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Fundamentals of Thermal Radiation

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Nano/Microscale Heat Transfer

Part of the book series: Mechanical Engineering Series ((MES))

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Abstract

This chapter contains an introduction to the electromagnetic wave theory, blackbody radiation, plane wave reflection, and refraction at the boundary between two semi-infinite media, evanescent waves and total internal reflection, and various models used to study the optical properties of different materials. A brief description on the typical experimental methods used to measure the spectral radiative properties is also presented. The materials covered in the following sections are intended to provide a sound background for more in-depth studies on the applications of thermal radiation to micro/nanosystems in subsequent chapters.

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Authors and Affiliations

  1. Marietta, GA, USA

    Zhuomin M. Zhang

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  1. Zhuomin M. Zhang
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Problems

Problems

  1. 8.1

    Write the wave equation in the 1D scalar form as \(\frac{{\partial^{2} \psi }}{{\partial x^{2} }} = \frac{1}{{c^{2} }}\frac{{\partial^{2} \psi }}{{\partial t^{2} }}\), where c is a positive constant. Prove that any analytical function f can be its solution as long as \(\psi (x,t) = f(x \pm ct)\). Plot \(\psi\) as a function of x for two fixed times t1 and t2. Show that the sign determines the direction (either forward or backward) and c is the speed of propagation. Develop an animated computer program to visualize wave propagation.

  2. 8.2

    Consider an electromagnetic wave propagating in the positive z-direction, i.e., \({\mathbf{k}} = k{\hat{\mathbf{z}}}\). Plot the vibration ellipse, and compare it with Fig. 8.2 for two cases: (1) \({\mathbf{a}} = 3{\hat{\mathbf{x}}}\) and \({\mathbf{b}} = {\hat{\mathbf{x}}} + 2{\hat{\mathbf{y}}}\) and (2) \({\mathbf{a}} = 3{\hat{\mathbf{x}}}\) and \({\mathbf{b}} = - 2{\hat{\mathbf{x}}} + {\hat{\mathbf{y}}}\). Consider the spatial dependence of the electric field at a given time, say \(\omega t = 2\pi m\), where m is an integer. Discuss how E will change with kz for the following two cases: (3) \(\text{Re} ({\mathbf{E}}_{0} ) = 3{\hat{\mathbf{x}}}\) and \(\text{Im} (\text{E}_{0} ) = 0\) and (4) \(\text{Re} ({\mathbf{E}}_{0} ) = 3{\hat{\mathbf{x}}}\) and \(\text{Im} ({\mathbf{E}}_{0} ) = - 3{\hat{\mathbf{y}}}\). The polarization is said to be right handed if the end of the electric field vector forms a right-handed coil or screw in space at any given time. Otherwise, it is said to be left handed. Discuss the handedness for all four cases.

  3. 8.3

    Integrate Eq. (8.17) over a control volume to show that the energy transferred through the boundary into the control volume is equal to the sum of the storage energy change and energy dissipation. Write an integral equation using Gauss’s theorem.

  4. 8.4

    Derive the wave equation in Eq. (8.20) for a conductive medium; show Eq. (8.9) is a solution if k is complex, as given in Eq. (8.21). Many books use \({\mathbf{E}} = {\mathbf{E}}_{0} {\text{e}}^{{{\text{i}}(\omega t - {\mathbf{k}} \cdot {\mathbf{r}})}}\) instead of Eq. (8.9) as the solution; how would you modify Eqs. (8.21) and (8.22)? Show that the complex refractive index must be defined as \(\tilde{n} = n - {\text{i}}\kappa\), where \(\kappa \ge 0\).

  5. 8.5

    Calculate the refractive index, the absorption coefficient, and the radiation penetration depth for the following materials, based on the dielectric function values at room temperature.

    1. (a)

      Glass (SiO2): \(\varepsilon = 2.1 + {\text{i}}0\) at 1 μm; \(\varepsilon = 1.8 + {\text{i0}} . 0 0 4\) at 5 μm.

    2. (b)

      Germanium: \(\varepsilon = 21 + {\text{i0}} . 1 4\) at 1 μm; \(\varepsilon = 16 + {\text{i0}} . 0 0 0 3\) at 20 μm.

    3. (c)

      Gold: \(\varepsilon = - 10 + {\text{i1}} . 0\) at 0.65 μm; \(\varepsilon = - 160 + {\text{i2}} . 1\) at 2 μm.

  6. 8.6

    Consider a metamaterial with \(\mu = - 1 + {\text{i}}0.01\) and \(\varepsilon = - 2 + {\text{i}}0.01\); determine the refractive index and the extinction coefficient. Calculate the radiation penetration depth. Do a quick Internet search on negative index materials, and briefly describe what you have learned.

