Keywords

1 Opening Remarks

As one of the three fundamental modes of heat transfer, radiative heat transfer maintains a pivotal position in the field of energy transport and consistently coexists with thermal conduction. Objects at temperatures above absolute zero inherently release thermal radiation, a principle expounded by the Stefan-Boltzmann law [1]. This phenomenon bears substantial importance within a multitude of disciplines.

However, the equations governing photon transport often exhibit significant nonlinearity and non-locality, imposing substantial computational demands even with the aid of abundant computing resources. In many cases, the diffusion approximation proves valuable in addressing computational challenges arising from various forms of thermal radiation [2]. While this approach reduces computational costs, it does entail some loss of precision. Among the diffusion approximations, the Rosseland approximation [1] is the most frequently employed, treating thermal radiation as photon diffusion and addressing the thermal coupling between radiation and conduction.

Concurrently, various radiation diffusion approximations, including Rosseland, are typically applicable only to situations involving far-field radiation [3]. In addition to far-field thermal radiation, extensive research has been conducted on near-field thermal radiation due to its enhanced near-field coupling effects [4], thus exhibiting different characteristics. These effects enable near-field radiation to overcome the limitations imposed by the Stefan-Boltzmann law in the far-field regime, thereby influencing a multitude of thermal technologies.

On the other hand, as composite materials [5] with deliberately engineered structures, metamaterials [6, 7] have drawn significant attention due to their extraordinary physical properties that surpass those of natural materials [8,9,10,11]. A considerable amount of work has already been done in the fields of optics, electricity, fluid, acoustics, and other related areas. This has also propelled the development of numerous theories, with one of the key focal points being transformation theory [12]. This theory was initially proposed in the field of electromagnetics, connecting the physical space of material parameters with the virtual space of ideal modulation, enabling the flexible manipulation of physical field variations. Transformation theory, as a fundamental methodology, has expanded from electromagnetic and matter waves to water waves and even to diffusion systems encompassing thermal, plasma and electrical conduction as well as fluid dynamics [13], it may even extend to the social sciences  [14, 15]. However, the conventional electromagnetic wave transformation theory is solely tailored for wave systems and cannot be directly applied to radiative heat transfer under the constraints of the Rosseland diffusion approximation. This necessitates researchers to innovate a novel transformation theory formulations to overcome this limitation.

In this chapter, we will commence by providing an overview of the integration of radiative heat transfer with thermal conduction under the framework of the Rosseland diffusion approximation, a method known as transformation multithermotics, making it amenable to transformation theory. Subsequently, we will briefly outline how this theory is harnessed for the design of thermal control devices related to radiative heat transfer, with a primary focus on thermal camouflage.Additionally, the design of metamaterials for thermal radiation beyond the scope of transformation theory is equally captivating. Considering the distinct natures of far-field and near-field radiative behaviors, we will separately elucidate recent research applications in radiative cooling and thermophotovoltaic devices within these two domains. Finally, we will conclude and offer insights into promising aspects of combining radiative heat transfer and thermal conduction.

2 Theory of Transformation Thermal Radiation under Rosseland Diffusion Approximation

2.1 Derivation of Rosseland Diffusion Approximation

To overcome the challenges of wave transformation theory’s inapplicability to thermal radiation, Xu and colleagues integrated the Rosseland diffusion approximation of thermal radiation with the heat conduction equation, establishing a novel transformation theory known as transformation multithermotics [16].

The Rosseland approximation assumes that the mean free path of photons is significantly smaller than the material thickness, allowing the transport of radiative heat to be approximated as a diffusion process. Therefore, this approximation is applicable to optically thick media. In the following, we will outline the derivation of the Rosseland approximation [1].

First write the differential equation for the local light intensity \(I_\nu \):

$$\begin{aligned} \cos \theta \frac{dI_\nu }{dz} = \beta _\nu (S_\nu - I_\nu ), \end{aligned}$$
(12.1)

where \(S_\nu \) is the source intensity, \(\beta _\nu = \left( \rho \alpha _\nu \right) ^{-1}\) is a coefficient measuring attenuation, \(\rho \) is mass density and \(\alpha _\nu \) is called Rosseland opacity. We assume that both \(S_\nu \) and \(I_\nu \) follow the Planck radiation formula:

$$\begin{aligned} I^{(0)}_{\nu } \approx S_\nu \approx B_\nu (T), \end{aligned}$$
(12.2)

where \(I^{(0)}_{\nu }\) is the zero-order solution of local intensity and it’s easy to check that \(B_\nu (\nu ,T)\) satisfies

$$\begin{aligned} \int _{0}^{\infty }B_\nu (\nu ,T)d\nu = \boldsymbol{n}^2\frac{\sigma }{\pi }T^4. \end{aligned}$$
(12.3)

Here \(\sigma \) is the Stefan-Boltzmann constant and \(\boldsymbol{n}\) is the relative refraction index. Let \(I_\nu \approx I^{(0)}_\nu + I^{(1)}_\nu \) and by combining Eqs. 12.1 and 12.2, we can obtain

$$\begin{aligned} \begin{aligned} I^{(1)}_\nu &= B_\nu (T)-\cos \theta \beta ^{-1}_\nu \frac{\partial B_\nu (T)}{\partial z} \\ &= B_\nu (T)-\cos \theta \beta ^{-1}_\nu \frac{\partial B_\nu (T)}{\partial T}\frac{\partial T}{\partial z}. \end{aligned} \end{aligned}$$
(12.4)

