium by forward modeling of solutions to the radiative transfer equation, w and p(g) must be specified. If the particles are spheres these quantities may be calculated from the size and refractive index using Lorenz-Mie theory. If the particles are not spherical, but are not too irregular, T-matrix theory can be used. However, the particles in most regoliths and laboratory powders are much larger than the wavelength, are irregular in shape, and may have reentrant surfaces; often they are filled with internal scatterers, such as impurities, voids, and crystalline boundaries. The only method by which the scattering parameters of such particles can be calculated exactly is the Monte Carlo ray tracing technique, which is too unwieldy and CPU-intensive to be practical for most applications. An approximate semi-empirical model for w is the equivalent slab model in which the single scattering albedo is given by w ¼ Se þ ð1 Se Þ
ð1 Si ÞY 1 Si Y
Y¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aða þ sÞDÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ri exp ð aða þ sÞDÞ ri þ exp ð
[18]
Here D is a length of the order of the mean particle size, a is the absorption coefficient, a ¼ 4pni =l
If there are no internal scatterers, s ¼ 0, ri ¼ 0, and Y ¼ exp (aD). Approximate expressions for the surface reflection coefficients are Se ¼ 0:0514 þ 0:9183rs ,
[19]
s is the internal scattering coefficient, and ri is the internal scattering factor pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a=ða þ sÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ri ¼ [20] 1 þ a=a þ s
Si ¼ 1
4 nr ðnr þ 1Þ2
[21]
where rs is the specular reflectivity eqn [1]. Equation [17] takes into account the two major processes that are involved in the scattering of light by a particle: surface scattering, which is Fresnel reflection from the outer surface, and volume scattering, which is light that has been refracted into the particle and scattered back out by either the internal scatterers or internal reflection from the surface. The first term on the right-hand side of eqn [17] is the surface scattering; the second term describes the volume scattering, including all orders of reflection from surfaces and internal scatterers. An empirical approximate expression for the single scattering albedo is w¼
1 1 þ aDe
[22]
where De is an effective particle diameter of the order of the mean particle size, but which also depends on the internal scattering coefficients. This expression is less accurate than eqn [17] and is valid only for aD <2. An isolated particle larger than the wavelength of light is a highly anisotropic scatterer because p(g) has a strong, narrow, Fraunhoffer diffraction peak in the forward direction at g¼180 caused by portions of wavefronts that pass by the particle interfering positively with each other. This peak is part of w and p(g) when they are calculated using Lorenz-Mie or T-matrix methods. However, when the same particle is part of a close-packed medium, the surrounding particles block this light, so that the diffraction peak does not exist and must be removed from such calculations. Thus, p(g) is much less anisotropic when in a powder than when in a sparsely packed medium, such as an aerosol, and may often be described to a sufficient approximation by the first few terms of a series of Legendre polynomials
[17]
where Se is the integral of the Fresnel reflection coefficients for light externally incident uniformly from all directions on a plane surface of a material with the same refractive index as the particle and Si is similarly defined for light incident from within the material, Y is the internal transmission factor
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pðgÞ ¼ 1 þ
1 X
bj Pj ð cos gÞ
[23]
1
where Pj(cosg) is the Legendre polynomial of order j. Since particle-scattering functions typically have both forward and backward scattering lobes, terms up to at least second order are required for a realistic description. Another widely used representation of p(g) is the doublelobed Henyey-Greenstein function: pðgÞ ¼
1þc 1 b21 1c þ 2 ð1 2b1 cos g þ b2 Þ3=2 2 1 [24]
1 b21 ð1 2b2 cos g þ b22 Þ
3=2
where 0 b1, b2 1, and c can have any value. Frequently it is sufficient to set b1 ¼ b2. The only other restriction on the b and c parameters is that p(g) 0.