  7. 8.7

    Find the magnetic field H for the wave given in Eq. (8.37). Show that the time-averaged Poynting vector is parallel to the x-axis. That is, the z-component of \(\left\langle {\mathbf{S}} \right\rangle\) for such a wave vanishes. Briefly describe the features of an evanescent wave.

  8. 8.8

    Write Planck’s distribution in terms of wavenumber \(\bar{\nu } = 1/\lambda\), i.e., the emissive power in terms of the wavenumber: \(e_{{b,\bar{\nu }}} (\bar{\nu },T)\). What is the most probable wavenumber in \({\text{cm}}^{ - 1}\)? Compare your answer with the most probable wavelength obtained from Wien’s displacement law in Eq. (8.45). Explain why the constants do not agree with each other. Cosmic background radiation can be treated as blackbody radiation at 2.7 K; what is the wavenumber corresponding to the maximum emissive power?

  9. 8.9

    Based on the geometric parameters provided in Example 8.3 and neglecting the atmospheric effect, calculate the total intensity of the solar radiation arriving at earth’s surface. Calculate the spectral intensity for solar radiation at 628 nm wavelength. A child used a lens to focus solar radiation to a small spot on a piece of paper and set fire this way. Does the beam focusing increase the intensity of the radiation? The lens diameter is 5 cm, and the distance between the lens and the paper is 2.5 cm. What are the focus size and the heat flux at the focus? Neglect the loss through the lens.

  10. 8.10

    For a surface at T = 1800 K, with an emissivity of 0.6, what are the radiance temperatures at λ = 0.65 μm and 1.5 μm? If a conical hole is formed with a half-cone angle of 15°, what is the effective emittance and the radiance temperature at λ = 0.65 μm?

  11. 8.11

    Derive Planck’s law for a medium with a refractive index \(n \ne 1\) in terms of the medium wavelength \(\lambda_{\text{m}}\), \(e_{{{\text{b,}}\lambda_{\text{m}} }} (\lambda_{\text{m}} ,T)\) from Eq. (8.43). Assume that n is not a function of frequency (i.e., the medium is nondispersive) in the spectral region of interest. How does it compare with Eq. (8.44)?

  12. 8.12

    Express Eq. (8.53) in terms of wavelength, \(s_{\lambda } (\lambda ,T)\). Find an expression of the entropy intensity for blackbody radiation, \(L_{\lambda } (\lambda ,T)\), and show that \(L_{\lambda } (\lambda ,T) = \frac{c}{4\pi }s_{\lambda } (\lambda ,T)\).

  13. 8.13

    Assume that all the blue light at λ in the range between 420 and 490 nm of solar radiation is scattered by the atmosphere and uniformly distributed over a solid angle of \(4\pi\) sr. What are the monochromatic temperatures of the scattered radiation at λ = 420 and 490 nm?

  14. 8.14

    A diode-pumped solid-state laser emits continuous-wave (cw) green light at a wavelength of 532 nm with a beam diameter of 1.1 mm. If the beam divergence is \(2 \times 10^{ - 7} \;{\text{sr}}\), what would be the spot size at a distance of 100 m from the laser (without scattering)? If the output optical power is 2 mW and the spectral width is \(\delta \lambda = 0.1\;{\text{nm}}\) (assuming a square function), what is the average intensity of the laser beam? Find the monochromatic radiation temperature of the laser when it is linearly polarized. Suppose the laser hits a rough surface and is scattered into the hemisphere isotropically. Find the radiation temperature of the scattered radiation and the entropy generation caused by scattering.

  15. 8.15

    In Example 8.5, the two plates are blackbodies. Assume that the plates are diffuse-gray surfaces with emissivities \(\varepsilon^{\prime}_{1} \;{\text{and}}\;\varepsilon^{\prime}_{ 2}\). Calculate the entropy generation rate in each plate per unit area. How will you determine the optimal efficiency for an energy conversion device installed at plate 2? For \(T_{1} = 1500\;{\text{K, }}T_{2} = 300\;{\text{K,}}\;{\text{and}}\;\varepsilon^{\prime}_{ 2} = 1\), plot the optimal efficiency versus \(\varepsilon^{\prime}_{1}\).

  16. 8.16

    The concept of dilute blackbody radiation can be used as an alternative method to calculate the entropy generation of a two-plate problem as in Problem 8.15. Assume that the multiply reflected rays are at not in equilibrium with each other. Rather, each ray retains its original entropy and can be treated as having an effective temperature of \(T_{1}\) or \(T_{2}\) depending on which plate the ray is emitted from. How would you evaluate the entropy transfer from plate 1 to 2 and the entropy generation by each plate then?

