Chapter 3 Theory of Thermal Radiation and Radiative Properties

CHAPTER 3

Theory of Thermal Radiation and Radiative Properties Zhuomin M. Zhang1 and Bong Jae Lee2

Contents 1. Introduction 2. Thermodynamics of Blackbody Radiation 2.1. Radiance, heat flux, irradiance, and exitance 2.2. Blackbody cavity and radiation pressure 2.3. Entropy and the Stefan2Boltzmann law 2.4. Planck’s law of blackbody radiation 3. Radiative Properties of Materials 3.1. Emissivity 3.2. Absorptivity 3.3. Kirchhoff’s law 3.4. Reflectivity 3.5. Effective or apparent emissivity 3.6. Atmospheric transmission effects 3.7. Semitransparent materials 4. Electromagnetic Wave Theory 4.1. Macroscopic Maxwell’s equation 4.2. Wave equation and wave propagation 4.3. Fresnel reflection and transmission coefficients 4.4. Reflectivity and emissivity 5. Optical Properties 5.1. The Drude model for a conductor 5.2. The Lorentz oscillator model for a nonconductor 5.3. Optical constants of a semiconductor 6. Radiative Properties of Layered Structures 6.1. The ray-tracing method 6.2. Thin-film optics

1

2

73 76 76 78 79 81 84 84 88 89 90 93 95 98 98 98 100 103 106 109 110 112 116 119 119 121

George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA 15261-3648, USA

Experimental Methods in the Physical Sciences, Volume 42 ISSN 1079-4042, DOI 10.1016/S1079-4042(09)04203-9

r 2010 Elsevier Inc. All rights reserved.

73

74

Zhuomin M. Zhang and Bong Jae Lee

6.3. Radiative properties of multilayer structures 6.4. Thin films on a thick substrate 7. Summary Acknowledgement References

123 125 128 129 129

1. Introduction Understanding the fundamentals of thermal emission and the radiative properties of materials is not only important to the establishment of radiation thermometry but also essential to the proper selection and application of radiometric temperature measurement techniques to industrial processes. This chapter describes the basic theory of thermal radiation and electrodynamics, as well as the definition and prediction of optical and radiative properties of solid materials. The nature of light has been a perplexing subject for ancient philosophers as well as for modern scientists. While earlier wave theory developed by Huygens in 1678 was able to explain light reflection and refraction, it was incomplete and was largely supplanted by the corpuscular theory of light formulated by Newton in 1704. However, Young’s doubleslit experiment in 1801 demonstrated interference phenomenon and hence the wave nature of light, as proposed by Huygens, reemerged as a description of light. Fresnel (178821827) further advanced the wave theory to explain polarization and interference effects. The discoveries of infrared (heat) radiation by Herschel in 1800 and ultraviolet (chemical) radiation by Ritter in 1801 broadened the understanding of the optical spectrum. Built on several separate relations, Maxwell in 1873 established the electromagnetic-wave theory, which was further confirmed by Hertz’s discovery of radio waves due to electrical vibration. The Michelson2Morley experiment in 1887 demonstrated that light waves can travel in vacuum without a medium with a speed independent of direction. Electromagnetic waves cover a broad range of frequencies, and are transverse waves, as illustrated in Figure 1 for a linearly polarized plane wave propagating along the y-axis. Here, k is the wavevector that points in the direction of propagation and l is the wavelength. The electric field vector E oscillates parallel to the z-axis and the magnetic field vector H oscillates parallel to the x-axis. Note that E, H, and k form a right-handed triplet. The surfaces of constant phase are parallel to the x2z plane. The speed of propagation is related to the wavelength lm (in the medium) and frequency n by: c ¼ lm n

(1)

The frequency is independent of the medium in which radiation propagates, but the speed of light and the wavelength are inversely proportional to the

75

Theory of Thermal Radiation and Radiative Properties

z E

k

y x

H

 Figure 1

Illustration of the propagation of a plane wave of wavelength l.

Frequency,  (Hz) 1022

1020

1018

1016

1014

1012

1010

108

106

Thermal Radiation -ray

x-ray

Optical Radiation

Micro-wave Radio-wave

Visible UV 10−6

10−4

10−2

IR 100

102

104

106

108

Wavelength,  (μm)

Figure 2 The electromagnetic-wave spectrum. The visible wavelengths are from about 380 to 750 nm. Thermal radiation covers the wavelength region from 0.1 to 1,000 mm.

refractive index n of the medium. Therefore, we have lm ¼ l/n and c ¼ c0/n, where l is used as the wavelength in vacuum and c0 is the speed of light in vacuum. The refractive index of air is about 0.03% greater than that of vacuum, and thus, it is often approximated to 1. It should be noted that c0 is a fundamental constant, which is fixed to an exact value [1]; see Appendix A for fundamental and other physical constants. In the 1983 version of the International System (SI) of Units [2], the unit of length (meter) is defined based on the speed of light in vacuum and the unit of time (second). In practice, monochromatic radiation refers to radiation that is confined to a very narrow spectral band, such as that from lasers or spectral lines emitted by gases. Polychromatic or broadband radiation, such as that emitted from a furnace or a heated plate, can be viewed as a superposition of monochromatic radiation. The electromagnetic spectrum covers a large spectral range from g-rays to radio waves, as shown in Figure 2. The visible spectrum covers the wavelength region from about 380 to 750 nm.

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Zhuomin M. Zhang and Bong Jae Lee

Thermal radiation often refers to the spectral region at wavelengths from approximately 0.1 to 1,000 mm, which covers most of the radiation emitted by objects at temperatures from about 20 to 15,000 K. In this definition, thermal radiation includes part of the ultraviolet (UV) region and the entire visible (VIS) and infrared (IR) regions. On contrary, the term optical radiation is used rather loosely and often is taken to include UV, VIS, and IR spectral regions, from extreme-deep UV (l ¼ 10 nm) to very far-IR (l ¼ 1,000 mm). Nevertheless, the wave approach to the fundamental nature of light is only one side of the story. In 1900, Planck introduced the concept of energy quanta in order to derive the law of thermal radiation (a more detailed discussion on thermal radiation will be presented in the next section). In 1905, Einstein used the concept of energy quantization to explain the phenomenon of photoelectric emission. Later in 1924, de Broglie hypothesized that a physical particle could also exhibit wave nature. Wave2particle duality became an established foundation of quantum mechanics and more broadly modern physics. According to this theory, radiation is made of discrete energy quanta, called photons. The energy of an individual photon is proportional to its frequency, i.e., E ¼ hn

(2)

where h is the Planck constant, and the frequency n is in Hertz. Although photons are massless, they possess momentum in addition to energy. The momentum of each photon is given as: p¼

hn h ¼ c lm

(3)

In vector form, p ¼ _k, where _ ¼ h=2p is the reduced Planck constant and the magnitude of the wavevector is k ¼ 2p/lm. More details on the mechanism of radiation emission, as well as how radiation propagates inside a medium and how it interacts with materials, will be presented in the remainder of this chapter. Section 2 discusses the thermodynamic theory of blackbody radiation. Section 3 defines radiative properties of materials and the relationship between different properties. Section 4 elaborates on the electromagnetic-wave theory and how to apply the macroscopic Maxwell equations to describe wave propagation inside a medium as well as wave reflection and refraction at the interface between different media. Section 5 addresses models of the frequency-dependent dielectric functions of various solid materials in the optical frequency region. Section 6 presents methods to obtain radiative properties of layered structures. A brief summary is given in Section 7.

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Theory of Thermal Radiation and Radiative Properties

2. Thermodynamics of Blackbody Radiation 2.1. Radiance, heat flux, irradiance, and exitance The spectral radiance is defined as the radiant power dq from a pencil cone, as shown in Figure 3, from a surface area dA in the direction s^ (defined by the zenith angle y and the azimuthal angle f), per unit wavelength interval, per unit solid angle, and per unit area projected onto the direction s^. Hence the spectral radiance is L l ðl; y; fÞ ¼

dq dA cosy dO dl

(4)

where dO is the element solid angle, and the unit of Ll is W/(m2 sr mm). Note that q is also called radiant flux, denoted by the symbol F in some chapters of this book. Using the spherical polar coordinates shown in Figure 4, by definition, we have dO ¼

dAn rdy r siny df ¼ ¼ siny dydf r2 r2

(5)

The total radiance is the integral of the spectral radiance over all wavelengths, i.e., Z

1

L¼

L l ðlÞdl

(6)

0

It should be noted that in most thermal radiation books, L and Ll are called intensity and spectral intensity, respectively [3,4]. However, the proper metrological terminology for these expressions is spectral or total radiance as given in Equation (4) or (6), respectively. This is the convention that is observed here and throughout this book. z

sˆ

θ

dΩ dA

y

φ x

Figure 3

Illustration of radiance and the solid angle.

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Zhuomin M. Zhang and Bong Jae Lee

z θ

dθ dAn

r dΩ dA

y dφ φ

x

Figure 4

Spherical coordinates showing the solid angle dO ¼ sin hdhd/.

The radiative heat flux is the rate of energy flow per unit area across an element area dA, W/m2, i.e., q00 ¼

dq ¼ dA

Z

2p

Z

p=2

L cosy siny dydf f¼0

(7)

y¼0

Note that dA can be either a real surface or an imaginary surface. For a vacuum enclosure, dA can be any surface element inside the volume. Assuming the enclosure is at thermal equilibrium (i.e., with isothermal walls at steady state), photon energies exist uniformly throughout the volume. Because photons travel with a speed of c0 in all directions, it can be shown from kinetic theory that [5,6] q00 ¼

uc 0 4

and

q00l ¼

ul c 0 4

(8)

where u and ul are the total (with units of J/m3) and spectral energy densities (with units of J/m3 mm), respectively. The radiative heat flux that leaves a surface is called exitance M, which includes emitted, reflected, and transmitted radiant power toward the hemisphere above the surface. If the radiance from a surface is independent of the direction, the surface is said to be diffuse and from Equation (7), one obtains, M ¼ pL

and

M l ¼ pL l

(9)

where M is the total exitance and Ml is the spectral exitance. In many books on heat transfer, the radiative heat flux leaving the surface solely by emission is called the emissive power and the exitance is also called radiosity. The terms self-exitance (emission from a surface neglecting any reflected and transmitted radiation) and exitance (outgoing radiative heat flux from a surface) are observed throughout this book.

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Theory of Thermal Radiation and Radiative Properties

The irradiance is the radiative heat flux incident upon a surface from all directions above the surface. The symbols E and El are used, respectively, for the total and spectral irradiance. Equation (7) can be used to calculate irradiance if the incoming radiance is prescribed as a function of the polar and azimuthal angles of incidence.

2.2. Blackbody cavity and radiation pressure The enclosure concept helped greatly in establishing the laws of blackbody radiation. In 1862, Kirchhoff coined the term ‘‘black body,’’ which is spelled as one word (blackbody) in current practice. A blackbody absorbs all radiation impinged onto it and emits the maximum amount of radiation at any given temperature. Theoretically, a blackbody can be realized by a vacuum enclosure whose walls are kept at a constant temperature, as shown in Figure 5(a). The nature of radiation inside the blackbody cavity is independent of the properties of the wall materials. Since radiation carries energy and momentum, the absorption and reflection of radiation by an object will result in a force, which is termed as radiation pressure. Consider radiation in a thermal-equilibrium enclosure, the momentum exchange is related to the energy exchange divided by the speed of light. Hence, the radiation pressure p (in pascal or N/m2) on a perfectly reflecting wall can be written as: 2 p¼ c0

Z

2p

f¼0

Z

p=2

Lcos2 y siny dydf

(10a)

y¼0

Here, the factor 2 arises because the reflected flux also contributes to the radiation pressure, and the additional cos y is needed since only the momentum component in the direction normal to the surface contributes to the radiation pressure. Because L is independent of the directions for a L (, T )

U, V, S, P L (, T ) L(T) LT()

L (, T ) T

(a)

Figure 5

T (b)

Illustration of (a) an isothermal enclosure and (b) a blackbody cavity.

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Zhuomin M. Zhang and Bong Jae Lee

blackbody cavity, Equation (10a) can be reduced to p¼

4pL u ¼ 3c 0 3

(10b)

where we have utilized Equations (8) and (9). In terms of the spectral properties, it can be shown that pl ¼ ul/3. In a blackbody cavity, p is independent of the properties of the wall and is uniform inside the medium. The same is true for the thermodynamic temperature T. Furthermore, the spectral distribution Ll should depend on l and T only. When a small hole is opened in a large blackbody cavity, as shown in Figure 5(b), the emitted radiation may be considered as diffuse and closely follows the blackbody distribution. For this to be true, the aperture should be small compared to the dimensions of the enclosure, while at the same time the diameter of the aperture must be much greater than the characteristic wavelength to avoid diffraction effects. In order for the cavity to behave like a blackbody, its smallest dimension should be much greater than the characteristic wavelength of thermal radiation [6].