Reflectance Methods and Applications
Reflectance Spectroscopy Spectra obtained by reflectance have a large number of uses, such as verification of solid state molecular orbital calculations by comparison with measurements and determination of composition by comparison of remotely measured spectra with libraries of known composition. If the composition of a particulate medium is known the particle size can be estimated, since the band depths and shapes depend on size. Figure 1 shows a typical bidirectional reflectance in which this quantity is plotted against phase angle. The dots show the measured values of a powder consisting of spherical glass particles of known size and refractive index. The line is the reflectance predicted from eqn [15], in which w and p(g) were calculated using Lorenz-Mie theory but with the diffraction peak removed. The overall agreement is quite satisfactory. The reflectance, whether bidirectional or hemispherical, is determined mainly by the single scattering albedo, which controls the height of the reflectance and the angular shape of the bidirectional reflectance. The single scattering phase function essentially fine-tunes them. Figure 2 plots the bidirectional and hemispherical reflectances, eqns [14]–[16], for a medium of isotropically scattering particles versus w. The figures show that the reflectances are nonlinear functions that increase monotonically with w. To extract quantitative absorption spectra from measurements of the reflectance spectra of powders it is necessary to first determine w by reverse modeling of solutions to the radiative transfer equation, such as eqn [15]. Ideally, the bidirectional reflectance is measured over as wide a range of phase angles as possible and w and p(g) are found by linear regression analysis. However, p(g) cannot be detemined from the hemispherical reflectance alone, and often the bidirectional 1.2
reflectance is measured at only a single set of angles. In these cases the only way to proceed is to assume that the scattering is isotropic (p(g) ¼ 1) and solve for w. The dependence of the single scattering albedo, eqn [17], on absorption coefficient is shown in Figure 3, in which w is plotted versus aD. When aD<< 1, w1, and the reflectance is high. As aD increases, w and the reflectance decrease. When aD >5 the second term in eqn [17] becomes negligible and w bottoms out at w ¼ Se. In this region, the reflectance is insensitive to a. As ni and a increase still further to the point where ni2 is no longer <<1, Se and w increases in accordance 1.2
1
0.8 Reflectance
934
0.6
0.4
Hemispherical reflectance
0.2 I/F
0
0
0.2
0.4
0.6
0.8
1
w Figure 2 The bidirectional reflectance radiance factor (I/F) at i ¼ 0, e ¼ 5 , and g ¼ 5 calculated from eqn [15], and the directionalhemispherical reflectance rh at i ¼ 0 from eqn [15].
1
1
0.8
I/F
0.8 0.6 0.6 w
0.4
0.4 0.2 0.2
0 0
80 40 Phase angle
120
Figure 1 Bidirectional reflectance (radiance factor I/F) for a powder of glass spheres with n ¼ 1.51þi 3.5105 and diameters between 2 and 50 mm, with the source fixed at i ¼ 60 . The detector was varied between 70 on either side of the normal in the vertical plane containing the source and the surface normal. The dots are the measured data and the line is predicted from eqn [15].
0
0
1
2
3
4
5
D Figure 3 Single scattering albedo versus aD. Solid line is eqn [17]; in this example nr ¼ 1.50, 4p D/l ¼ 5, and s ¼ 0. Dashed line is the approximate expression [22] in which De ¼ 2.595 D.
Reflectance Methods and Applications
In a nonlinear mixture, the different types of particles are together in intimate contact. Because multiple scattering is a nonlinear process the reflectances do not combine linearly. Mixtures may be taken into account by generalizing eqns [9]. Denote the various types of particles by subscript j. Then eqns [8] become X X Nj sj QEj , S ¼ Nj sj QSj E¼
By reflectance
1200
By transmittance Absorption coefficient (cm−1)
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1000 800 600
j
400
j
and
200 0 0
0.5
1.0 1.5 Wavelength (micrometers)
2.0
2.5
Figure 4 Comparison of the spectral absorption coefficient of a cobalt silicate glass measured by transmission of a thin section (solid line) and by reflectance of the glass in powder form (open circles). Reproduced from Hapke B and Wells E (1981) Bidirectional reflectance spectroscopy. 2. Experiments and observations. Journal of Geophysical Research 86: 3055–3060, with permission from the American Geophysical Union.