  17. 8.17

    Calculate the entropy generation rate per unit volume for Example 2.7. Further, calculate the entropy generated at each surface, assuming that surface 2 is at 300 K.

  18. 8.18

    The conversion efficiency of thermophotovoltaic devices is wavelength dependent, and the optical constants are wavelength dependent as well. Perform a literature search to find some recent publications in this area. Use the entropy concept to determine the ultimate efficiency of a specific design. Based on your analysis, propose a few suggestions for further improvement of the particular design you have chosen.

  19. 8.19

    Derive the Fresnel reflection coefficient for a TM wave, following the derivation given in the text for a TE wave .

  20. 8.20

    Show that \(\rho^{\prime}_{\lambda ,s} + \alpha^{\prime}_{\lambda ,s} = 1\), where \(\rho^{\prime}_{\lambda ,s}\) is given in Eq. (8.73) and \(\alpha^{\prime}_{\lambda ,s}\) is given in Eq. (8.75). Discuss why the z-component of the time-averaged Poynting vector must be continuous at the boundary but not the x-component.

  21. 8.21

    For nonmagnetic lossy media with \(\varepsilon_{1} = \varepsilon^{\prime}_{1} + {\text{i}}\varepsilon^{\prime\prime}_{1}\) and \(\varepsilon_{2} = \varepsilon^{\prime}_{2} + {\text{i}}\varepsilon^{\prime\prime}_{2}\), expand Eq. (8.70b) and compare your results with Eq. (8.71).

  22. 8.22

    For plane waves incident from air to a nonmagnetic material with \(\varepsilon = - 2 + {\text{i}}0\) (negative real), show that the reflectivity is always 1 regardless of the angle of incidence and the polarization. What can you say about \(k_{2z}\) and \(\left\langle {S_{2z} } \right\rangle\)? Is the wave in the medium a homogeneous wave or an evanescent wave?

  23. 8.23

    The refractive index of glass is approximately 1.5 in the visible region. What is the Brewster angle for glass when light is incident from air? Calculate the reflectance and plot it against the incidence angle for p-polarization, s-polarization, and random polarization. Redo the calculation for incidence from glass to air, and plot the reflectance against the incidence angle. At what angle does total internal reflection begin and what is this angle called?

  24. 8.24

    Denote the incidence angle at which the ratio of the reflectance for TM and TE waves is minimized as \(\theta_{\text{M}}\). For radiation incident from air to a medium with \(n = 2\) and \(\kappa = 1\), determine \(\theta_{\text{M}}\) and compare it with the principle angle \(\theta_{\text{P}}\), at which the phase difference between the two reflection coefficients equals to \(\pi /2\). [Hint: Use graphs to prove the existence of \(\theta_{\text{M}}\) and \(\theta_{\text{P}}\).]

  25. 8.25

    For incidence from glass with n = 1.5 to air, calculate the Goos–Hänchen phase shift δ for both TE and TM waves . Plot δ as a function of the incidence angle \(\theta_1\).

  26. 8.26

    Show that the normal component of the time-averaged Poynting vector is zero in both the incident and transmitting media when total internal reflection occurs. Furthermore, derive Eq. (8.92).

  27. 8.27

    Calculate the Goos–Hänchen lateral shift upon total internal reflection from a dielectric with n = 2 to air. Plot the lateral shift for both TE and TM waves as a function of \(\theta_{1}\). Discuss the cause and the physical significance of the lateral beam shift.

  28. 8.28

    A perfect conductor can be understood based on the Drude free-electron model by neglecting the collision term. The dielectric function becomes \(\varepsilon (\omega ) = 1 - \omega_{\text{p} }^{2} /\omega^{2}\), where \(\omega_{\text{p} }\) is the plasma frequency. For radiation incident from air to a perfect conductor, calculate the phase shift when \(\omega = \omega_{\text{p}} /2\) for TE and TM waves as a function of the incidence angle. Use Eq. (8.93) to calculate the lateral beam shift for a TM wave and modify it for a TE wave . Do you expect a sign difference between the TE and TM waves?

  29. 8.29

    Calculate and plot the emissivity (averaged over the two polarizations) versus the zenith angle for the materials and wavelengths given in Problem 8.5. Calculate and tabulate the normal and hemispherical emissivities for all cases.

  30. 8.30

    Calculate the optical constants and the radiation penetration depth for either gold or silver at room temperature, using the Drude model, and plot them as functions of wavelength. In addition, calculate the reflectivity and plot it against wavelength. Compare the results using the Hagen–Ruben equation. How will the scattering rate and the plasma frequency change if the temperature is raised to 600 K?