2.3. Entropy and the Stefan2Boltzmann law Boltzmann in 1884 conceived a thought experiment involving radiation as the working fluid in a piston-cylinder arrangement during a reversible process. The first law of thermodynamics can be written as: dU ¼ dQ  pdV

(11)

where U is the internal energy, V the volume of the cylinder or blackbody cavity, and dQ the infinitesimal amount of heat transferred into the cylinder. In a reversible process, dQ ¼ TdS, so that the entropy change is dS ¼

dU þ pdV T

(12a)

Note that dU ¼ d(uV) ¼ Vdu + udV and p ¼ u/3. Thus, we have dS ¼

V du 4u dT þ dV T dT 3T

(12b)

The coefficients on the right-hand side (RHS) are nothing but (qS/qT )V and (qS/qV )T, respectively. The derivative of (V/T )(du/dT ) with respect to V gives (q2S/qVqT ) ¼ (1/T )(du/dT ); and the derivative of (4u/3T ) with respect to T is (q2S/qTqV ) ¼ (4/3T )(du/dT )(4u/3T 2 ). Because (q2S/qVqT ) ¼ (q2S/qTqV ), one obtains du 4u ¼ dT T

(13)

Theory of Thermal Radiation and Radiative Properties

81

It can be seen that u is proportional to T to the fourth power or upT4. In terms of the radiance and exitance, it can be shown by applying Equations (8) and (9) that Lb ¼

sT 4 p

and

M b ¼ sT 4

(14)

where s is a fundamental constant called the Stefan2Boltzmann constant, and the subscript ‘‘b’’ denotes blackbody. This is the Stefan2Boltzmann law, first postulated by Stefan in 1879 based on his analysis of previously measured data and later derived by Boltzmann in 1884 from thermodynamics. Boltzmann’s thought experiment also allowed him to conceptually design a perfect Carnot engine with radiation as the working fluid. Subsequent analysis showed that the temperature that describes the radiation field was the same as the thermodynamic temperature based on the Carnot cycle. Furthermore, it is consistent with the definition of thermodynamic temperature for an equilibrium system according to 1 ¼ T



@S @U

 (15) V

Note that the entropy of blackbody radiation can be expressed as: S¼

16sT 3 V 3c 0

(16)

2.4. Planck’s law of blackbody radiation By treating the radiation field as standing waves, Wien derived a formula for the spectral distribution of blackbody radiation in 1896, called Wien’s formula, which is applicable to the short-wavelength region of the blackbody spectrum but deviates toward long wavelengths. In the subsequent years, Planck attempted to rigorously derive the blackbody distribution and finally succeeded in 1900, by using statistical entropy to obtain the mean energy of an oscillator. The most important contribution of this work was the counter-intuitive assumption of discrete energies (hv) that the oscillator could possess. Also in 1900, Rayleigh used the equipartition theorem to show that the blackbody emission should be proportional directly to temperature but inversely to the fourth power of wavelength. Jeans derived a more complete expression in 1905. The Rayleigh2Jeans formula agreed with experiments at sufficiently high temperatures and long wavelengths, where Wien’s formula failed, but disagreed with experiments at short wavelengths. In 1924, Bose used quantum statistics (independent particles) to derive Planck’s law. The theory of quantum statistics was further advanced by Einstein to form Bose2Einstein statistics. The following derivation of

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Zhuomin M. Zhang and Bong Jae Lee

Planck’s law is based on the blackbody cavity shown in Figure 5(a) using quantum statistics. Under thermodynamic equilibrium, the spectral radiance Ll(l,T) must be a function of l and T only. Let nndn be the number of particles (photons) per unit volume in the frequency interval between n and n + dn, then nn dn ¼ f BE ðnÞDðnÞdn

(17a)

Here, fBE(n) is the Bose2Einstein distribution given by [6] f BE ðnÞ ¼

1 ehn=kT  1

(17b)

where k is the Boltzmann constant, and D(n) is the density of states. The number of states per unit volume in the momentum space with a spherical shell from p to p + dp is equal to the volume of the spherical shell divided by Planck’s constant to the third power. Therefore, DðnÞdn ¼

4pp2 dp 4pn2 dn ¼ c 30 h3

(17c)

8phn3  1Þ

(17d)

The spectral energy density is un ¼ 2hnnn ¼

c 30 ðehn=kT

where the factor of 2 arises from the two polarizations or polarization degeneracy. Note that Ln ¼ unc0/4p, we have the spectral radiance for a blackbody: L b;n ðn; T Þ ¼

2hn3 c 20 ðehn=kT  1Þ

(18)

This is Planck’s law of blackbody radiation. As Lb,ldl ¼ Lb,ndn the spectral radiation of blackbody in terms of wavelength can be written as: L b;l ðl; T Þ ¼

l

5

c 1L c =lT 2 ðe

 1Þ

(19)

where c 1L ¼ 2hc 20 is the first radiation constant for spectral radiance, and c2 ¼ hc0/k is the second radiation constant. Strictly speaking, the cavity must be evacuated, because any medium except for vacuum is dispersive and the speed of light depends on the frequency. For a gas or dielectric medium whose refractive index n can be assumed constant in the thermal radiation spectrum, Equation (19) can be modified by multiplying a factor n2 while treating l as the wavelength in vacuum [4,6], viz. L b;l ðl; T Þ ¼

n2 c 1L l5 ðec2 =lT  1Þ

(20a)

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Theory of Thermal Radiation and Radiative Properties

Alternatively, if the wavelength is taken as that in the medium lm, Planck’s law becomes L b;lm ðlm ; T Þ ¼

c 1L 5 c 2 =ðnlm T Þ 2 n lm ½e

(20b)

 1

since L b;lm ðlm ; T Þdlm ¼ L b;l ðl; T Þdl [4]. As shown in Figure 6 for blackbody radiation in vacuum, the Planck distribution has a peak and approaches zero at extremely short or long wavelengths. By setting the derivative of the Planck function to zero, we obtain the important Wien displacement law: lmax T ¼ c 3

(21)

where lmax is the wavelength corresponding to the peak radiance, and c3 is known as the third radiation constant. It can be seen from Figure 6 and Equation (21) that the maximum emission peak shifts toward shorter wavelengths as the temperature increases. Historically, Wien derived Equation (21) in 1893 by a thought experiment, similar to what Boltzmann did in deriving Equation (14). In the long wavelength limit when c2/lT{1, since ex1Ex for x{1, the RHS of Equation (19) approaches L b;l ðl; T Þ 

c 1L T c 2 l4

(22)

109 Spectral Radiance (W m−2 μm−1 sr−1)

10000 K 107

Lb, (, T) =

6000 K

c1L 5

(ec2/T−1)

105

2000 K 103

1000 K 101

300 K

10-1 10-3 10-5 10-1

T = 50 K

100

101

102

Wavelength (μm)

Figure 6

Planck’s blackbody distribution at different temperatures.

103

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Zhuomin M. Zhang and Bong Jae Lee

which is the Rayleigh2Jeans formula. When plotted on a log2log scale for a given temperature, Equation (22) is a straight line with a slope of 4. This is indeed the case as can be seen from Figure 6 at wavelengths much longer than lmax. In the limit of extremely long wavelengths, the spectral radiance is proportional to temperature. The Rayleigh2Jeans formula fails as the wavelength is reduced to near the emission peak of the blackbody curve and in particular in the short-wavelength region, where the Rayleigh2 Jeans equation predicts infinite energy density at zero wavelength. This is clearly a failure of the theory to accurately reflect the underlying physical processes and thus is termed the ‘‘ultraviolet catastrophe.’’ At short wavelengths, when c2/lTc1, the RHS of Equation (19) can be approximated by: L b;l ðl; T Þ 

c 1L 5 c 2 =lT le

(23)

This is called Wien’s formula, which gives approximately the correct result all the way to l ¼ lmax, where the relative error is 0.7%. Hence, for many industrial applications, Wien’s formula is an excellent approximation. Note that Wien’s formula always underpredicts the spectral radiance. When lWlmax, the error becomes large, as shown in Figure 7. At very long wavelengths, Wien’s formula asymptotically approaches c1Ll5 and becomes independent of temperature; this is physically incorrect. Furthermore, Equation (19) can be integrated over the whole spectrum to obtain the Stefan2Boltzmann law, Equation (14), using the relation Z

1 0

ex

x3 p4 dx ¼ 1 15

(24)

It can be shown that, for blackbody emission, nearly 25% of the energy is at lolmax and W95% of energy falls in the spectral region between 0.5lmax and 50lmax. Once the spectral radiance is known, integration over the wavelength band, the solid angle, and area gives the total power. As an example, consider two coaxial disks, each with a diameter of 1 cm, facing each other and separated at the distance of 1 m. One disk is the source, while the other is the detector. Assuming that both are blackbodies, and supposing that a narrow-band filter is placed in between so as to allow only radiation from 1,200 nmrlr1,220 nm to pass through without loss (i.e., transmittance t ¼ 100%), the goal is to estimate the total radiant power received by the cold detector (i.e., negligible emission) from the source at a temperature of 1,400 K. Since the dimensions of the surfaces are much smaller than the distance, we can assume all the radiation is from the normal direction. The solid angle subtended by the source as viewed from the detector is dO ¼ As/ L2 ¼ 7.85  105 sr. Neglecting the effect of the filter, the radiant power

85

Theory of Thermal Radiation and Radiative Properties

Percentage Error of Wien's Formula (%)

101

100

10-1

10-2

λmaxT

10-3 0

1000

2000

3000

4000

5000

6000

λT (μm-K)

Figure 7

Relative error of Wien’s formula compared with Planck’s law.

reaching the detector is  qwithout filter ¼ LðT ÞdO cosy dA ¼

 sT 4 dO cosy dAd ¼ 428 mW p

(25)

where cos y ¼ 1. With the filter, we have qwith filter ¼ tL l ðl; T ÞdO cosy dAdl   W  ð7:85  105 srÞ ¼ 9:39  103 sr m2 mm  ð7:85  105 m2 Þ  ð2:0  102 mmÞ ¼ 1:16 mW

ð26Þ

In this example only a small fraction of the total radiation is transmitted through the narrow-band filter.

3. Radiative Properties of Materials 3.1. Emissivity For real materials, the rate of emitted energy per unit area cannot exceed that of a blackbody at the same temperature. The ratio between the spectral radiance emitted by a real surface to that of a blackbody at the same

86

Zhuomin M. Zhang and Bong Jae Lee

temperature is called the directional spectral emissivity, 0l ðl; y; f; T Þ 

L em;l ðl; y; f; T Þ L b;l ðl; T Þ

(27)

Here, subscript ‘‘em’’ is used for the emitted. Note that in general the emissivity also depends on the surface temperature. It can be viewed as the ratio of the emitted energy between the surface and that of a blackbody at the same temperature per unit time, per unit solid angle, per unit wavelength interval, and per unit projected area toward the direction defined by the zenith angle y and azimuthal angle f. The directional total emissivity is defined based on the total radiance as follows: 0t ðy; f; T Þ 

L em ðy; f; T Þ p ¼ L b ðT Þ sT 4

Z

1 0

0l L b;l ðl; T Þdl

(28)

The ratio of the exitance defined in Equation (9) emitted from an object to that from a blackbody at the same temperature is called the hemispherical spectral emissivity, hl ðl; T Þ

M em;l ðl; T Þ 1  ¼ pL b;l p

Z

2p

Z

0

p=2

0l cosy siny dydf

(29)

0

where Mem,l is due to emission only and is called self-exitance. Similarly, the hemispherical total emissivity is defined as: ht ðT Þ 

M em ðT Þ p ¼ 4 sT sT 4

Z

1

hl L b;l ðl; T Þdl

(30a)

0

which can also be expressed as: ht ðT Þ 

1 p

Z 0

2p

Z 0

p=2

0t cosy siny dydf

(30b)

In this chapter, the prime sign signifies a directional property, or directional-hemispherical property, and superscript ‘‘h’’ designates a hemispherical property for clarity. In other chapters of this book and in many other publications, the prime or superscript is often omitted and readers need to distinguish directional from hemispherical properties from the context. Many industrial materials have spectrally selective surfaces for which the emissivity is a complicated function of temperature, wavelength, the direction of emission, polarization, as well as surface conditions, such as finishing, roughness, contamination, coatings, etc. For perfectly smooth surfaces, the emissivity can be predicted from the electromagnetic-wave theory, as will be discussed in the following sections. Based on the definition of blackbody and emissivity, the value of emissivity of real surfaces is between 0 and 1. The range of the normal total emissivities for a

Theory of Thermal Radiation and Radiative Properties

87

Normal Total Emissivities at 25 °C

Plants, water, skin, wall paints, and anodized finishes 0.88 – 0.98 Graphites, minerals, soil, and glasses 0.76 – 0.95 Oxides and ceramics 0.4 – 0.8 Oxidized metals 0.25 – 0.68 Unpolished metals 0.1 – 0.4 Polished metals 0.04 – 0.13 Highly polished metals, foils or films 0.02 – 0.07

Figure 8 Ranges of the normal total emissivity of common materials at room temperature [3,7].

variety of materials near room temperature is shown in Figure 8. Generally speaking, conductors have a higher reflectance and lower emissivity in the VIS and IR. Pure metals usually have a very low emissivity. The nature of surface finish and oxidization can greatly affect its emissivity. Some spectrally selective materials that appear highly reflecting in the visible may have a high emissivity (such as white paint and paper) in the mid-IR spectrum. It is important to bear in mind that at near room temperature, the most important spectral region is from 5 to 50 mm. A comprehensive compilation of radiative properties of solids is given in Ref. [7]. Additional information on the emissivity of real materials can be found from Refs. [3,4,8]. For metallic materials or electrical conductors, the emissivity averaged over two polarizations is nearly constant for y less than about 401, increases to a maximum near 801, and finally drops to zero at the grazing angle. For nonconductors, the polarization-averaged emissivity is nearly constant for y less than about 701 and decreases sharply to zero when y approaches 901. The hemispherical emissivity is higher (r30%) than the normal emissivity for conductors and slightly lower (r5%) than the normal emissivity for nonconductors. Note the assumption of diffuse (i.e., angle independent), emission is a good first-order approximation in many engineering applications. In these cases, the hemispherical emissivity may be assumed equal to the normal emissivity. This assumption is not applicable for structured surfaces for which the emissivity may strongly depend on the direction. For a diffuse emitter, the emissivity is independent of the viewing direction; thus, 0l ¼ hl

and

0t ¼ ht

(31)

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Zhuomin M. Zhang and Bong Jae Lee

A gray or grey surface is one such that its emissivity and absorptivity are independent of wavelength. The following relations between the emissivities exist for a gray surface: 0l ¼ 0t

and

hl ¼ ht

(32)

In the literature, both emissivity and emittance are used interchangeably. While it has been suggested that emissivity should be used for perfectly smooth surfaces and emittance should be used for real materials [4], most people do not make such a distinction and tend to use emissivity for all sorts of materials. For semitransparent materials, reflectance, transmittance, and absorptance are most frequently used. Absorptivity is often used interchangeably with absorptance. Reflectivity is commonly used as the fraction of energy reflected at the interface between two semi-infinite media. Emissivity determination is one of the central problems in radiation thermometry as discussed in Chapter 1. The measurement and modeling of emissivity become extremely important in many applications measuring surface temperature using radiometric methods, such as rapid thermal processing (RTP) of semiconductors and the steel and glass industries. Further discussion on theoretical modeling of emissivity for opaque and semitransparent materials will be given later in this chapter.