with eqn [21]. This illustrates a seemingly paradoxical property of absorption bands seen in reflectance: relatively weak bands (aD <5) give rise to reflectance minima, but strong bands (ni2 >0.1) cause reflectance maxima. Figure 3 also plots the approximate expression for w, eqn [22]. To retrieve quantitative spectra of a(l) from w(l), the parameters in eqn [17] must be specified. Sometimes these are known or can be estimated from other information about the material, but often they are completely unknown. In this case a relative absorbance spectrum a(l)De can be obtained using eqn [22] for w, keeping in mind that this expression is valid only in the region of w where aDe is small. If a is independently known at one or a few wavelengths, this can be used to calibrate De by fitting the spectrum at those wavelengths. Figure 4 compares the spectrum of the absorption coefficient of a silicate glass doped with Ca2þ measured from the transmittance of a thin section and retrieved from the diffuse reflectance of the glass in powder form. The approximate eqn [22] for w was used with De adjusted to match the transmission data in the blue region of the spectrum. The agreement is excellent except at the highest values of a, where eqn [22] begins to lose accuracy.
w¼
X j
!, Nj sj QSj
X
! Nj sj QEj
[25]
j
If the particles are roughly spherical, the bulk density of the jth type of particle is Bj ¼ rj Nj pD3j =6
[26]
where rj is the solid density of the jth type of particle, and Dj is their diameter. When the particles are much larger than the wavelength, QEj1 to a sufficient approximation, and w becomes 0 1 0 1 pD2j QSj pD2j QEj B B j j B C,B C 4 C BX 4 C BX B C B C w¼B C B C B j pD3j C B j pD3j C @ A @ A rj rj [27] 6 6 0 1,0 1 X Bj X Bj A ¼@ wj A @ rj Dj rj Dj j j where wj ¼ QSj/QEj. Equation [27] is the mixing formula for intimate mixtures and may be inserted into the appropriate equations to calculate the reflectance.
See also: ATR and Reflectance IR Spectroscopy, Applications; Plasmon-Controlled Fluorescence Methods and Applications; Surface Plasmon Resonance, Applications; Surface Plasmon Resonance, Instrumentation; Surface Plasmon Resonance, Theory.
Further Reading Mixtures Most particulate media encountered in nature or the laboratory are mixtures of different types of particles. The particles may differ in terms of size, shape, composition, degree of surface roughness, and internal structure. There are two types of mixtures: linear or checkerboard, and nonlinear or intimate. Linear mixtures occur when different types of deposits are side-by-side within the field of view of the detector, such as a field of soil next to a snow pack. In that case the reflectance is simply the average of the individual reflectances weighted by area.
Bohren C and Huffman D (1983) Absorption and Scattering of Light by Small Particles. New York: Wiley. Chandrasekhar S (1960) Radiative Transfer. New York: Dover. Hapke B (1993) Theory of Reflectance and Emittance Spectroscopy. Cambridge: Cambridge University Press. Lenoble J (ed.) (1985) Radiative Transfer in Scattering and Absorbing Atmospheres Standard Computational Procedures. Hampton: Deepak Publishing. Mishchenko M, Travis L, and Lacis A (2002) Scattering, Absorption and Emission of Light by Small Particles. Cambridge: Cambridge University Press. Pieters C and Englert P (1993) Remote Geochemical Analysis: Elemental and Mineralogical Composition. Cambridge: Cambridge University Press.
Introduction The amount of light reflected or scattered by a solid or particulate medium changes with wavelength. These spectral variations are caused by wavelength-dependent absorption and not only give rise to the sensation of color, but can also be used to infer such properties of the material as composition, complex refractive index, and electronic or molecular structure. The relative amount of light scattered into different directions depends on the physical structure, such as particle size and spacing, of the material. Both the spectral and directional variations of the reflectance can be measured without physically contacting the medium, and hence, they form a major tool in the study of planetary surfaces by remote sensing. Virtually all spacecraft that fly by or orbit the earth or other planets carry imaging spectrometer array detectors for this purpose. Absorption spectra are commonly measured by transmittance, for which a thin slab of material is sliced off and both sides polished. One advantage of measuring spectra by diffuse reflectance is convenience: the only preparation required is that the sample be in powder form. Another is that if the absorbance is too strong, it may be difficult to prepare and handle a section thin enough to pass light through, whereas the powder can simply be ground to a finer size. On the negative side, the process of diffuse reflectance involves multiple scattering of photons from one particle to another and is highly nonlinear. Thus, fairly complex reflectance models are required to extract quantitative spectral absorption coefficients. As with gases, absorption by solids usually occurs in discrete bands, the exception being metals, which absorb over a broad range of wavelengths. However, most solid state bands are much broader than their gaseous counterparts, with Dl/l 10% being typical, where l is the wavelength. For this reason the bands often overlap, complicating their recognition and sometimes producing an apparent absorption continuum. Solid state bands arise from a variety of causes. Absorption in metals and semiconductors is due to photon-induced motion of conduction band electrons, which then collide with impurities or irregularities and transfer this energy to the solid state lattice. In insulators, ultraviolet, visible, and near-infrared photons are absorbed by electronic transitions, while at mid-infrared and radio frequencies absorption occurs by changes in the vibrational and rotational states of the ions of the lattice. The electronic transitions are of three types: valence to conduction band jumps, electronic excitation, in which an electron remains localized on one ion while changing its orbital state, and charge transfer, in which its localization state moves from one ion to another nearby.