  31. 8.31

    Calculate the normal emissivity of MgO from 2000 to 200 cm–1 (5 to 50 μm) using the Lorentz model with two oscillators having the following parameters: \(\varepsilon_{\infty } = 3.01\); \(\omega_{1} = 401\;{\text{cm}}^{ - 1}\), \(\gamma_{1} = 7.62\;{\text{cm}}^{ - 1}\), and \(S_{1} = 6.6\); \(\omega_{2} = 640\;{\text{cm}}^{ - 1}\), \(\gamma_{2} = 102.4\;{\text{cm}}^{ - 1}\), and \(S_{2} = 0.045\). Can you develop a program to calculate the hemispherical emissivity and plot it against the normal emissivity for a comparison?

  32. 8.32

    Find the Brewster angles for light incident from air to a NIM with (a) \(\varepsilon_{2} = - 2\) and \(\mu_{2} = - 2\), (b) \(\varepsilon_{2} = - 1\) and \(\mu_{2} = - 4\), and (c) \(\varepsilon_{2} = - 8\) and \(\mu_{2} = - 0.5\).

  33. 8.33

    Use the online resources posted on the author’s webpage [54] to calculate the absorption coefficient and normal reflectivity of intrinsic doped silicon for \(0.5\;\upmu{\text{m}} < \lambda < 25\;\upmu{\text{m}}\).

  34. 8.34

    First reproduce Fig. 8.21 for the dielectric function at 400 K and then calculate the dielectric function at 300 K for the same doping concentrations. Furthermore, calculate the real and imaginary parts of the refractive index of n-type doped silicon with a dopant concentration of \(10^{19} \;{\text{cm}}^{ - 1}\) and plot them versus angular frequency.

  35. 8.35

    Suppose a NIM can be described by Eqs. (8.135) and (8.136) with the following parameters: \(\omega_{\text{p} } = 4.0 \times 10^{14}\) rad/s (i.e., \(\lambda_{\text{p}} = 4.71\;\upmu{\text{m}}\)), \(\omega_{0} = 2.0 \times 10^{14}\) rad/s (i.e., \(\lambda_{0} = 9.42\;\upmu{\text{m}}\)), \(\gamma = 0\), and \(F = 0.785\). Assume a wave is propagating in such a medium in the region of \(n < 0\) with a wavevector \({\mathbf{k}} = k_{x} {\hat{\mathbf{x}}}\), where \(k_{x} = k = \left| n \right|\omega /c_{0}\). Show that the group velocity is in the negative x-direction. Also show that the Poynting vector is in the same direction as the group velocity.

  36. 8.36

    Suppose a NIM can be described by Eqs. (8.135) and (8.136) with the following parameters: \(\omega_{\text{p}} = 4.0 \times 10^{14}\) rad/s (i.e., \(\lambda_{\text{p}} = 4.71\;\upmu{\text{m}}\)), \(\omega_{0} = 2.0 \times 10^{14}\) rad/s (i.e., \(\lambda_{0} = 9.42\;\upmu{\text{m}}\), and \(F = 0.5\). Calculate and plot the refractive index and the extinction coefficient in the spectral region from 2 to 15 μm, for \(\gamma = 0\), 1012, and 1013 rad/s.

  37. 8.37

    What is a detector? What is a bolometer? What is a radiometer? If you are asked to buy a detector for infrared radiation measurement for the wavelength range between 2 and 16 μm, discuss how you would select a detector and why. [Hint: Do some online search.]

  38. 8.38

    A bolometer uses a thin YBCO film on a sapphire substrate whose area is 2 mm × 2 mm, operating at 90 K. The thickness of the sapphire plate is 25 μm. The thermal conductance between the detector element and a heat sink is \(G = 8.4 \times 10^{ - 5} \;{\text{W/K}}\). The resistance \(R_{0} (90{\text{ K}}) = 200\;\Omega\) and \(\beta = 1.5{\text{ K}}^{ - 1}\). Assume the absorptance \(\alpha = 0.7\). Calculate the time constant for different bias currents, \(I = 0. 1 , { 0} . 2\;{\text{and}}\; 0. 3\;{\text{mA}}\). Calculate and plot the detector responsivity as a function of modulation frequency \(\omega_{f}\) between 0.1 and 10 Hz for each bias current value given above. Neglect the heat capacity of the YBCO film. The density and specific heat of sapphire at the operating temperature are \(\rho = 3970\;{\text{kg/m}}^{3}\) and \(c_{p} = 102\;{\text{J/kg}}\,{\text{K}}\).

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Zhang, Z.M. (2020). Fundamentals of Thermal Radiation. In: Nano/Microscale Heat Transfer. Mechanical Engineering Series. Springer, Cham. https://doi.org/10.1007/978-3-030-45039-7_8

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