3.2. Absorptivity The directional spectral absorptivity depends on the temperature, as well as the direction and wavelength of the incident radiation. Unless the incident radiance is extremely intense to cause nonlinear absorption, a0l is considered as an inherent material property that is independent of the environment. The directional spectral absorptivity is the ratio of the absorbed spectral radiance to that incident on the surface of the object: a0l ðl; y; fÞ 

L abs;l ðl; y; fÞ L i;l ðl; y; fÞ

(33)

In general, a0l ðl; y; fÞ also depends on the temperature of the object. The hemispherical spectral absorptivity is defined based on the ratio of the absorbed irradiance to the irradiance coming onto the surface from all directions in the upper hemisphere; therefore, ahl ðlÞ 

E abs;l ðlÞ El ðlÞ

(34a)

Theory of Thermal Radiation and Radiative Properties

89

where Z

2p

Z

p=2

L i;l ðl; y; fÞ cosy siny dydf

El ðlÞ ¼ 0

(34b)

0

and Z

2p

Z

p=2

a0l ðl; y; fÞL i;l ðl; y; fÞ cosy siny dydf

Eabs;l ðlÞ ¼ 0

0

(34c)

In general, ahl ðlÞ depends on the angular distribution of the radiation incident onto the surface (i.e., incoming radiation). If the incoming radiation is diffuse (or isotropic), then the radiance is independent of the direction. In this case, we have ahl ðlÞ

1  p

Z 0

2p

Z 0

p=2

a0l ðl; y; fÞ cosy siny dydf

(35)

The directional total absorptivity a0t is the fraction of absorbed total radiance, but it is not used very often. The hemispherical total absorptivity is defined according to aht 

R1 R1 h Eabs;l ðlÞdl a E l ðlÞdl E abs ¼ R0 1 l ¼ 0R 1 E E ðlÞdl l 0 0 E l ðlÞdl

(36)

It should be mentioned that only a0l is an inherent property of the materials, whereas a0t , ahl , and aht are not, because they also depend on the directional distribution and/or spectral distribution of the incident radiation. Note that all the above definitions can be tied to the ratio of energy absorbed to that of the incident energy, as described in Ref. [4].

3.3. Kirchhoff’s law Consider an object placed in a blackbody enclosure, such as the one shown in Figure 5(a). Under thermodynamic equilibrium, the emitted radiant power from the object is ht As M b ðT s Þ ¼ ht As pL b ðT s Þ, where As and Ts are the surface area and the temperature, respectively, and Lb is the blackbody radiance. On contrary, the absorbed power by the object is aht As EðT w Þ, where Tw is the wall temperature and E(Tw) ¼ pLb(Tw) is the irradiance on the object. Noting that Ts ¼ Tw and, under equilibrium, the absorbed power must equal to the emitted power, we have aht ¼ ht

(37a)

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It can be shown that the equality also applies to directional and spectral properties [4]. Thus, under thermal equilibrium, we also have a0t ¼ 0t ;

ahl ¼ hl ;

and

a0l ¼ 0l

(37b)

The equalities given in Equations (37a) and (37b) are for thermal equilibrium and are generally not applicable when the incoming radiation is not from an environment that is in thermal equilibrium with the object. This is because the absorptivities depend on the incoming radiation from surroundings, except for a0l which is an inherent property of the material that is independent of the condition of the incoming radiation. Hence, the following equation holds for most practical conditions, a0l  0l

(38)

This equation known as Kirchhoff’s law states that the directional spectral absorptivity is equal to the directional spectral emissivity [325]. The emissivity definition is solely based on the spontaneous emission from the surface and is an inherent property of the material that does not depend on the surroundings. However, the absorptivity is defined based on the net absorbed energy by treating stimulated or induced emission as negative absorption. Under these definitions, Kirchhoff’s law given in Equation (38) is always valid in terms of the directional spectral absorptivity and emissivity for any given polarization. The only assumptions are (a) the material under consideration is at a uniform temperature, at least within several penetration depths near its surface and (b) the external electromagnetic field is not too strong to alter the material’s inherent properties, as in a nonlinear interaction. For a diffuse surface, ahl ¼ a0l and hl ¼ 0l ; hence, we have ahl ¼ hl . If a surface is gray, then a0t ¼ a0l and 0t ¼ 0l ; thus we have a0t ¼ 0t . In practice, a surface may be considered ‘‘gray’’ if the emissivity is independent of the wavelength over the spectral regions of importance. If a surface is diffuse-gray, then there is no spectral and directional dependence of the absorptivity or emissivity. This means that Equations (37a) and (37b) hold regardless of whether the surface is in an isothermal enclosure or not. Kirchhoff’s law has important applications in determination of the radiative properties. For example, the emissivity can be measured indirectly by measuring the absorptivity, or reflectivity as discussed below.

3.4. Reflectivity Let us first consider an opaque surface, where radiation cannot penetrate very deep inside the material and there is no radiation transmission. For a mirror-like perfectly smooth surface, the reflected rays are in the specular direction, as shown in Figure 9(a). This requires that the root-mean-square (rms) surface roughness be less than one-hundredth of the wavelength

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Specular reflection

Diffuse reflection

BRDF from rough surface

(a)

(b)

(c)

Figure 9 Schematics of surface reflection: (a) specular or mirror-like, (b) diffuse, and (c) bidirectional reflectance distribution function (BRDF) from a rough surface.

[9211]. Another ideal surface is a perfectly diffuse surface, as shown in Figure 9(b), where the reflected radiance is independent of the direction. Such an ideal surface is more difficult to realize even with surface roughness. In practice, there exists a bidirectional distribution of the reflected radiance from a rough surface, as shown in Figure 9(c). Consider radiation incident on a surface within an element solid angle dO with a spectral radiance Li,l at an incidence angle y. The incoming radiant power is given by qi,l ¼ Li,ldO cos ydAdl. The fraction of this power absorbed by the material is determined by the directional spectral absorptivity a0l , while the remaining is reflected toward the hemisphere. The directional-hemispherical spectral reflectivity r0l is defined as the ratio of the reflected energy over the hemisphere to the incident energy from a given direction. Hence, r0l ðl; T ; y; fÞ ¼ 1  a0l ðl; T ; y; fÞ

(39a)

Similar relations can be drawn for the directional-hemispherical total reflectivity and the directional total absorptivity, r0t ðT ; y; fÞ ¼ 1  a0t ðT ; y; fÞ. By using Kirchhoff’s law, it can be seen that for an opaque surface, we have 0l ðl; T ; y; fÞ ¼ 1  r0l ðl; T ; y; fÞ

(39b)

This equation provides a powerful indirect means for determining the directional spectral emissivity. Spectrophotometers, such as dispersive instruments with gratings or filters and Fourier transform infrared (FTIR) spectrometers, often equipped with a suitable reflectance accessory or an integrating sphere, can be used to measure the spectral reflectivity [12219]. Lasers are also commonly used for reflectivity measurements at single or multiple wavelengths. In general, it is necessary to employ two directions to describe the reflection by real surfaces. The bidirectional reflectance distribution function (BRDF), also known as bidirectional reflectance, is a fundamental radiative property that describes the redistribution of reflected energy.

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z sˆi

dΩi

dΩr

i

sˆr

r i r x

Figure 10 Geometry of the incident and reflected beams.

BRDF is defined as a ratio of the reflected spectral radiance to the incident spectral irradiance, according to the geometry depicted in Figure 10 for the incident and reflected beams [4,19221]: f r ðl; s^i ; s^r Þ ¼ f r ðl; yi ; fi ; yr ; fr Þ ¼

dL r L i cosyi dOi

(40)

where Li is the spectral radiance of the incident radiation within the solid angle dOi in the direction s^i , and dLr is the spectral radiance of reflected light in the direction s^r . An important principle of the BRDF is reciprocity [4], which states that the BRDF remains the same when the incidence and reflection directions are interchanged. The reciprocity principle can be expressed as: f r ðl; yi ; fi ; yr ; fr Þ ¼ f r ðl; yr ; fr ; yi ; fi Þ

(41)

The directional-hemispherical properties can be obtained from integration of BRDF over the reflecting hemisphere. The directional-hemispherical spectral reflectance can be expressed as: R0l ðl; yi ; fi Þ

Z ¼ 2p

f r ðl; s^i ; s^r Þ cosyr dOr

(42)

where 2p indicates the solid angle of the upper hemisphere. Here, R0l is used instead of r0l , which is reserved for smooth opaque surfaces. For an opaque surface, R0l ¼ r0l , and thus Equation (42) is applicable to any surface. It should be noted that the hemispherical-directional reflectance depends on the angular distribution of the incoming radiance. When the incoming radiance is independent of the direction, the hemisphericaldirectional spectral reflectance is equal to the directional-hemispherical

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spectral reflectance. For a diffuse surface, the reflected radiance is uniform in all directions and BRDF is related to the directional-hemispherical reflectance by f r ¼ R0l =p. For a specular surface, BRDF is a delta function around the specular reflection angle [6,9]; in this case, the specular reflectance is the same as the directional-hemispherical reflectance, or simply the reflectance denoted by r0l . While the BRDF of a rough surface can be predicted by solving Maxwell’s equations with the knowledge of surface roughness and optical properties of each medium, this is usually impractical due to the difficulties in determining the actual surface topography as well as in obtaining the solution of complete 3D vector equations at the boundary. Hence, approximation methods, such as the Rayleigh2Rice perturbation theory, the Kirchhoff approximation, and the geometric optics approximation, are often employed since the rigorous electromagnetic solution requires tremendous memory space and CPU time, and boundary conditions are very rarely known with sufficient precision. Notice that these approximations are only applicable with certain ranges of roughness parameters and the wavelength [9,11]. Monte Carlo models have been developed to simulate semiconductor, metal, and coated microrough surfaces [21,22] based on geometric optics in which a rough surface can be viewed as consisting of microfacets. The slope or orientation of microfacet is important in modeling the BRDF of rough surfaces. Two surface realization methods have been used: one is the perturbation method that incorporated the slope distribution obtained by analyzing the topographic data measured with an atomic force microscope (AFM); the other is directly applied to the surface topographic data from an AFM. The results agree reasonably well with experiments and demonstrate the importance of incorporating the anisotropic slope distribution in the BRDF modeling [22]. A typical setup for BRDF measurements is shown in Figure 11. The laser beam passes through a polarizer and a beam splitter: one beam goes to the reference detector that monitors the laser power and the other reaches the sample. The sample and signal detector can be rotated independently. The BRDF measurement equation is [20] f r ¼ CI

Vs V m cosyr DOr

(43)

where Vs and Vm are outputs of the signal and reference detectors, respectively, and DOr is the reflection solid angle. The instrument constant CI compensates the beam-splitter ratio and the difference in the responsivities of the two detectors. In essence, Vs is proportional to the power reaching the detector reflected from the sample and Vm is proportional to the power that is incident on the sample. Note that the

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Goniometric Table

BeamPolarizer splitter

Signal Detector

Position Controller

Laser Computer Sample Reference Detector Pre-amp

Lock-in

Figure 11 Schematic of the experimental setup for measuring BRDF [20].

measured BRDF values are always finite because the solid angle in the experiment is finite. Measurements and modeling of BRDF and directional-hemispherical reflectance are important for many industrial applications. One of the examples is for temperature measurement in RTP [23]. Radiation thermometers are commonly used to monitor the semiconductor wafer temperature during RTP [24]. The emissivity of the wafer must be determined in advance to correct the thermometer reading. The uncertainty of the radiometric temperature measurement largely depends on how accurately the emissivity of the silicon wafer is known. Furthermore, chamber reflections must be considered in order to determine the radiant flux reaching the radiometer through the optical fiber or lightpipe [25]. Since the back surface of the silicon wafer is usually rough, knowledge of the BRDF is crucial to the emissivity modeling or heat transfer analysis in an RTP chamber. For this reason, there have been a number of studies that dealt with the modeling and measurements of BRDF of rough silicon wafers with or without thin-film coatings [21,22,26228].

3.5. Effective or apparent emissivity One of the methods to improve the accuracy of temperature measurement is to enhance the emissivity through the introduction of a cavity on the workpiece, usually by drilling a hole, as illustrated in Figure 12. Suppose the temperature of the workpiece is T, the area of the opening is A1, and the wall area of the cavity is A2. Assume all surfaces are diffuse with a spectral emissivity of el (which is independent of the direction). Without the cavity, the power emitted by the surface in the wavelength interval dl is elpLb,l(l,T)A1dl, as in the case of Figure 12(a). There will be more power

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Detector

Detector

A1, 

T

A2, 

T (a)

(b)

Figure 12 Illustration of effective emissivity: (a) the detector views a flat workpiece of area A1, and (b) the detector views a cavity on the workpiece with a surface area of A2.

emitted from the opening with the cavity as in the case of Figure 12(b). One can trace the multiple reflections to find out the power emitted by the wall (surface A2) that escapes through the opening; i.e., "

#     A1 A1 A1 A1 A1 2 2 ð1  l Þ þ l pL b;l ðl; T ÞA2 dl þ 1 1 ð1  l Þ þ    A2 A2 A2 A2 A2 (44a)

For a blackbody of surface area A1, the radiant power emitted toward the hemisphere is qb ¼ pL b;l ðl; T ÞA1 dl

(44b)

The ratio of Equation (44a) to Equation (44b) defines the effective emissivity, which is eff ¼

l 1  ð1  A1 =A2 Þð1  l Þ

(45)

It can be seen that eeff -1 as A1/A2-0, as expected. More complicated analysis of the effective emissivity of cavities can be performed using the radiation network method or a Monte Carlo method [3,4,29].

3.6. Atmospheric transmission effects A major challenge in practicing radiation thermometry is the environmental absorption and scattering by the medium in the optical path. If the medium is at high temperature, emission from the gases must also be considered. This section briefly reviews the equations governing the radiative transfer in the atmosphere without considering wave effects. Usually, gases (such as Ar, N2, and O2) do not absorb or emit radiation in the VIS and near-IR region unless their temperature is extremely high. Polar gases, such as carbon

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Zhuomin M. Zhang and Bong Jae Lee

dioxide, water vapor, and hydrocarbon gases, emit and absorb radiation in specific wavelength bands over a wide range of temperatures. Gas molecules and aerosols in the atmosphere, pulverized coal in a combustor, soot in flames, dust, and other particulates also scatter radiation. As shown in Figure 13, at steady state, the spectral radiance L l ðl; x; s^Þ also depends on the location or spatial coordinate x and directional vector s^. As the ray travels in the medium, the radiance is attenuated by absorption and scattering. If the emission of the medium is not negligible, the radiance is enhanced by emission. Furthermore, rays from all directions can be scattered toward the traveling direction, which is called in-scattering. The macroscopic description of the spectral radiance is known as the radiative transfer equation (RTE) [4]: dL l sl ¼ al L b;l ðl; T Þ  bl L l þ dx 4p

Z

L l ðl; x; s^0 ÞFl ðs^0 ; s^ÞdO0

(46)

4p

where al and sl are the absorption and scattering coefficients, respectively; bl ¼ al + sl is the attenuation (or extinction) coefficient; and Fl ð^s0 ; s^Þ is the scattering phase function (Fl1 for isotropic scattering). Note that ˆ L(,  + d, s)

dξ

(a) Attenuation by absorption: dL = −aL(, , ˆs)

ˆ L(, , s) ˆ L(,  + d, s)

dξ

(b) Enhancement by emission: dL = aLb,(, T)

T ˆ L(, , s)

out-scattering

dξ

ˆ L(, , s)

ˆ L(,  + d, s)

(c) Effect of scattering

in-scattering

Figure 13 Schematic of radiative transfer in a participating medium: (a) attenuation by absorption, (b) enhancement by emission, and (c) effect of scattering.