Specular Reflection: Kramers-Kronig Spectroscopy If the medium is continuous and homogeneous and is separated from a vacuum by a smooth plane interface, light
Encyclopedia of Spectroscopy and Spectrometry, Third Edition
incident on the surface will be reflected in a specular or mirror-like manner, and the amount reflected is governed by the Fresnel reflection coefficients. For vertically incident light of arbitrary polarization, the specular reflectivity of the intensity is rs ðnÞ ¼
ðnr 1Þ2 þ n2i ðnr þ 1Þ2 þ n2i
[1]
corresponding to the reflection coefficient for the amplitude of the electric field, Ereflected 1 nr ini ¼ Eincident 1 þ nr þ ini
[2]
the where n(n) ¼ nr(n)þini(n)pisffiffiffiffiffiffi ffi complex index of refraction, n is the frequency, and i ¼ 1. If ni21, nr can be calculated from a measurement of rs(n) directly using eqn [1]. If the imaginary component is not negligible, both components of n can still be obtained from rs(n) using the Kramers-Kronig method. The reflection coefficient can be written in the form [3] Ereflected =Eincident ¼ exp ði’Þ pffiffiffiffiffiffiffiffi where ðnÞ ¼ rðnÞ and tan ’ðnÞ ¼ 2ni =½n2r 1 þ n2i . Solving for the components of n, nr ¼
1 2 2 sin ’ , ni ¼ 1 2 cos ’ þ 2 1 2 cos ’ þ 2
[4]
Assuming that n is analytic, the quantities and ’ must satisfy the Cauchy relations: ’ðnÞ ¼
2n p
Z1
ln ½ðnÞ ¼
ln ½ðn0 Þ 0 dn , n0 2 n2
0
2 p
Z1 0
n0 ’ðn0 Þ 0 dn n0 2 n2
[5]
The reflectivity rs is measured over as wide a range of wavepffiffiffiffi lengths as possible, from which ¼ rs is found. A suitable theory, such as the Drude or Lorentz model, is used to extrapolate the measured data to zero and infinity, and the result inserted into eqn [5] for ’; then nr and ni as functions of frequency or wavelength are found from eqns [4].