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97

Lb,l(l,T) in Equation (46) should be obtained from Equation (20a) to include the effect of the refractive index while l is the vacuum wavelength. The RHS of Equation (46) is composed of three terms: (1) the first accounts for the contribution of emission, which depends on the local gas temperature T(x); (2) the second is the attenuation by absorption and scattering in the considered volume; and (3) the third is the contribution of in-scattering from all directions (integration over the solid angle 4p) to the direction s^. If only absorption is considered, Equation (46) reduces to dLl/dx ¼ alLl or L l ¼ L l;0 expðal xÞ

(47)

where Ll,0 is the radiance at x ¼ 0. The radiance will exponentially decay inside the medium. Furthermore, the radiation penetration depth or photon mean free path is defined as dl ¼ a1 l . At a distance x ¼ dl, the radiance is attenuated by e1. In general, Equation (46) is an integro-differential equation, subject to often complicated boundary conditions. The integration of the spectral intensity over all wavelengths and all directions gives the radiative heat flux. Unless the temperature field is prescribed, the RTE is coupled with the heat conduction or convection equation to completely describe the heat flow for a particular set of boundary conditions. Solutions of Equation (46) are rarely needed in radiometric temperature measurements. However, the characteristics of the absorption and scattering of materials should be known in order to estimate the impact of the environment. Gas radiation is more important for radiation temperature measurements in combustion furnaces and in remote sensing of fires, as will be discussed in other chapters of this book. Scattering by individual gas molecules is usually negligible because their diameters are much smaller than the wavelengths of interest. When radiation travels through a gas, some of the energy may be absorbed. The absorption of radiant energy (or photons) raises the energy levels of individual molecules within the gas. Conversely, gas molecules may spontaneously lower their energy levels and emit photons. These changes in energy levels are called radiative transitions. The most important transitions for radiation thermometry are bound2bound transitions between vibrational energy levels coupled with rotational transitions, because they are prevalent in the near-IR. The photon energy must be exactly the same as the difference between two energy levels in order for the photon to be absorbed or emitted; therefore, the quantization of the energy levels results in discrete spectral lines for absorption and emission. The rotational lines superimposed on a vibrational line give a band of closely spaced spectral lines, called the vibration2rotation spectrum. A large number of quantum states in polyatomic gases can produce spectral lines that are very closely

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Zhuomin M. Zhang and Bong Jae Lee

spaced to each other; some lines may overlap as a consequence of broadening. The results are called vibration2rotation bands. Band models are, therefore, developed to facilitate the computation of the total radiative properties. More elaborated discussions on the absorption and emission mechanisms as well as radiative properties of gases can be found from Refs. [4,30232]. Small particles are important for radiative transfer in the open atmosphere and in combustion chambers. Soot particles ranging from 5 to 100 nm in size are formed during combustion processes and emit thermal radiation (often stronger than the emission from combustion gases) in a continuous spectrum over the IR and some VIS regions. In pulverized coal combustion furnaces, coal, char, and fly ash particles are important as well as soot particles. Particles can also scatter electromagnetic wave or photons, causing a change in the direction of propagation. In the early 20th century, Mie developed a theory for scattering by a spherical particle based on Maxwell’s equations, known as the Mie scattering theory, which can be used to predict the scattering phase function [30]. In the case when the particle sizes are small compared with the wavelength the formulation reduces to the simple expression obtained by Rayleigh, known as Rayleigh scattering. In the Rayleigh scattering limit, the wavelength of the scattered light remains unchanged and the scattering efficiency is inversely proportional to l4. The absorption efficiency, however, is proportional to the inverse of l. The scattering phase function is nearly isotropic. Soot particles usually fall in the Rayleigh limit. For spheres whose diameters are much greater than the wavelength, geometric optics can be applied. If the scattering by one particle is affected by the presence of other particles, the scattering is called dependent scattering, which requires more complicated treatments to obtain the scattering phase function [4,30].

3.7. Semitransparent materials If a material is not opaque, the bidirectional transmittance distribution function or BTDF is defined as: f t ðl; s^i ; s^r Þ ¼

dL t L i cosyi dOi

(48)

which is similar to the BRDF given in Equation (40), with the replacement of the reflected radiance by the transmitted radiance. The directionalhemispherical spectral transmittance can be calculated by integrating the BTDF over the hemisphere: T 0l ¼

Z 2p

f t ðl; s^i ; s^t Þ cosyt dOt

(49)

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99

From energy conservation, the directional spectral absorptance of a semitransparent material is given by: A0l ¼ 1  R0l  T 0l

(50)

Semitransparent materials are important as optical lenses, windows, and filters for radiation thermometer applications. Further discussion on how to calculate radiative properties of multilayer films will be given in Section 6. The transmittance of a filter over a spectral band needs to be defined based on the radiance distribution and the wavelength interval, i.e., R l2

T 0filter

T 0l L l dl ¼ lR1 l2 l1 L l dl

(51)

4. Electromagnetic Wave Theory 4.1. Macroscopic Maxwell’s equation Microscopically, a medium consists of molecules or atoms that are made of electrons and nuclei. In general, nuclei are in random motion due to thermal excitation, and electrons are in orbitals around nuclei. Therefore, the microscopic electromagnetic fields generated by these charged particles fluctuate, over distances on the order of 1010 m, with periods of near 1013 s for nuclear vibrations and 1017 s for electron orbital motion [33]. However, all the microscopic fluctuations are averaged out in most of measuring devices, resulting in reasonably smooth and slowly varying macroscopic quantities. The macroscopic electromagnetic fields are described by macroscopic Maxwell’s equations given as follows: rE¼

@B @t

rH¼Jþ

@D @t

(52)

(53)

rD¼r

(54)

rB¼0

(55)

Here, B is the magnetic induction field, D the electric displacement vector, J the electric current density in units (A/m2), and r the charge density in the medium. Equation (52) is Faraday’s law of induction, which shows the interdependence of electric and magnetic phenomena. It states that a

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Zhuomin M. Zhang and Bong Jae Lee

time-varying magnetic field produces an electric field in a coil; in other words, any closed electric field line generates a time-varying magnetic field. Equation (53) is Ampere’s law with Maxwell’s correction [33,34]. It states that through any closed magnetic field line, there is an electric current density or an electric displacement or both. Gauss’s law given in Equation (54) describes how the divergence of electric field is affected by the charge. On contrary, Equation (55) is the Gauss’s law of magnetism, which states that the divergence of magnetic field is zero and implies that there exist no magnetic monopoles. Note that D ¼ e0E + P, where e0 is the electric permittivity of vacuum, and P is the polarization vector that is the dipole moment in a unit volume. Polarization is related to the response of permanent or induced dipole moment to the electric field, see Refs. [35,36] for a detailed discussion. Because P ¼ e0(e1)E, where e is the relative permittivity or dielectric function of the medium, the electric displacement vector can be expressed by the electric field for a linear isotropic medium as D ¼ ee0E. Note that e is a function of frequency for a dispersive medium and becomes a dyadic tensor for an anisotropic medium. Furthermore, from Ohm’s law, J ¼ sE, where s is the electric conductivity. At optical frequencies, both e and s are complex variables and the distinction between a conductor and a dielectric diminishes [6]. The magnetic induction is related to the magnetic field H by B ¼ m0(H + M), where m0 is the magnetic permeability of vacuum and M is the magnetization vector, that is, magnetic moment in unit volume. With the relation M ¼ (m1)H, where m is the relative magnetic permeability, the magnetic induction can be expressed by the magnetic field for a linear isotropic medium as B ¼ mm0H. Magnetic materials, whose relative permeability is different from unity, rarely exist in nature at optical frequencies. Metamaterials with nanostructures can possess a negative m at some optical frequencies, resulting in certain unique optical and thermal radiative properties for emerging applications, such as superlens that can provide images with resolution below the diffraction limit, invisible cloaking, enhanced transmittance, and coherent thermal emission [37,38].

4.2. Wave equation and wave propagation Let us consider the situation of free space for simplicity, such that r ¼ 0, s ¼ 0, e ¼ 1, and m ¼ 1. After substituting the constitutive relations into Maxwell’s equations and then combining Equations (52) and (53) in terms of the electric field, we obtain r 2 E ¼ m0  0

@2 E @t 2

(56)

Theory of Thermal Radiation and Radiative Properties

101

where the vector identity r  (r  E) ¼ r(r  E)r2E ¼ r2E has been used. Equation (56) is the wave equation, whose solution is a function of all four variables x, y, z, and t. A special kind of solution for the wave equation is a plane wave, whose time dependence is harmonic (i.e., sin ot or cos ot) with the angular frequency o. In the one-dimensional case, the solutions of the wave equation are linear combinations of sin(kx7ot) and cosðkx  otÞ, where k ¼ o/c0 ¼ 2p/l. By extending the above solution to three dimensions, a monochromatic plane wave solution of the wave equation in Equation (56) can be written as: E ¼ E0 eiðkrotÞ

(57)

where E0 is the amplitude vector, r ¼ x^x þ y^y þ z^z the position vector, and k ¼ kx x^ þ ky y^ þ kz z^ the wavevector. The physical electric field is Re(E). Since any time-dependent function can be expressed as a Fourier expansion of time-harmonic components, Equation (57) can be integrated over all frequencies to obtain the total electric field at given time and position. The surface perpendicular to the wave propagation direction is called a wavefront and can be determined by k  r ¼ constant. In vacuum, the wavefront travels in the k direction at the speed c0 ¼

o 1 ¼ pffiffiffiffiffiffiffiffiffi k m0 0

(58)

which is called phase speed. The magnetic field can be obtained from the electric field using Maxwell’s equations. In vacuum, both the electric and magnetic fields are perpendicular to the wavevector k and satisfy the following relations: k  E ¼ om0 H

(59a)

k  H ¼ o0 E

(59b)

Consequently, E, H, and k are orthogonal to each other and form a righthanded triplet, as depicted in Figure 1. The discussion of electromagnetic wave propagation in free space can be extended to any medium with a relative permittivity e and permeability m. In the case of a lossless medium, both e and m are real. The refractive index of the medium is defined as: n¼

pffiffiffiffiffi m

(60)

The phase velocity is reduced by a factor n; thus, the phase speed in the medium becomes: c¼

c0 n

(61)

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and the wavelength in the medium pffiffi becomes lm ¼ l/n. For nonmagnetic materials, it is obvious that n ¼ . Because the phase speed is inversely proportional to the refractive index, electromagnetic waves of different frequencies will travel at different phase speeds in a dispersive medium. The wave group is a superposition of waves traveling in different speeds and, therefore, will continuously change shape as it propagates through the medium. The group amplitude is modulated as a wave with its own front, called group front, which travels at its own velocity. The modulation of the group amplitude forms an envelope or wave packet. The velocity at which the wave packet travels is called the group velocity, given by vg ¼

@o k @k k

(62)

in an isotropic medium [33,39]. Generally speaking, the group velocity is considered as the velocity of energy transport. In some cases, the refractive index of a material can be less than unity depending on the frequency region, resulting in the phase velocity greater than c0 at that frequency. However, the wave packet or energy of the polychromatic wave propagates at the group velocity, which is always less than c0. Metamaterials may possess a negative refractive index and thus a negative phase velocity [6]; but the group velocity is always positive [40]. If the conductivity s is included, the wave equation for a nonmagnetic material becomes: r2 E ¼ m0 s

@E @2 E þ m0 0 2 @t @t

(63)

Substituting Equation (57) into Equation (63), it can be shown that k2 ¼ iom0 s þ o2 m0 0

(64)

suggesting that the wavevector should be complex. The complex dielectric function can be defined as: ~ ¼ 0 þ i00 ¼  þ i

s o0

(65)

Accordingly, the complex refractive index n~ ¼ n þ ik is related to the complex dielectric function by ~ ¼ ðn þ ikÞ2 . The imaginary part k of the complex refractive index is called the extinction coefficient. In addition, the refractive index and the extinction coefficient are called the optical constants. It can be seen from Equations (64) and (65) that k2 ¼ ~o2 m0 0 or k ¼ n~ o=c 0 . It should be noted that some texts use ~ ¼ 0  i00 for the dielectric function and n~ ¼ n  ik for the complex refractive index. In this case, the choice of time-harmonic term is exp(iot), such that Equation (57) must be rewritten as E ¼ E0ei(otk  r) [6].

Theory of Thermal Radiation and Radiative Properties

103

Equation (57) represents a plane wave whose electric field vector is parallel to E0. Such a wave is said to be linearly polarized since the oscillation direction does not change with time. Mathematically, two orthogonal linearly polarized waves can be superimposed to represent ^ that an arbitrarily polarized wave. Consider two unit vectors, a^ and b, are real and perpendicular to each other. The electric field vector can be expressed as: ^ iðkrotÞ E ¼ ðE1 a^ þ E2 bÞe

(66)

where E1 and E2 are complex quantities, which may be expressed as E1 ¼ |E1|eif1 and E2 ¼ |E2|eif2. It can be shown that ReðEÞ ¼ jE1 j cos y1 a^ þ jE2 j cos y2 b^

(67)

where y1 ¼ k  r  ot + f1 and y2 ¼ k  r  ot + f2. The simplest case is when y1 ¼ y2, that is E1 and E2 are in the same phase. In this case, the electric field is linearly polarized. If the phases of E1 and E2 are different, the electromagnetic wave is generally elliptically polarized. For the special case when |E1| ¼ |E2|, the wave becomes circularly polarized. Since thermally emitted light is randomly polarized, polarization effects are usually not considered in radiative heat transfer. This assumption, however, may be invalid for structured surfaces or upon reflection from a surface due to the effect of polarization. The energy density of the electromagnetic fields in a lossless medium is given by: 1 u ¼ ðE  D þ B  HÞ 2

(68a)

The time-averaged energy density for the time-harmonic fields becomes: 1 u ¼ ðE  D þ B  H Þ 4

(68b)

where  denotes the complex conjugate. Furthermore, the vector S ¼ E  H represents the energy flux and is called the Poynting vector, which indicates the instantaneous energy flux. In reality, the time-averaged Poynting vector S is employed to describe the radiant power through a unit area. The time-averaged Poynting vector can be expressed in terms of the complex electric and magnetic fields as: 1 S ¼ ReðE  H Þ 2

(69)

In an absorbing medium, it can be shown that the amplitude of either the electric or magnetic field decays exponentially according to exp(2pkx/l), where x is the coordinate along the direction of propagation. Hence, the

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Zhuomin M. Zhang and Bong Jae Lee

Poynting vector decays proportionally to exp(4pkx/l) ¼ exp(alx), where al ¼ 4pk/l is the absorption coefficient.