Diffuse Reflectance from a Particulate Medium The reflectances of disordered particulate media, such as powders in the laboratory, soils, or planetary regoliths, are problems of particular theoretical and practical interest. Most reflectance measurements are one of two types: bidirectional reflectance, denoted by rbd, in which the surface is illuminated from one direction by highly collimated light, such as sunlight
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or a laser, and is observed by a detector that subtends a small angle as seen from the surface; and directional-hemispherical reflectance, denoted by rh, in which the surface is illuminated by collimated light, but the total light scattered into all directions is integrated and measured using an integrating sphere. The latter type of reflectance is also known as the hemispherical reflectance, hemispherical albedo, and plane albedo. Other types of reflectance, including hemispherical-directional and bihemispherical, are also possible. In planetary photometry the total fraction of incident sunlight scattered into all directions by a solar system object is called the Bond albedo, and is equal to the bihemispherical reflectance. The most elementary bidirectional reflectance is an empirical expression known as Lambert’s law, 1 rL ði, e, gÞ ¼ cos i p
scattered by an element of the medium from direction O0 to direction O. The first term on the right-hand side of eqn [7] gives the decrease of I caused by scattering and absorption as the wave propagates through the medium, the second term describes the increase of I due to light traveling in direction O0 scattered into direction O, and the third term is the source term and describes how I is increased as incident irradiance is scattered from direction O0 into direction O. Define the scattering coefficient S, single scattering phase function p(y), and single scattering albedo w by G(O0 ,O) ¼ Sp (y), where y is the angle between O0 and O, and p(y) is normalized so that 1 2
where # is the angle between O and the positive z axis, O0 is the direction of the incident irradiance, i, E is the extinction coefficient, and G(O0 ,O) is the average probability of light being
pðyÞ sin ydy ¼ 1
[8]
0
[6]
where i is the angle between the direction to the source of illumination and the normal to the sample surface, e is the angle between the direction to the detector and the normal, and g is the phase angle, the angle between the directions to the source and detector subtended at the surface. This equation is widely used because of its simplicity. Although no material is known to obey Lambert’s Law exactly, several purified powdered forms of high albedo materials, such as titanium dioxide, barium sulfate, and polytetrafluoroethylene, come close. Thus, these materials can provide a convenient empirical reflectance standard. For mechanical reasons many reflectance spectrometers cannot measure the incident irradiance J directly. Instead, the measured radiance I of light scattered by the material of interest is ratioed to the radiance scattered at i ¼ e ¼ 0 by the standard, which is assumed to be a Lambert surface. A quantity F is defined as F ¼ J/p and the ratio of reflectances is called the radiance factor, denoted by I/F. At present the exact solution of Maxwell’s equations for scattering by a plane electromagnetic wave incident on a disordered semi-infinite medium of interacting particles can be found numerically only for systems of a few particles of simple shapes. Hence, approximate methods and models must be used. The most widely used of these models assume that the powder can be treated as a continuous medium containing embedded scattering and absorption centers in which plane waves or photons diffuse through it in a manner that can be described by a form of the Boltzmann transport equation known as the equation of radiative transfer. Let a particulate medium be illuminated from above by collimated light of irradiance J from a distant source of electromagnetic radiation. The surface of the medium is the z ¼ 0 plane and is examined by a distant detector located in the empty space above the surface. Let the radiance within and above the surface be I(z, O), where O is the direction of propagation. Then the equation of radiative transfer is Z @Iðz, OÞ 1 ¼ EIðz, OÞ þ Iðz, O0 ÞGðO0 , OÞdO0 cos # @z 4p O¼4p 1 Ez þ J GðO0 , OÞ exp [7] 4p cos i
Zp
and w ¼ S/E; w is the fraction of light scattered by an average scattering element of the medium. For a particulate medium of identical particles E ¼ N s QE S ¼ Ns QS and w ¼ QS =QE
[9]
where N is the number of particles per unit volume, s is their rotationally averaged cross-sectional area, QE is the extinction efficiency (which includes both scattering and absorption), and QS is the scattering efficiency. If the particles are large compared with the wavelength, the scattering elements of the medium can be identified with the individual particles so that w is the fraction of light scattered by a particle. If the particles are smaller than the wavelength, the nature of the scattering element is rather nebulous, but is probably defined by the clumps and statistical irregularities in the packing of the particles. With these definitions, eqn [7] can be put into the form Z @Iðz, OÞ w cos # Iðz, O0 ÞpðyÞdO0 þ J ¼ Iðz, OÞ þ @t 4p O¼4p w t pðy0 Þ exp [10] 4p cos i where y0 is the angle between O0 and O, and t is the optical depth: Z1 t¼
Edz0
[11]
z