4.3. Fresnel reflection and transmission coefficients The solution of the wave equation considered so far is for waves traveling in an unbounded medium. However, radiation is often incident from one medium to another. Consider a plane wave at an angular frequency o incident from medium 1 to medium 2, which are separated by a smooth interface, as shown in Figure 14. Each medium is assumed to be homogenous and isotropic, so that e and m describe the material response to external electric and magnetic fields, respectively. It is a common practice to analyze the wave propagation for two separate polarizations. In the case of s polarization as shown in Figure 14(a), the electric field is perpendicular to the x2z plane. This is also called a transverse electric (TE) wave, for which the incident electric field can be expressed as: Ei ¼ y^ E i eiðk1x xþk1z zÞ

(70)

where Ei is the amplitude of the incident wave at the origin. The timeharmonic term exp(iot) is omitted for simplicity. Note that k21x þ k21z ¼ k21 ¼ m1 1 o2 =c 20 . Boundary conditions require that the wavevector of the reflected wave to be kr ¼ k1x x^  k1z z^ . Hence, the reflected electric field can be expressed as Er ¼ y^ E r eiðk1x xk1z zÞ

(71)

where Er is the amplitude of the reflected wave. Similarly, the transmitted electric field is given by: Et ¼ y^ E t eiðk2x xþk2z zÞ

Figure 14 Reflection and transmission of a linearly polarized plane wave at the interface between two semi-infinite media: (a) s polarization; (b) p polarization.

(72)

Theory of Thermal Radiation and Radiative Properties

105

where Et is the amplitude of the transmitted wave. It should be noted that k22x þ k22z ¼ k22 ¼ m2 2 o2 =c 20 . The electric field in medium 1 is a superposition of the incidence and the reflected fields. At the interface (z ¼ 0), boundary conditions require that the tangential component of the electric field (Ey) must be continuous. Thus, we have k1x ¼ k2x ¼ kx

(73)

Ei þ Er ¼ Et

(74)

and

Note that Equation (73) results in Snell’s law, that is, n1sin y1 ¼ n2sin y2 for two dielectrics. The magnetic field can be obtained using Maxwell’s equation, H ¼ (iomm0)1r  E. The requirement of the tangential component of magnetic field (Hx) to be continuous at the boundary yields the following equation: k1z k2z ðEi  E r Þ ¼ Et m1 m2

(75)

The Fresnel reflection and transmission coefficients for s polarization are defined according to the ratios of the electric fields as r s12 ¼ E r =Ei and t s12 ¼ Et =E i , respectively. From Equations (74) and (75), it can be shown that [6,39] r s12 ¼

Er k1z =m1  k2z =m2 ¼ Ei k1z =m1 þ k2z =m2

(76)

t s12 ¼

Et 2k1z =m1 ¼ Ei k1z =m1 þ k2z =m2

(77)

and

For p polarization or transverse magnetic (TM) waves, the magnetic field is in the y-direction, as shown in Figure 14(b). The Fresnel reflection and transmission coefficients are defined based on the ratios of the magnetic fields. The same procedure can be applied to determine the magnetic fields and then the electric fields in both media. After the boundary conditions are implemented, we obtain p

r 12 ¼

H r k1z =1  k2z =2 ¼ H i k1z =1 þ k2z =2

(78)

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Zhuomin M. Zhang and Bong Jae Lee

and p

t 12 ¼

Ht 2k1z =1 ¼ H i k1z =1 þ k2z =2

(79)

For nonmagnetic materials, the Fresnel reflection and transmission coefficients reduce to

r 12 ¼

and t 12

8 n~ 1 cosy1  n~ 2 cosy2 > > < n~ cosy þ n~ cosy ;

for s polarization

n~ 2 cosy1  n~ 1 cosy2 > > ; : n~ 2 cosy1 þ n~ 1 cosy2

for p polarization

1

1

2

2

8 2~n1 cosy1 > > < n~ cosy þ n~ cosy ; 1 1 2 2 ¼ 2~ n cosy > 2 1 > ; : n~ 2 cosy1 þ n~ 1 cosy2

(80)

for s polarization (81) for p polarization

Note that the Fresnel coefficients for s and p polarizations are different even at normal incidence because they are defined based on the electric and magnetic fields, respectively. Ellipsometry is a major technique for optical property measurements and material diagnostics, especially in the UV, VIS, and near-IR spectral regions [41]. Ellipsometers utilize the polarization effects to determine the ratio of the reflection coefficients for p polarization and s polarization: rp ¼ tanðcÞeiD rs

(82)

where c and D are defined as ellipsometric angles (or parameters). For incidence on the surface of a semi-infinite medium, measurements of c and D allow the determination of n and k because the reflection coefficients are related to optical constants by Equation (80). For a dielectric film on an opaque substrate, ellipsometers can be used to measure the refractive index and the film thickness, if the refractive index of the substrate is previously determined. These quantities can be used to deduce surface temperature as will be discussed in Chapter 7.

4.4. Reflectivity and emissivity The reflectivity at the interface between two media can be expressed by the time-averaged Poynting vector. For s polarization, referring to Figure 14(a), the z component of the incident Poynting vector at the interface is   1 k1z S 1z ¼ Re ðE þ E ÞðE  E Þ i r i r om0 m 1 2

(83)

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107

The reflected wave and the incident wave are in general coupled such that the energy flow cannot be separated into a reflected flux and an incident flux [42]. Under the assumption that medium 1 is nonabsorbing and k1z is real, Equation (83) can be rewritten as [6]: S 1z ¼ S iz  S rz

(84)

where S iz ¼

k1z k1z jE i j2 and S rz ¼ jE r j2 2om0 m1 2om0 m1

(85)

The spectral reflectivity r0l at the interface is defined as the ratio of the reflected flux to the incident flux. Hence, r0l ¼

S rz jEr j2 ¼ S iz jEi j2

(86a)

Alternatively, Equation (86a) can be expressed as: r0l ¼ jr s12 j2

(86b)

where r12 is the Fresnel reflection coefficient given in Equation (76). For p polarization, the same procedure can be followed to show that r0l ¼

jH r j2 p ¼ jr 12 j2 jH i j2

(87)

From Equation (80), it can be shown that for normal incidence from air (n1E1) to a medium with n~ 2 ¼ n2 þ ik2 , the reflectivity is independent of polarization and can be expressed as: r0l;n ¼

ðn2  1Þ2 þ k22 ðn2 þ 1Þ2 þ k22

(88)

The emissivity of a surface can be calculated from Equation (39b), that is, subtracting the reflectivity from unity. However, the reflectivity should be averaged over the two polarizations calculated from Equations (86) and (87) for incidence from air to the medium. Knowledge of the optical constants of materials is important for the prediction of their radiative properties. Figure 15(a) shows the reflectivity as a function of the incidence angle when radiation is incident from air (n1 ¼ 1) to a dielectric medium with n2 ¼ 1.5 for both polarizations. By carefully examining Equation (80), it can be seen that the reflection coefficient can be zero for p polarization p only. The angle at which r 12 ¼ 0 is called the Brewster angle (also known as

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Zhuomin M. Zhang and Bong Jae Lee

1 (a) n1 = 1.0, n2 = 1.5

Reflectivity

0.8

s polarization p polarization

0.6

0.4 θB = 56.3°

0.2

0

0

30

60

90

Angle of Incidence (deg)

1

θc = 41.8°

0.8

Reflectivity

s polarization p polarization

0.6

0.4

θB = 35.6°

0.2

0

(b) n1 = 1.5, n2 = 1.0

0

30

60

90

Angle of Incidence (deg)

Figure 15 Reflectivity at the interface between air and a prism (n ¼ 1.5) for both polarizations: (a) radiation incident from air; (b) radiation incident from the prism.

the polarizing angle): yB ¼ tan1(n2/n1), which is 56.31 when n1 ¼ 1 and n2 ¼ 1.5. At the Brewster angle, the propagating directions of the reflected and transmitted waves form a right angle (i.e., y1 + y2 ¼ p/2). If the medium is absorbing, the reflectivity for p polarization will show a minimum, which is greater than zero [39]. The Brewster angle can be employed to build polarizers and to achieve nearly perfect transmission for p-polarized laser light.

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Theory of Thermal Radiation and Radiative Properties

In Figure 15(b), the reflectivity is plotted when the radiation is incident from a dielectric medium to air. In this case, the Brewster angle is yB ¼ 35.61. When the incidence angle is greater than a certain value, the reflectivity becomes unity for both polarizations. This phenomenon is called total internal reflection, which occurs when the radiation is incident from an optically denser medium to an optically rarer medium. The onset angle of total internal reflection is called the critical angle, given by yc ¼ sin1(n2/n1) according to Snell’s law. Total internal reflection is important for optical fiber or lightpipe thermometers [24,25] and near-field radiative transfer due to photon tunneling [43]. Surface roughness can modify the emissivity. Figure 16 shows the emissivity of silicon at the wavelength l ¼ 635 nm near room temperature. It should be noted that silicon wafers of approximately 400-mm thickness are opaque at this wavelength due to strong band-gap absorption. The emissivity data were deduced from the directional-hemispherical reflectance measured with an integrating sphere by Lee et al. [18]. Except at the emission angle of 401, the measurement results for the smooth surface agree well with those predicted using the optical constants from Refs. [44,45]. The rough surface has an rms roughness of 630 nm and exhibits an anisotropic slope distribution (sample no. 2 in Ref. [18]). While the emissivity is a strong function of polarization, the polarization-averaged emissivity does not change significantly up to the Brewster angle. At smaller angles, the emissivity is enhanced by surface roughness. However, the 1 Silicon λ = 635 nm

Emissivity

0.8

0.6 Theory, p 0.4

Theory, s Smooth, p Smooth, s

0.2

Rough, p

Rough, s 0 0

30

60

90

Angle of Emission (deg)

Figure 16 Comparison of calculated and measured emissivity of silicon versus the angle of emission for both polarizations at the wavelength of 635 nm.

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Zhuomin M. Zhang and Bong Jae Lee

emissivity of the rough surface is lower for p polarization at large angles; this is due to the polarization mixing caused by the randomly oriented microfacets. The polarization-averaged emissivity is enhanced by 0.0520.06 due to anisotropic surface roughness. This example illustrates that emissivity at oblique incidence is polarization dependent. Furthermore, the emissivity variation due to surface roughness must be compensated in order for the radiation thermometer to provide accurate measurements of surface temperatures.

5. Optical Properties Several simple dielectric function models are described to account for free-electron absorption, phonon vibrations, and interband absorption. These models are convenient for prediction of the VIS and IR optical constants of conductors, dielectrics, and semiconductors. In the case of nonmagnetic materials, the complex dielectric function is related to the complex refractive index by: ~ðoÞ ¼ 0 þ i00 ¼ ðn þ ikÞ2

(89)

It should be mentioned that the real and imaginary parts of the dielectric function are intrinsically related based on the Kramers2Kronig dispersion relation, which is a consequence of the causality and very useful in determining the frequency-dependent dielectric function of real materials [30,33].

5.1. The Drude model for a conductor In a conductor, electrons in the outermost orbits are ‘‘free’’ to move in accordance with the external electric field. The dielectric function of a conductor can be modeled by considering the electron movement under the electric field. From Newton’s second law, the equation of motion for an electron can be expressed by assuming the damping force to be proportional to its velocity as: me ðx€ þ g_xÞ ¼ eE

(90)

where me is the electron mass, e the electron charge, and g the strength of the damping and called the scattering rate. The damping term is mainly due to the collision of electrons with lattice vibrations, and the inverse of the scattering rate is called the relaxation time. The electric field E in Equation (90) is assumed to be macroscopic field by neglecting the local field effects [30,36]. For a harmonic excitation, that is, E ¼ E0eiot, the solution of

Theory of Thermal Radiation and Radiative Properties

111

Equation (90) is x¼

e=me E oðo þ igÞ

(91)

Note that the electric current density J is related to the electron velocity by: ~ J ¼ ne e_x ¼ sE

(92)

where ne is the number density of electrons and s~ the complex conductivity. From Equation (92), the conductivity can be expressed in terms of the frequency and scattering rate as: ~ sðoÞ ¼

ne e2 =me s0 g ¼ g  io g  io

(93)

where s0 ¼ nee2/(gme) is the dc conductivity. The complex dielectric function of a metal can be calculated by using Equation (65) as follows: ~ ðoÞ ¼ 1 

o2p s0 g=0 ¼   1 o2 þ iog o2 þ iog

(94)

where eN accounts for high-frequency contributions and op is the plasma frequency given by: rffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffi s0 g ne e2 ¼ op ¼ 0 me 0

(95)

The value of eN is usually on the order of unity and has little influence on the optical constants of conductors in the IR region. The plasma frequency is in the UV region for most of the metals. When ooop, the real part of the dielectric function is negative such that nok. At very low frequencies, the real part of the dielectric function is much smaller than the imaginary part and, therefore, nEk. Metals become highly reflective in the VIS and IR region. The values of scattering rate and plasma frequency for several metals can be found in Ref. [46]. Figure 17 shows the normal reflectivity calculated from Equation (88) of gold, using the Drude model with parameters of eN ¼ 1, op ¼ 7.25  104 cm1, and g ¼ 2.16  102 cm1 [46], as well as using the experimentally determined optical constants from Ref. [44]. In the IR wavelengths, the reflectivity is very high and the emissivity is below 0.023 at lW1 mm. In this region, the reflectivity predicted by the Drude model agrees well with that calculated from optical constants obtained from experiments. However, there exist significant deviations in the VIS and UV regions. The simple Drude model significantly overpredicts the reflectivity and the resulting

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Zhuomin M. Zhang and Bong Jae Lee

1

0.8

Reflectivity

Au

0.6

Drude model [46]

0.4

Tabulated data [44]

0.2

0 0. 1

1

10

Wavelength (μm)

Figure 17 Normal reflectivity of gold calculated from the simple Drude model and from the experimentally determined optical constants.

reflectivity exhibits a sharp edge at the plasma frequency. There exist complicated interband transitions in the VIS region. The interband transition at wavelength around 500 nm causes significant absorption when the photon energy is higher than the band-gap energy. Etchegoin et al. [47] used a critical-point model to include interband transitions and obtained good agreement of the optical constants of gold at wavelengths from 200 to 1,000 nm. Although discrepancies exist in the shorter wavelength region, the Drude model is easy to apply and in the absence of measurement data provides insight on the dielectric function of conductors in the IR and microwave spectral regions. The measured optical constants of a large number of metals are tabulated in Ref. [44]. While the data vary from sample to sample as well as between measurements performed by different groups, the tabulated data should be considered as more reliable than those predicted by the Drude model.