No exact analytic solutions to eqn [10] in closed form have been obtained. However, because this equation is important in theories of stellar and planetary atmospheres, including the earth’s, it has been studied extensively and numerical solutions tabulated for many cases of interest. The simplest case is that of isotropically scattering particles, p(y) ¼ 1, for which the radiance emerging from the surface in the direction toward the detector is given by Iði, e, gÞ ¼ J
w m Hðw, m0 ÞHðw, mÞ 4p m0 þ m
[12]
where m ¼ cose, m0 ¼ cosi, and H(w,x) are the AmbartsumianChandrasekhar H-functions given by the solutions to the integral equation
Reflectance Methods and Applications
Hðw, xÞ ¼ 1 þ
Z1 w Hðw, x0 Þ 0 dx xHðw, xÞ 2 x þ x0
[13]
0
These functions are tabulated in many references. An approximate expression for them, accurate to better than 3%, is Hðw, xÞ
1 þ 2x pffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2x 1 w
[14]
If the particles of the medium do not scatter isotropically, but are not too anisotropic, a useful approximate solution for the bidirectional reflectance is rbd ði, e, gÞ ¼
Iði, e, gÞ J w m0 ¼ ½pðgÞ þ Hðw, m0 ÞHðw, mÞ 1 4p m0 þ m
[15]
where H(w, x) is given by eqn [14], and g ¼ py is the phase angle. The directional-hemispherical reflectance for isotropic particles is rh ¼
1 Jm0
Zp=2 Iði, e, gÞ cos e sin ede ¼ 1
pffiffiffiffiffiffiffiffiffiffiffiffi 1 wHðw, m0 Þ [16]
0
To predict the reflectance of a medium by forward modeling of solutions to the radiative transfer equation, w and p(g) must be specified. If the particles are spheres these quantities may be calculated from the size and refractive index using Lorenz-Mie theory. If the particles are not spherical, but are not too irregular, T-matrix theory can be used. However, the particles in most regoliths and laboratory powders are much larger than the wavelength, are irregular in shape, and may have reentrant surfaces; often they are filled with internal scatterers, such as impurities, voids, and crystalline boundaries. The only method by which the scattering parameters of such particles can be calculated exactly is the Monte Carlo ray tracing technique, which is too unwieldy and CPU-intensive to be practical for most applications. An approximate semi-empirical model for w is the equivalent slab model in which the single scattering albedo is given by w ¼ Se þ ð1 Se Þ
ð1 Si ÞY 1 Si Y
Y¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aða þ sÞDÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ri exp ð aða þ sÞDÞ ri þ exp ð
[18]
Here D is a length of the order of the mean particle size, a is the absorption coefficient, a ¼ 4pni =l
If there are no internal scatterers, s ¼ 0, ri ¼ 0, and Y ¼ exp (aD). Approximate expressions for the surface reflection coefficients are Se ¼ 0:0514 þ 0:9183rs ,
[19]
s is the internal scattering coefficient, and ri is the internal scattering factor pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a=ða þ sÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ri ¼ [20] 1 þ a=a þ s
Si ¼ 1
4 nr ðnr þ 1Þ2
[21]
where rs is the specular reflectivity eqn [1]. Equation [17] takes into account the two major processes that are involved in the scattering of light by a particle: surface scattering, which is Fresnel reflection from the outer surface, and volume scattering, which is light that has been refracted into the particle and scattered back out by either the internal scatterers or internal reflection from the surface. The first term on the right-hand side of eqn [17] is the surface scattering; the second term describes the volume scattering, including all orders of reflection from surfaces and internal scatterers. An empirical approximate expression for the single scattering albedo is w¼
1 1 þ aDe
[22]
where De is an effective particle diameter of the order of the mean particle size, but which also depends on the internal scattering coefficients. This expression is less accurate than eqn [17] and is valid only for aD <2. An isolated particle larger than the wavelength of light is a highly anisotropic scatterer because p(g) has a strong, narrow, Fraunhoffer diffraction peak in the forward direction at g¼180 caused by portions of wavefronts that pass by the particle interfering positively with each other. This peak is part of w and p(g) when they are calculated using Lorenz-Mie or T-matrix methods. However, when the same particle is part of a close-packed medium, the surrounding particles block this light, so that the diffraction peak does not exist and must be removed from such calculations. Thus, p(g) is much less anisotropic when in a powder than when in a sparsely packed medium, such as an aerosol, and may often be described to a sufficient approximation by the first few terms of a series of Legendre polynomials
[17]
where Se is the integral of the Fresnel reflection coefficients for light externally incident uniformly from all directions on a plane surface of a material with the same refractive index as the particle and Si is similarly defined for light incident from within the material, Y is the internal transmission factor
933
pðgÞ ¼ 1 þ
1 X
bj Pj ð cos gÞ
[23]
1
where Pj(cosg) is the Legendre polynomial of order j. Since particle-scattering functions typically have both forward and backward scattering lobes, terms up to at least second order are required for a realistic description. Another widely used representation of p(g) is the doublelobed Henyey-Greenstein function: pðgÞ ¼
1þc 1 b21 1c þ 2 ð1 2b1 cos g þ b2 Þ3=2 2 1 [24]
1 b21 ð1 2b2 cos g þ b22 Þ
3=2
where 0 b1, b2 1, and c can have any value. Frequently it is sufficient to set b1 ¼ b2. The only other restriction on the b and c parameters is that p(g) 0.