5.2. The Lorentz oscillator model for a nonconductor The electrons in a nonconductor are bound to molecules and cannot move freely. In contrast to free electrons, a bound charge experiences a binding force, which can be viewed due to an imaginary spring with a resonance frequency o0. The binding force can be expressed as mo20 x, where x represents the displacement of a bound charge and m is the mass associated with the oscillator. Consequently, the equation of motion for an oscillator

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Theory of Thermal Radiation and Radiative Properties

with a positive charge of e, subject to spring-damping forces, can be expressed as follows [30]: mð€x þ g_x þ o20 xÞ ¼ eE

(96)

where g is the damping coefficient or scattering rate. With the assumption of harmonic electric field, exp(iot), the solution of Equation (96) is x¼

e=m E o20  o2  iog

(97)

Note that the dipole moment p due to a bound charge is p ¼ ex ¼

o20

e2 =m E  o2  iog

(98)

If the number density of the oscillator is n, then the polarization (dipole moment per unit volume) is P ¼ np. There exist different kinds of oscillators in a real material, such as bound electrons or lattice ions. From the response of a single-charge oscillator to a time-harmonic electric field, we can extend the response function to a collection of oscillators. Assume that there are N types of oscillators. The mass, resonance frequency, scattering rate, and number density for the jth oscillator are mj, oj, gj, and nj, respectively. Then, the dielectric function can be expressed as a superposition of the contributions from each type of oscillators. Therefore, ~ ðoÞ ¼ 1 þ

N X

o2p;j

j

o2j  o2  iogj

(99)

where o2p;j ¼ nj e2 =ðmj 0 Þ and op, j may be viewed as the plasma frequency of the jth oscillator. In the derivation of the above equation, the constitutive relation P ¼ 0 ð~  1ÞE has been used. As in the Drude model, eN accounts for high-frequency contributions. The Lorentz model was first derived based on the classical theory at the beginning of the 20th century. Later, it was confirmed by quantum mechanics. The resonance frequency corresponds to the transverse optical phonon mode. Since the parameters for the Lorentz model are more difficult to be modeled as compared to those for the Drude model, they are usually determined from fitting the measured reflectance. Figure 18 shows the dielectric function, optical constants, and normal emissivity of c-axis crystalline Al2O3 (sapphire) based on the Lorentz parameters obtained by Barker [48] for ordinary rays in which the electric field is perpendicular to c-axis. There are four oscillators located at wavelengths of 15.7, 17.6, 22.6, and 26.0 mm, with

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Zhuomin M. Zhang and Bong Jae Lee

300

Dielectric Function

(a)

ε′

200

ε′′ 100

0

Al2O3

-100 0

5

10

15

20

25

30

35

Wavelength (μm)

14 12

Optical Constants

(b) 10

n

8

κ

6 4 2 0 0

5

10

15

20

25

30

35

Wavelength (μm) 1 (c)

Emissivity

0.8 0.6 0.4

Opaque 0.2 0

0.5 mm

0

5

10

15

20

25

30

35

Wavelength (μm)

Figure 18 (a) Dielectric function, (b) optical constants, and (c) normal emissivity of Al2O3.

Theory of Thermal Radiation and Radiative Properties

115

corresponding wavenumbers of 635, 569, 442, and 385 cm1. There is a sharp increase in 00 ¼ Imð~Þ at almost the resonance frequency. The real part of the dielectric function 0 experiences oscillation across the resonance frequency. Regarding the dielectric function, the normal dispersion is associated with an increase in 0 with frequency, and the anomalous dispersion is the reverse. Normal dispersion can be found everywhere except in the vicinity of the resonance frequency oj. In addition, in the region where anomalous dispersion occurs, the imaginary part of the dielectric function becomes comparable or larger than the real part of the dielectric function. Therefore, the material becomes highly reflective near the resonance frequency, and the radiation inside the material is rapidly attenuated or dissipated. The spectral region with a large imaginary part of the dielectric function is also called the region of resonance absorption. Because phonon absorption only affects a relatively narrow band, the extinction coefficient is rather small for lo10 mm and lW28 mm. The normal emissivity plotted in Figure 18(c) is calculated for a semiinfinite (opaque) medium, using method discussed previously, and for a 0.5-mm-thick film, using the formula that will be detailed in Section 6.1. The emissivity oscillates with wavelength due to the variation of optical constants. However, a resonance peak in k does not necessarily correspond to a maximum emissivity. This is because a large k will result in a large reflectance, especially when kcn (i.e., when 0 o0), and consequently, a low emissivity. Furthermore, there exists a wavelength at which the emissivity becomes unity, that is, at l ¼ 9.5 mm for sapphire. This wavelength is called the Christiansen wavelength at which n ¼ 1 and k{1 so that the reflectance is nearly zero [30,49]. On contrary, the photon penetration depth dl ¼ 1/al ¼ l/(4pk) is less than a few tens of micrometers, small enough for a slab with a thickness of 100 mm to be considered as opaque at this wavelength. Hence, the emissivity will be almost 1 and the material behaves like a blackbody at the Christiansen wavelength. This phenomenon is called the Christiansen effect. Because of this unique characteristic, the Christiansen effect has been used to calibrate radiation thermometers and to determine IR emissivity of dielectric materials [49252]. Interestingly, the Christiansen wavelength depends little on the angle of incidence and temperature [50,52]. In addition, it is nearly the same for both the ordinary and extraordinary rays for a uniaxial crystal [44]. Table 1 lists the Christiansen wavelength for a number of insulators, mostly taken from Ref. [44], except that of Si3N4 is from Ref. [49], those of LaAlO3 and LaGaO3 are from Ref. [53], and that of ZrO2 is from Ref. [54]. In some polar semiconductors, such as GaAs, InSb, and ZnSe, the Lorentz model can also describe the contribution of the phonon vibration to the dielectric function in the IR. At the wavelength corresponding to n ¼ 1, due to free-carrier contributions, the extinction coefficient is too large to obtain a near-zero reflectance or close-to-unity emissivity.

116 Table 1

Zhuomin M. Zhang and Bong Jae Lee

Christiansen wavelength of several materials [44,49,53,54].

Name

Chemical formula

Christiansen wavelength (mm)

Aluminum oxide Aluminum oxynitride Barium fluoride Barium titanate Beryllium oxide Calcium fluoride Cesium bromide Lanthanum aluminate Lanthanum gallate Lithium tantalate Magnesium fluoride Magnesium oxide Potassium bromide Silicon carbide (6 H) Silicon dioxide Silicon nitride Strontium titanate Titanium dioxide (rutile) Yttrium oxide Zirconia

Al2O3 AlON BaF2 BaTiO3 BeO CaF2 CsBr LaAlO3 LaGaO3 LiTaO3 MgF2 MgO KBr SiC SiO2 Si3O4 SrTiO3 TiO2 Y2O3 ZrO2

9.5 9.5 20.9 12.8 8.0 16.0 78.0 12.5 14.0 10.7 12.6 11.8 50.0 10.0 7.4 8.3 11.5 11.8 14.5 13.0

5.3. Optical constants of a semiconductor The optical constants of a semiconductor largely depend on its crystalline structure, temperature, and impurity levels. In this section, emphasis is given to single-crystal silicon because of its common use in the semiconductor industry. There have been numerous studies on the optical and radiative properties of silicon materials. The fundamental physics has been well established regarding the crystalline structure, electron and phonon dispersion, and scattering mechanisms [55]. The absorption processes associated with the band-gap absorption, free-carrier absorption, and lattice absorption are well understood. Model expressions, which relate the optical constants or sometimes the dielectric function to the wavelength and temperature, have been developed to cover certain spectral and temperature ranges with good accuracy. However, even though numerous functional expressions for the optical constants of silicon exist, none of them covers the entire range of wavelengths and temperatures of interest. Several functional expressions for optical constants of lightly doped silicon can be found in the review paper by Timans [23]. Measurement data of the optical constants of lightly doped silicon (doping concentration less than 1015 cm3) over wide wavelength regions

Theory of Thermal Radiation and Radiative Properties

117

at room temperature can be found in the handbook edited by Palik [44]. However, few experimental data exist at high temperatures. Sato [56] was the first to comprehensively study the emissivity of silicon wafers from direct measurements as well as from the reflectance and transmittance measurements in the temperature range between room temperature and 8001C. The refractive index values based on his experimental data are consistently higher than those obtained from recent studies. In the following section, several functional expressions are selected to illustrate the optical constants of silicon in the VIS and near-IR region at different temperatures. Figure 19(a) shows the refractive index of silicon at 251C and 8001C, in the wavelength region from 0.5 to 1.5 mm. The refractive index decreases slightly as the wavelength increases. At room temperature, the refractive index values based on each expression agree well with the data compiled in Ref. [44]. Since no experimental data exist at wavelengths between 0.84 and 1.1 mm, the expressions of Jellison and Modine [57] and Li [58] are separately extrapolated to this wavelength region, which is between the two vertical dash-dotted lines. The refractive index obtained from these expressions differs in the extrapolated region, and the difference becomes larger at higher temperatures. A difference in the refractive index may cause an error in the prediction of radiative properties and in the temperature measurement by radiation thermometers. The absorption coefficient al of lightly doped silicon is plotted for silicon in Figure 19(b) at 251C and 8001C, in the wavelength region from 0.5 to 1.5 mm. The squares represent the values converted from the extinction coefficient in Ref. [44] and the circles indicate the data by Saritas and McKell [59]. Since no theoretical functions exist in the wavelength region between 0.84 and 1.1 mm, the Jellison and Modine expression [57] is extrapolated to longer wavelengths, while the expression of Timans [60] coupled with the free-carrier absorption formula [61] is extrapolated to shorter wavelengths. The agreement between the calculated extinction coefficient values and the measured data from both Palik’s handbook [44] and Saritas and McKell [59] is very good at room temperature. However, as in the case of the refractive index, the disagreement between the expressions increases with temperature. Although it is not shown in the figure, lattice absorption occurs in the wavelength range between 6 and 25 mm. It should be noted that the effect of lattice absorption is negligible in most RTP applications compared to the band-gap absorption as well as free-carrier absorption. For heavily doped Si at room temperature, ionized donors or acceptors produce free carriers, which can greatly enhance the absorption in the IR region. The Drude free-electron model has been used to describe the freecarrier absorption of doped Si in many studies [23,62265]. Fu and Zhang [64] combined the interband absorption, lattice vibration, and free-electron

118

Zhuomin M. Zhang and Bong Jae Lee

4.4 Jellison-Modine [57] Li [58]

4.2

Refractive Index

Palik [44] 4.0

3.8 800°C

3.6

25°C

(a) 3.4 0.5

0.7

0.9

1.1

1.3

1.5

Wavelength (μm) 105

Jellison-Modine [57] Timans [60] Palik [44] Saritas et al. [59]

Absorption Coefficient (cm−1)

800°C 104

103

25°C

102

101

(b) 100 0.5

0.7

0.9

1.1

1.3

1.5

Wavelength (μm)

Figure 19 Optical constants of lightly doped silicon at selected temperatures [45]: (a) refractive index; (b) absorption coefficient.

absorption to model the dielectric function of silicon from low to high doping concentrations at temperatures from 300 to 1,000 K. More recently, Basu et al. [65] compared different models for the mobility and carrier concentration of heavily doped silicon near room temperature.

Theory of Thermal Radiation and Radiative Properties

119

An improved Drude model was developed for doping concentration exceeding 1018 cm3 and validated by reflectance and transmittance measurements in the wavelength region from 2 to 20 mm with an FTIR spectrometer [65].

6. Radiative Properties of Layered Structures The radiative properties of surfaces can be strongly affected by thinfilm coatings [6]. If the multiply reflected waves interfere with each other, the radiative properties for a nondispersive medium exhibit evenly spaced fringes when plotted in terms of the wavenumber (1/l). In this case, the multiply reflected waves are coherent and the film is said to be ‘‘thin.’’ On contrary, a ‘‘thick’’ layer refers to the case when the effect of interference between multiply reflected waves can be neglected in calculating the radiative properties. In other words, the waves are incoherent. In general, when the thickness is comparable to the wavelength, interference effects become important. However, this does not guarantee complete coherence in reality because of the nature of the source and imperfect surfaces (e.g., roughness and nonparallelism), resulting in partially coherent waves. Partial coherence theory considers the intermediate regime between the coherent limit and incoherent limit [66,67]. In the following, the radiative properties of films are discussed by using the ray-tracing method for thick films and thin-film optics for thin films. Because all radiative properties are presented on spectral basis, the subscript l is omitted for simplicity in notation. Furthermore, discussions are limited to the case of optically smooth surfaces.

6.1. The ray-tracing method Figure 20 shows a thick film or slab placed between two dielectrics (media 1 and 3). The refractive index and the extinction coefficient of the film are n2 and k2, respectively, and the thickness is d2cn2l. It should be noted that if the extinction coefficient is comparable to the refractive index, the transmittance of a thick film will be negligible and the film can be regarded as a semi-infinite medium. Therefore, for a thick film, it is usually appropriate to assume that the layer is slightly absorbing, that is, k2{n2. The refractive index of medium 1 or 3 is indicated by n1 and n3, respectively. In this figure, rij represents the reflectivity at the interface between medium i and medium j (i, j ¼ 1, 2, or 3) and can be calculated for each polarization using Equation (86b) or (87).

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Zhuomin M. Zhang and Bong Jae Lee

n1

1

12

(1 − 21)

21

(1 − 12) 2

n2 , 2

d2

23

n3

3

(1 − 23)

Figure 20 Schematic of the ray-tracing method for calculating the reflectance and transmittance by a thick film.

By tracing the energy of all reflected rays, the directional-hemispherical spectral reflectance of a thick film can be expressed as: R0l ¼ r12 þ r23 ð1  r12 Þð1  r21 Þt2i þ ð1  r12 Þð1  r21 Þr21 r223 t4i þ    ¼ r12 þ

r23 ð1  r12 Þ2 t2i 1  r12 r23 t2i

ð100Þ

where ti is the internal transmissivity given by:     4pk2 d 2 a2 d 2 ¼ exp  ti ¼ exp  l cos y2 cos y2

(101)

Here, y2 is the angle of refraction inside the slab. In general, the angle of refraction is complex due to the absorption. For a slightly absorbing medium with k2{1, however, y2 can be determined using Snell’s law by neglecting absorption [68]. It should be noted that the absorption by the layer is accounted by the internal transmissivity. Similarly, the directionalhemispherical spectral transmittance can be obtained as: T 0l ¼

ð1  r12 Þð1  r23 Þti 1  r12 r23 t2i

(102)

For randomly polarized incidence, the radiative properties should be averaged over s and p polarizations. The directional spectral absorptance can be obtained from A0l ¼ 1  R0l  T 0l and it is equal to the directional spectral emittance (or emissivity) according to Kirchhoff’s law. In the case of emissivity of a slab in air, we obtain 0l ¼

ð1  rÞð1  ti Þ 1  rti

(103)

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Theory of Thermal Radiation and Radiative Properties

where r is the reflectivity at the air2medium interface. This equation has been used to calculate the normal emissivity of a semitransparent Al2O3 slab (see Figure 18).

6.2. Thin-film optics When the thickness of each layer is comparable or less than the wavelength of incident radiation, the wave interference effects inside a layer become important. Figure 21 shows a thin film with thickness d2. Here, rij and tij represent the Fresnel reflection and transmission coefficient, respectively, at the interface between medium i and medium j (i, j ¼ 1, 2, or 3). In order to account for wave interference effects, the amplitude and phase of electric field (for s polarization) or magnetic field (for p polarization) need to be taken into account. After superposition of multiply reflected waves considering phase shifts upon traveling in the layer, the reflection and transmission coefficients of the film can be expressed as: r ¼ r 12 þ

t 12 t 21 r 23 ei2k2z d2 1  r 21 r 23 ei2k2z d2

(104)

and t¼

t 12 t 23 eik2z d2 1  r 21 r 23 ei2k2z d2

(105)

The above equations are known as Airy’s formulae [39,69]. Based on the reflection and transmission coefficients, the directional-hemispherical spectral reflectance and transmittance can be calculated as: R0l ¼ jrj2 1, 1

1

(106)

t21

r12

x z

t12

r21

2

d2

2, 2 r23

3, 3

3

t23

Figure 21 Illustration of the Fresnel reflection and transmission coefficients at the boundaries of a thin film.

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Zhuomin M. Zhang and Bong Jae Lee

1

Transmittance

0.8 0.6 0.4 Thin-film optics

0.2

Free spectral range

Ray-tracing method 0 1000

1200

1400

1600

1800

2000

Wavenumber (cm−1)

Figure 22

Normal transmittance of a 30-mm-thick silicon wafer.

8 Reðk3z =m3 Þ 2 > > jtj ; > < Reðk1z =m1 Þ 0 Tl ¼ > Reðk3z =3 Þ 2 > > : Reðk = Þ jtj ; 1z 1

for s polarization (107) for p polarization

Again, A0l ¼ 1  R0l  T 0l and Kirchhoff’s law states that 0l ¼ A0l . For randomly polarized incidence, the radiative properties should be separately calculated based on Fresnel coefficients for s and p polarizations and then averaged over two polarizations. Figure 22 shows the transmittance of a silicon wafer calculated based on either the thin-film optics (solid line) or the ray-tracing method (dashed line). The first and last media are assumed to be air (n ¼ 1). The optical constants of silicon are taken from the tabulated values [44] and the wafer thickness is set to be 30 mm in the calculation. When wave interference in the film is important, the resulting transmittance oscillates periodically as a function of wavenumber. The period of oscillation, that is, the separation between two consecutive interference maxima (or minima), is called the free spectral range Dn (in cm1) that is given by: Dn ¼

1 2n2 d2 cosy2

(108)

where n2 is the refractive index of silicon and d2 is its thickness. In contrast, the ray-tracing method (or geometric optics) completely ignores the interference effects and results in no fringe. In many practical applications, partial coherence must be taken into consideration to predict the radiative properties. In reality, partial coherence is often caused by the limited resolution of the instrument, beam divergence, as well as nonparallelism and surface roughness of the sample [6,9,10].

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Theory of Thermal Radiation and Radiative Properties

1 = 1, 1 = 1

1 x

2, 2

2

d2

N−1, N−1

N−1

dN−1

N, N

N z

Figure 23 structure.

Geometry for calculating the radiative properties of a multilayer

6.3. Radiative properties of multilayer structures The matrix formulation provides a convenient way to calculate the radiative properties of multilayer structures [6,69,70]. For the jth medium (refer to Figure 23), the relative electric permittivity is ej and the relative magnetic permeability is mj. The thickness of the jth layer is denoted by dj. It is assumed that e1 ¼ 1 and m1 ¼ 1, that is, the top semi-infinite medium is air or vacuum. The electromagnetic wave is incident from air at an angle of incidence y1 and is reflected from or transmitted through the following layers. For s-polarized waves, the y component of electric field in the jth medium can be expressed as a summation of forward and backward waves in the z-direction:  (  ik z A1 e 1z þ B1 ek1z z eiðkx xotÞ ; j ¼ 1  Ey ðx; zÞ ¼  ikjz ðzzj1 Þ þ Bj ekjz ðzzj1 Þ eiðkx xotÞ ; Aj e

j ¼ 2; 3; . . . ; N

(109)

where Aj and Bj are the amplitudes of forward and backward waves in the jth layer, respectively, z1 ¼ 0 and zj ¼ zj1 + dj( j ¼ 2, 3, y, N1), and kx and kjz are the parallel and perpendicular components of the wavevector, respectively. Notice that kx is conserved throughoutqthe layers due to the ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

phase-matching condition, and kjz is given by kjz ¼ mj j o2 =c 20  k2x . The magnetic field can be obtained from the electric field using Maxwell’s equation. From the boundary conditions at the interface, field amplitudes

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Zhuomin M. Zhang and Bong Jae Lee

of adjacent layers can be related as: Aj Bj

! ¼

ðPj D1 j Djþ1 Þ

Ajþ1 Bjþ1

! ; j ¼ 1; 2; . . . ; N  1

(110)

Here, Pj is the propagation matrix given by:  P1 ¼ I ¼

1 0

0 1

 (111a)

and Pj ¼

eikjz dj

0

0

eikjz dj

! ;

j ¼ 2; 3; . . . ; N  1

(111b)

In addition, Dj is the dynamical matrix given as: Dj ¼

!

1

1

kjz =mj

kjz =mj

;

j ¼ 1; 2; . . . ; N

(112)

By applying Equation (110) to all layers, we obtain A1 B1

! ¼

N1 Y

Pj D1 j Djþ1

j¼1

AN BN

! (113)

Consequently, the spectral radiative properties of the N-layer system are B1 B 1 A21

(114)

ReðkNz =mN Þ AN A N Reðk1z =m1 Þ A21

(115)

R0l ¼

and T 0l ¼

For p-polarized waves, the same procedure can be applied using the magnetic field with mj’s being replaced by ej’s. A more detailed derivation suitable for both nonmagnetic and magnetic materials can be found from Ref. [70]. Figure 24 shows the normal spectral transmittance of a one-dimensional multilayer structure as a wavelength-selective filter with a center wavelength at 550 nm. The structure is composed of alternating SiC (nHE2.69) and SiO2 (nLE1.47) thin films. In order to suppress the

125

Theory of Thermal Radiation and Radiative Properties

1

M1

0.6

0.4

M2

Transmittance

0.8

0.2

0 0.3

0.4

0.5

0.6

0.7

0.8

0.9

Wavelength (μm)

Figure 24 Normal transmittance of a multilayer structure for a spectrally selective filter, which is constructed by two sets of four unit cells of SiC/SiO2 films with different periods.

transmission above and below the center wavelength, two sets of SiC/SiO2 photonic crystal are used with different periods. The filter performance is optimized by varying the thickness of SiC and SiO2 layers. For the top structure M1, the thicknesses of SiC and SiO2 are set to be 66 and 121 nm, respectively. For the bottom structure M2, the SiC and SiO2 thicknesses are set to be 33 and 61 nm, respectively. Due to the wave interference effects, the filter provides higher transmittance at wavelengths between 500 and 600 nm. Hence, the SiC/SiO2 structure can be used for filter radiometers [71].

6.4. Thin films on a thick substrate In many cases, one needs to consider structures of thin films coated on a thick substrate, where waves inside the films must be treated as coherent but waves in the substrate can be treated as incoherent. Figure 25 shows the geometry of a thick substrate with thin-film coatings on both sides. Here, rta is the reflectance of the top multilayer films for rays incident from air, assuming that the substrate extends to infinity; whereas rts is the reflectance for rays incident from the substrate to the top films. Note that the transmittance tt of the top multilayer film is the same regardless of whether the ray is incident from air or the substrate, when absorption inside the substrate is negligibly small. Similarly, rbs and tb are the reflectance and transmittance, respectively, for the multilayer films at the bottom of

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Zhuomin M. Zhang and Bong Jae Lee

ta

1

ds

s

t

t

Air

ts

bs

b

1

Substrate

Air

Figure 25 Schematic of thin-film coatings on both sides of a thick substrate.

the substrate for rays incident from the substrate. The transfer matrix formulation discussed in the previous section can be separately applied to calculate the reflectance and transmittance of the thin films at the top and bottom, by treating the substrate as a nonabsorbing dielectric. Nevertheless, the absorption of the substrate can be accounted for by introducing the internal transmissvity ti, as done in the ray-tracing method discussed previously. Consequently, the radiative properties of the substrate with thin-film coatings in the semitransparent region can be expressed as [23,45]: R0l ¼ rta þ

t2i t2t rbs 1  t2i rts rbs

(116)

ti tt tb 1  t2i rts rbs

(117)

0l ¼ 1  R0l  T 0l

(118)

T 0l ¼

and

Lee and Zhang [72] developed a software tool, named Rad-Pro for Radiative Properties, using VBA programming in Excel. It allows users to easily calculate and plot the directional, spectral, and temperature dependence of the radiative properties of silicon wafers with various thin-film coatings. In Rad-Pro, users can choose to use the formulation for coherent and incoherent, as well as other options, such as the polarization state of incident radiation and the option that treats the silicon substrate as a semi-infinite medium (opaque). This software tool and user

127

Theory of Thermal Radiation and Radiative Properties

1 Thin films coated on Si substrates 0.8 Reflectance

(v) Curves: Calculation Marks: Measurement

0.6

(i)

0.4

(iv)

0.2

(ii) (iii) 0 0.5

0.6

0.7

0.8

0.9

1

Wavelength (μm)

Figure 26 Near-normal reflectance of silicon substrates with coatings [45]: (i) bare substrate with 4-nm native SiO2, (ii) 128-nm Si3N4 film, (iii) 306-nm SiO2 film, (iv) 868-nm SiO2 film, and (v) 50-nm polysilicon and 170-nm SiO2 films.

manual are downloadable free of charge at the following website: www.me.gatech.edu/Bzzhang. A few examples are given below to illustrate the effect of thin films on the radiative properties. Figure 26 compares the calculated reflectance with that measured at near-normal incidence, in the spectral region from 0.5 to 1 mm, using a calibrated spectrophotometer at the National Institute of Standards and Technology [45]. In this region, the silicon wafer is opaque. Excellent agreement exists between the predicted and the measured values. The reflectance with one layer coating never exceeds that of the bare silicon. For the nitride-coated wafer, there exists a reflectance minimum of o0.003 at l ¼ 1 mm. For the oxide-coated wafer, interference effects can be clearly seen and the period decreases with increasing film thickness. The variation is the largest in the reflectance spectra for the silicon wafer with a 50 nm polysilicon film on top of a 170 nm silicon-dioxide coating. Figure 27 shows the calculated normal emissivity for a 700-mm-thick silicon wafer at different temperatures at wavelengths from 0.5 to 5 mm. The silicon wafer is assumed to be lightly doped and polished on both sides. The optical constants of polysilicon are assumed to be the same as those of silicon. Note that silicon wafers are semitransparent at wavelengths longer than the band-gap wavelength lg. Hence, the emissivity depends on the thickness of the wafer. At 251C, free-carrier absorption is negligibly small so that the emissivity at wavelengths longer than lgE1.1 mm is almost zero. At 5001C, thermally excited carriers can result in significant absorption and subsequently a large emissivity beyond 1.1 mm. With multilayer coatings,

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1 300 nm SiO2, 25 °C 300 nm SiO2, 500 °C

0.8

Emissivity

800 nm SiO2, 500 °C 0.6

0.4

0.2 Polysilicon/SiO2 0 0.5 0.6

0.8

1

3

5

Wavelength (μm)

Figure 27 Calculated normal emissivity of coated silicon of 700-mm thickness at different temperatures. In the case of the polysilicon and SiO2 coatings, the thicknesses are 100 and 300 nm, respectively.

the emissivity can be close to unity at certain wavelengths and very small at some other wavelengths. The correction of radiometric temperature measurement caused by the uncertainty in emissivity posts a significant challenge in lightpipe thermometers used in RTP industry. Additional discussion on the application of lightpipe thermometry in semiconductor processing can be found from Chapter 3 in the companion volume, Vol. 43 of this series.

7. Summary Beginning with a discussion of the historical perspectives in the development of the thermal radiation theory, this chapter describes the thermodynamics foundation of the Stefan2Boltzmann law and the Planck law of blackbody radiation. Definitions of the radiative properties and their relationships are then presented, for example, emissivity, absorptivity, and reflectivity for opaque materials, as well as the reflectance, transmittance, and absorptance for semitransparent materials. Brief discussions on effective emissivity and radiative transfer in participating media are also given. Maxwell’s electromagnetic wave theory is introduced for the prediction of reflection and transmission at the interfaces as well as in multilayer structures. Furthermore, frequency-dependent dielectric function models are described for metals, dielectric materials or insulators, and semiconductors, in the VIS and IR regions. The fundamental concepts and basic

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129

theories described in this chapter are essential for the practice of radiation thermometry described in subsequent chapters.