Reflectance Methods and Applications
Reflectance Spectroscopy Spectra obtained by reflectance have a large number of uses, such as verification of solid state molecular orbital calculations by comparison with measurements and determination of composition by comparison of remotely measured spectra with libraries of known composition. If the composition of a particulate medium is known the particle size can be estimated, since the band depths and shapes depend on size. Figure 1 shows a typical bidirectional reflectance in which this quantity is plotted against phase angle. The dots show the measured values of a powder consisting of spherical glass particles of known size and refractive index. The line is the reflectance predicted from eqn [15], in which w and p(g) were calculated using Lorenz-Mie theory but with the diffraction peak removed. The overall agreement is quite satisfactory. The reflectance, whether bidirectional or hemispherical, is determined mainly by the single scattering albedo, which controls the height of the reflectance and the angular shape of the bidirectional reflectance. The single scattering phase function essentially fine-tunes them. Figure 2 plots the bidirectional and hemispherical reflectances, eqns [14]–[16], for a medium of isotropically scattering particles versus w. The figures show that the reflectances are nonlinear functions that increase monotonically with w. To extract quantitative absorption spectra from measurements of the reflectance spectra of powders it is necessary to first determine w by reverse modeling of solutions to the radiative transfer equation, such as eqn [15]. Ideally, the bidirectional reflectance is measured over as wide a range of phase angles as possible and w and p(g) are found by linear regression analysis. However, p(g) cannot be detemined from the hemispherical reflectance alone, and often the bidirectional 1.2
reflectance is measured at only a single set of angles. In these cases the only way to proceed is to assume that the scattering is isotropic (p(g) ¼ 1) and solve for w. The dependence of the single scattering albedo, eqn [17], on absorption coefficient is shown in Figure 3, in which w is plotted versus aD. When aD<< 1, w1, and the reflectance is high. As aD increases, w and the reflectance decrease. When aD >5 the second term in eqn [17] becomes negligible and w bottoms out at w ¼ Se. In this region, the reflectance is insensitive to a. As ni and a increase still further to the point where ni2 is no longer <<1, Se and w increases in accordance 1.2
1
0.8 Reflectance
934
0.6
0.4
Hemispherical reflectance
0.2 I/F
0
0
0.2
0.4
0.6
0.8
1
w Figure 2 The bidirectional reflectance radiance factor (I/F) at i ¼ 0, e ¼ 5 , and g ¼ 5 calculated from eqn [15], and the directionalhemispherical reflectance rh at i ¼ 0 from eqn [15].
1
1
0.8
I/F
0.8 0.6 0.6 w
0.4
0.4 0.2 0.2
0 0
80 40 Phase angle
120
Figure 1 Bidirectional reflectance (radiance factor I/F) for a powder of glass spheres with n ¼ 1.51þi 3.5105 and diameters between 2 and 50 mm, with the source fixed at i ¼ 60 . The detector was varied between 70 on either side of the normal in the vertical plane containing the source and the surface normal. The dots are the measured data and the line is predicted from eqn [15].
0
0
1
2
3
4
5
D Figure 3 Single scattering albedo versus aD. Solid line is eqn [17]; in this example nr ¼ 1.50, 4p D/l ¼ 5, and s ¼ 0. Dashed line is the approximate expression [22] in which De ¼ 2.595 D.