ACKNOWLEDGEMENT Partial support was provided by the National Science Foundation (CBET-0500113, 0828701).

REFERENCES [1]

[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

P. J. Mohr, B. N. Taylor, and D. B. Newell, ‘‘The 2006 CODATA Recommended Values of the Fundamental Physical Constants,’’ Rev. Mod. Phys. 80, 6332730. Available: http://physics.nist.gov/constants [2008, November 17]. National Institute of Standards and Technology, Gaithersburg, MD 20899 (2008). International Bureau of Weights and Measures (BIPM), The International System of Units (SI), Paris, France, 8th ed. www.bipm.org/utils/common/pdf/si_brochure_ 8_en.pdf (2006). F. P. Incropera, D. P. DeWitt, T. L. Bergman, and A. S. Lavine, Fundamentals of Heat and Mass Transfer, Wiley, New York, 6th ed. (2007). R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer, Taylor & Francis, New York, 4th ed. (2002). M. Planck, The Theory of Heat Radiation, Dover Publications, New York (1959). Z. M. Zhang, Nano/Microscale Heat Transfer, McGraw-Hill, New York, (2007). Y. S. Touloukian and D. P. DeWitt, ‘‘Thermal Radiative Properties,’’ in: Thermophysical Properties of Matter, TPRC Data Series, edited by Y. S. Touloukian and C. Y. Ho, IFI Plenum, New York, Vols. 729 (197021972). D. R. Lide, CRC Handbook of Chemistry and Physics, CRC Press, Boca Raton, FL, 89th ed. (2008). Z. M. Zhang, C. J. Fu, and Q. Z. Zhu, ‘‘Optical and Thermal Radiative Properties of Semiconductors Related to Micro/Nanotechnology,’’ Adv. Heat Transfer 37, 1792296 (2003). B. J. Lee, V. P. Khuu, and Z. M. Zhang, ‘‘Partially Coherent Spectral Radiative Properties of Dielectric Thin Films with Rough Surfaces,’’ J. Thermophys. Heat Transfer 19, 3602366 (2005). P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, Artech House, Norwood, MA (1987). J. M. Chalmers and P. R. Griffiths (eds.), Handbook of Vibrational Spectroscopy, Wiley, New York, Vol. 5 (2002). L. M. Hanssen and K. A. Snail, ‘‘Integrating Spheres for Mid- and Near-Infrared Reflectance Spectroscopy,’’ In Ref. [12], Vol. 2, pp. 117521191 (2002). P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectrometry, Wiley, New York (1986). Z. M. Zhang, L. M. Hanssen, R. U. Datla, and H. D. Drew, ‘‘An Apparatus for Infrared Transmittance and Reflectance Measurements at Cryogenic Temperatures,’’ Int. J. Thermophys. 17, 144121454 (1996). Z. M. Zhang, L. M. Hanssen, and R. U. Datla, ‘‘Polarization-Dependent Angular Reflectance of Silicon and Germanium in the Infrared,’’ Infrared Phys. Technol. 37, 5392546 (1996).

130

Zhuomin M. Zhang and Bong Jae Lee

[17]

A. R. Kumar, Z. M. Zhang, V. A. Boychev, D. B. Tanner, L. R. Vale, and D. A. Rudman, ‘‘Far-Infrared Transmittance and Reflectance of YBa2Cu3O7d Films on Si Substrates,’’ J. Heat Transfer 121, 8442851 (1999). H. J. Lee, A. C. Bryson, and Z. M. Zhang, ‘‘Measurement and Modeling of the Emittance of Silicon Wafers with Anisotropic Roughness,’’ Int. J. Thermophys. 28, 9182932 (2007). P. Y. Barnes, E. A. Early, and A. C. Parr, Spectral Reflectance, NIST Special Publication 250-48, U.S. Government Printing Office, Washington, DC (1998). Y. J. Shen, Q. Z. Zhu, and Z. M. Zhang, ‘‘A Scatterometer for Measuring the Bidirectional Reflectance and Transmittance of Semiconductor Wafers with Rough Surfaces,’’ Rev. Sci. Instrum. 74, 488524892 (2003). H. J. Lee, B. J. Lee, and Z. M. Zhang, ‘‘Modeling the Radiative Properties of Semitransparent Wafers with Rough Surfaces and Thin-Film Coatings,’’ J. Quant. Spectrosc. Radiat. Transfer 93, 1852194 (2005). H. J. Lee, Y.-B. Chen, and Z. M. Zhang, ‘‘Directional Radiative Properties of Anisotropic Rough Silicon and Gold Surfaces,’’ Int. J. Heat Mass Transfer 49, 448224495 (2006). P. J. Timans, ‘‘The Thermal Radiative Properties of Semiconductors,’’ in Advances in Rapid Thermal and Integrated Processing, edited by F. Roozeboon, Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 352101 (1996). Z. M. Zhang, ‘‘Surface Temperature Measurement Using Optical Techniques,’’ in Annual Reviews of Heat Transfer, edited by C. L. Tien, Begell House, New York, Vol. 11, pp. 3512411 (2000). Y. H. Zhou, Y.-J. Shen, Z. M. Zhang, B. K. Tsai, and D. P. DeWitt, ‘‘A Monte Carlo Model for Predicting the Effective Emissivity of the Silicon Wafer in Rapid Thermal Processing Furnaces,’’ Int. J. Heat Mass Transfer 45, 194521949 (2002). Y. H. Zhou and Z. M. Zhang, ‘‘Radiative Properties of Semitransparent Silicon Wafers with Rough Surfaces,’’ J. Heat Transfer 125, 4622470 (2003). Q. Z. Zhu and Z. M. Zhang, ‘‘Anisotropic Slope Distribution and Bidirectional Reflectance of a Rough Silicon Surface,’’ J. Heat Transfer 126, 9852993 (2004). H. J. Lee and Z. M. Zhang, ‘‘Measurement and Modeling of the Bidirectional Reflectance of SiO2 Coated Si Surfaces,’’ Int. J. Thermophys. 27, 8202839 (2006). Z. M. Zhang and Y. H. Zhou, ‘‘An Effective Emissivity Model for Rapid Thermal Processing Using the Net-Radiation Method,’’ Int. J. Thermophys. 22, 156321575 (2001). C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley, New York (1983). C. L. Tien, ‘‘Thermal Radiation Properties of Gases,’’ Adv. Heat Transfer 5, 2532324 (1968). L. S. Rothman, D. Jacquemart, A. Barbe, et al., ‘‘The HITRAN 2004 Molecular Spectroscopic Database,’’ J. Quant. Spectrosc. Radiat. Transfer 96, 1392204 (2005). J. D. Jackson, Classical Electrodynamics, Wiley, New York, 3rd ed. (1999). D. J. Griffiths, Introduction to Electrodynamics, Prentice Hall, Upper Saddle River, NJ, 3rd ed. (1999). N. W. Ashcroft and N. D. Mermin, Solid State Physics, Saunders College, Philadelphia, PA (1976). M. A. Omar, Elementary Solid State Physics: Principles and Applications, Addison-Wesley, New York (1975). V. M. Shalaev, ‘‘Optical Negative-Index Metamaterials,’’ Nat. Photonics 1, 41248 (2007).

[18] [19] [20] [21] [22] [23] [24] [25]

[26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]

Theory of Thermal Radiation and Radiative Properties

[38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59]

131

C. J. Fu and Z. M. Zhang, ‘‘Thermal Radiative Properties of Metamaterials and Other Nanostructured Materials: A Review,’’ Front. Ener. Power Eng. China 3, 11226 (2009). M. Born and E. Wolf, Principles of Optics, Cambridge University Press, Cambridge, UK, 7th ed. (1999). Z. M. Zhang and K. Park, ‘‘On the Group Front and Group Velocity in a Dispersive Medium upon Refraction from a Nondispersive Medium,’’ J. Heat Transfer 126, 2442249 (2004). R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, North-Holland, Amsterdam, The Netherlands (1977). Z. M. Zhang and B. J. Lee, ‘‘Lateral Shift in Photon Tunneling Studied by the Energy Streamline Method,’’ Opt. Express 14, 996329970 (2006). B. J. Lee, K. Park, and Z. M. Zhang, ‘‘Energy Pathways in Nanoscale Thermal Radiation,’’ Appl. Phys. Lett. 91, 153101 (2007). E. D. Palik, Handbook of Optical Constants of Solids I, II, & III, Academic Press, San Diego, CA (1998). B. J. Lee, Z. M. Zhang, E. A. Early, D. P. DeWitt, and B. K. Tsai, ‘‘Modeling Radiative Properties of Silicon with Coatings and Comparison with Reflectance Measurements,’’ J. Thermophys. Heat Transfer 19, 5582569 (2005). M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander, and C. A. Ward, ‘‘Optical Properties of the Metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the Infrared and Far Infrared,’’ Appl. Opt. 22, 109921119 (1983). P. G. Etchegoin, E. C. Le Ru, and M. Meyer, ‘‘An Analytic Model for the Optical Properties of Gold,’’ J. Chem. Phys. 125, 164705 (2006). A. S. Barker, Jr., ‘‘Infrared Lattice Vibration and Dielectric Dispersion in Corundum,’’ Phys. Rev. 132, 147421481 (1963). J. R. Markham, P. R. Solomon, and P. E. Best, ‘‘An FT-IR Based Instrument for Measuring Spectral Emittance of Material at High Temperature,’’ Rev. Sci. Instrum. 61, 370023708 (1990). J. R. Markham, P. E. Best, P. R. Solomon, and Z. Z. Yu, ‘‘Measurement of Radiative Properties of Ash and Slag by FT-IR Emission and Reflection Spectroscopy,’’ J. Heat Transfer 114, 4582464 (1992). O. Rozenbaum, D. De Sousa Meneses, Y. Auger, S. Chermanne, and P. Echegut, ‘‘A Spectroscopic Method to Measure the Spectral Emissivity of Semi-Transparent Materials up to High Temperature,’’ Rev. Sci. Instrum. 70, 402024025 (1999). B. Rousseau, J. F. Brun, D. De Sousa Meneses, and P. Echegut, ‘‘Temperature Measurement: Christiansen Wavelength and Blackbody Reference,’’ Int. J. Thermophys. 26, 127721286 (2005). Z. M. Zhang, B. I. Choi, M. I. Flik, and A. C. Anderson, ‘‘Infrared Refractive Indices of LaAlO3, LaGaO3, and NdGaO3,’’ J. Opt. Soc. Am. B 11, 225222257 (1991). P. Calvani, M. Capizzi, S. Lupi, P. Maselli, and E. Agostinelli, ‘‘High Tc Superconducting Thin Films: An Analysis of Reflectance Spectra,’’ Physica C 180, 1162119 (1991). S. M. Sze, Physics of Semiconductor Devices, Wiley, New York, 2nd ed. (1981). T. Sato, ‘‘Spectral Emissivity of Silicon,’’ Jpn. J. Appl. Phys. 6, 3392347 (1967). G. E. Jellison and F. A. Modine, ‘‘Optical Functions of Silicon at Elevated Temperatures,’’ J. Appl. Phys. 76, 375823761 (1994). H. H. Li, ‘‘Refractive-Index of Silicon and Germanium and Its Wavelength and Temperature Derivatives,’’ J. Phys. Chem. Ref. Data 9, 5612658 (1980). M. Saritas and H. D. McKell, ‘‘Absorption Coefficient of Si in the Wavelength Region between 0.8021.16 mm,’’ J. Appl. Phys. 61, 492324925 (1987).

132

Zhuomin M. Zhang and Bong Jae Lee

[60]

P. J. Timans, ‘‘Emissivity of Silicon at Elevated Temperatures,’’ J. Appl. Phys. 74, 635326364 (1993). P. Vandenabeele and K. Maex, ‘‘Influence of Temperature and Backside Roughness on the Emissivity of Si Wafers during Rapid Thermal Processing,’’ J. Appl. Phys. 72, 586725875 (1992). P. J. Hesketh, J. N. Zemel, and B. Gebhart, ‘‘Polarized Spectral Emittance from Periodic Micromachined Surfaces. 2. Doped Silicon-Angular Variation,’’ Phys. Rev. B 37, 10803210813 (1988). J. P. Hebb, ‘‘Pattern Effects in Rapid Thermal Processing’’, Ph.D. Dissertation, Massachusetts Institute of Technology, Cambridge, MA (1997). C. J. Fu and Z. M. Zhang, ‘‘Nanoscale Radiation Heat Transfer for Silicon at Different Doping Levels,’’ Int. J. Heat Mass Transfer 49, 170321718 (2006). S. Basu, B. J. Lee, and Z. M. Zhang, ‘‘Infrared Radiative Properties of Heavily Doped Silicon at Room Temperature,’’ J. Heat Transfer (accepted). G. Chen and C. L. Tien, ‘‘Partial Coherence Theory of Thin-Film Radiative Properties,’’ J. Heat Transfer 114, 6362643 (1992). K. Fu, P.-F. Hsu, and Z. M. Zhang, ‘‘Unified Analytical Formulation of Thin-Film Radiative Properties Including Partial Coherence,’’ Appl. Opt. 45, 6532661 (2006). Z. Zhang, ‘‘Optical Properties of a Slightly Absorbing Film for Oblique Incidence,’’ Appl. Opt. 38, 2052207 (1999). P. Yeh, Optical Waves in Layered Media, Wiley, New York (1988). C. J. Fu, Z. M. Zhang, and D. B. Tanner, ‘‘Energy Transmission by Photon Tunneling in Multilayer Structures Including Negative Index Materials,’’ J. Heat Transfer 127, 104621052 (2005). M. Durak and F. Samadov, ‘‘Realization of a Filter Radiometer-Based Irradiance Scale with High Accuracy in the Region from 286 nm to 901 nm,’’ Metrologia 41, 4012406 (2004). B. J. Lee and Z. M. Zhang, ‘‘Rad-Pro: Effective Software for Modeling Radiative Properties in Rapid Thermal Processing,’’ in Proceedings of the 13th IEEE International Conference on Advanced Thermal Processing of Semiconductors-RTP 2005, Santa Barbara, CA, pp. 2752281 (2005).

[61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72]