Reflectance Methods and Applications
In a nonlinear mixture, the different types of particles are together in intimate contact. Because multiple scattering is a nonlinear process the reflectances do not combine linearly. Mixtures may be taken into account by generalizing eqns [9]. Denote the various types of particles by subscript j. Then eqns [8] become X X Nj sj QEj , S ¼ Nj sj QSj E¼
By reflectance
1200
By transmittance Absorption coefficient (cm−1)
935
1000 800 600
j
400
j
and
200 0 0
0.5
1.0 1.5 Wavelength (micrometers)
2.0
2.5
Figure 4 Comparison of the spectral absorption coefficient of a cobalt silicate glass measured by transmission of a thin section (solid line) and by reflectance of the glass in powder form (open circles). Reproduced from Hapke B and Wells E (1981) Bidirectional reflectance spectroscopy. 2. Experiments and observations. Journal of Geophysical Research 86: 3055–3060, with permission from the American Geophysical Union.
with eqn [21]. This illustrates a seemingly paradoxical property of absorption bands seen in reflectance: relatively weak bands (aD <5) give rise to reflectance minima, but strong bands (ni2 >0.1) cause reflectance maxima. Figure 3 also plots the approximate expression for w, eqn [22]. To retrieve quantitative spectra of a(l) from w(l), the parameters in eqn [17] must be specified. Sometimes these are known or can be estimated from other information about the material, but often they are completely unknown. In this case a relative absorbance spectrum a(l)De can be obtained using eqn [22] for w, keeping in mind that this expression is valid only in the region of w where aDe is small. If a is independently known at one or a few wavelengths, this can be used to calibrate De by fitting the spectrum at those wavelengths. Figure 4 compares the spectrum of the absorption coefficient of a silicate glass doped with Ca2þ measured from the transmittance of a thin section and retrieved from the diffuse reflectance of the glass in powder form. The approximate eqn [22] for w was used with De adjusted to match the transmission data in the blue region of the spectrum. The agreement is excellent except at the highest values of a, where eqn [22] begins to lose accuracy.
w¼
X j
!, Nj sj QSj
X
! Nj sj QEj
[25]
j
If the particles are roughly spherical, the bulk density of the jth type of particle is Bj ¼ rj Nj pD3j =6
[26]
where rj is the solid density of the jth type of particle, and Dj is their diameter. When the particles are much larger than the wavelength, QEj1 to a sufficient approximation, and w becomes 0 1 0 1 pD2j QSj pD2j QEj B B j j B C,B C 4 C BX 4 C BX B C B C w¼B C B C B j pD3j C B j pD3j C @ A @ A rj rj [27] 6 6 0 1,0 1 X Bj X Bj A ¼@ wj A @ rj Dj rj Dj j j where wj ¼ QSj/QEj. Equation [27] is the mixing formula for intimate mixtures and may be inserted into the appropriate equations to calculate the reflectance.
See also: ATR and Reflectance IR Spectroscopy, Applications; Plasmon-Controlled Fluorescence Methods and Applications; Surface Plasmon Resonance, Applications; Surface Plasmon Resonance, Instrumentation; Surface Plasmon Resonance, Theory.
Further Reading Mixtures Most particulate media encountered in nature or the laboratory are mixtures of different types of particles. The particles may differ in terms of size, shape, composition, degree of surface roughness, and internal structure. There are two types of mixtures: linear or checkerboard, and nonlinear or intimate. Linear mixtures occur when different types of deposits are side-by-side within the field of view of the detector, such as a field of soil next to a snow pack. In that case the reflectance is simply the average of the individual reflectances weighted by area.
Bohren C and Huffman D (1983) Absorption and Scattering of Light by Small Particles. New York: Wiley. Chandrasekhar S (1960) Radiative Transfer. New York: Dover. Hapke B (1993) Theory of Reflectance and Emittance Spectroscopy. Cambridge: Cambridge University Press. Lenoble J (ed.) (1985) Radiative Transfer in Scattering and Absorbing Atmospheres Standard Computational Procedures. Hampton: Deepak Publishing. Mishchenko M, Travis L, and Lacis A (2002) Scattering, Absorption and Emission of Light by Small Particles. Cambridge: Cambridge University Press. Pieters C and Englert P (1993) Remote Geochemical Analysis: Elemental and Mineralogical Composition. Cambridge: Cambridge University Press.