Then we can calculate the radiative heat flux (density) \(\boldsymbol{J}_{rad}\):

$$\begin{aligned} \begin{aligned} \boldsymbol{J}_{rad} &= \int _{0}^{\infty }d\nu \int _{\Omega _0}d\Omega I_\nu (T)\cos \theta \\ &= \int _{0}^{\infty }d\nu \int _{\Omega _0}d\Omega \cos \theta [2B_\nu (T)-\cos \theta \beta ^{-1}_\nu \frac{\partial B_\nu (T)}{\partial T}\frac{\partial T}{\partial z}] d\Omega \\ &= \int _{0}^{\infty }d\nu \int _{\Omega _0}d\Omega \cos ^2\theta [-\beta ^{-1}_\nu \frac{\partial B_\nu (T)}{\partial T}\frac{\partial T}{\partial z}]\sin \theta d\theta d\psi \\ &= -2\pi \int _{0}^{\infty }d\nu \beta ^{-1}_\nu \frac{\partial B_\nu (T)}{\partial T}\frac{\partial T}{\partial z} \int _{-1}^{1}\mu ^2d\mu \\ &= -\frac{4\pi }{3}\frac{\partial T}{\partial z} \int _{0}^{\infty }\frac{\partial B_\nu (T)}{\partial T}\frac{1}{\beta _\nu }d\nu , \end{aligned} \end{aligned}$$
(12.5)

where \(\Omega _0\) is the unit hemispherical surface. The \(B_\nu (T)\) is isotropic and make a variable replacement \(\mu = \cos \theta \). We use \(\beta \) to represent the average absorption coefficient:

$$\begin{aligned} \beta ^{-1}\int _{0}^{\infty }\frac{\partial B_\nu (T)}{\partial T}d\nu = \int _{0}^{\infty }\frac{\partial B_\nu (T)}{\partial T}\frac{1}{\beta _\nu }d\nu , \end{aligned}$$
(12.6)

then

$$\begin{aligned} \beta ^{-1} = \frac{\pi \int _{0}^{\infty }\frac{\partial B_\nu (T)}{\partial T}\frac{1}{\beta _\nu }d\nu }{4\boldsymbol{n}^2\sigma T^3}. \end{aligned}$$
(12.7)

Finally, the radiative flux \(\boldsymbol{J}_{rad}\) according to the Rosseland diffusion approximation is

$$\begin{aligned} \begin{aligned} \boldsymbol{J}_{rad} &= -\frac{16}{3}\beta ^{-1}\boldsymbol{n}^2\sigma T^3\cdot \nabla T \\ &= \boldsymbol{\tau }T^3\cdot \nabla T, \end{aligned} \end{aligned}$$
(12.8)

where denote \(\boldsymbol{\tau }= -16/3\beta ^{-1}\boldsymbol{n}^2\sigma \). Hence, the thermal radiation equation can be rephrased as a diffusion equation, wherein the thermal conductivity varies according to \(T^3\). Consequently, we have the opportunity to utilize the principles of thermal conduction modulation for the purpose of regulating heat radiation. The subsequent discourse on the transformation theory provides an exemplary illustration of this concept.

2.2 Transformation Theory of Thermal Radiation

Next, we commence by showcasing the preservation of the fundamental equation in multithermotics as it transitions from a curvilinear space X to a physical space \(X^\prime \). This process will lead us to unveil the principles governing the transformation of multithermotics.

Assuming heat transfer is a passive process, the constitutive equation followed by multithermotics is as follows

$$\begin{aligned} \rho C\frac{\partial T}{\partial t}+\nabla \cdot (\boldsymbol{J}_{rad}+\boldsymbol{J}_{con}) = 0, \end{aligned}$$
(12.9)

where \(\rho \) and C are the density and heat capacity of participating media, respectively. The conductive flux is determined by the Fourier law \(\boldsymbol{J}_{con} = -\kappa \cdot \nabla T\), where \(\kappa \) is thermal conductivity.

Express the dominant equation in its component form. In curvilinear space with a contravariant basis \([\boldsymbol{g}^u, \boldsymbol{g}^v, \boldsymbol{g}^w]\), a covariant basis \([\boldsymbol{g}_u, \boldsymbol{g}_v, \boldsymbol{g}_w]\), and corresponding contravariant components \([\boldsymbol{x}^u, \boldsymbol{x}^v, \boldsymbol{x}^w]\), the radiative term can be rewritten as

$$\begin{aligned} \begin{aligned} \nabla \cdot [\boldsymbol{\tau }T^3\cdot \nabla T)] &= \boldsymbol{g}^w \cdot \frac{\partial }{\partial x^w}[\tau ^{uv}T^3\boldsymbol{g}_u\otimes \boldsymbol{g}_v\cdot \boldsymbol{g}^l \frac{\partial T}{\partial x^l}] \\ &= \boldsymbol{g}^w \cdot \frac{\partial }{\partial x^w}[\tau ^{uv}T^3\boldsymbol{g}_u\boldsymbol{g}_u\frac{\partial T}{\partial x^v}] \\ &= \frac{\partial \tau ^{v T^3}}{\partial x^u} \frac{\partial T}{\partial x^v}+\frac{\partial ^2 T}{\partial x^u \partial x^v}T^3+\boldsymbol{g}^w\cdot \frac{\partial \boldsymbol{g}_u}{\partial x^w}[\tau ^{uv}T^3\frac{\partial T}{\partial x^v}] \\ &= \partial _u[\tau ^{uv}T^3\partial _v T]+\Gamma ^w_{wu}\tau ^{uv}T^3\partial _v T \\ &= \partial _u[\tau ^{uv}T^3\partial _v T]+\frac{1}{\sqrt{g}}(\partial _u\sqrt{g})\tau ^{uv}T^3\partial _v T \\ &= \frac{1}{\sqrt{g}}\partial _u[\sqrt{g}\tau ^{uv}T^3\partial _v T] , \end{aligned} \end{aligned}$$
(12.10)

where \(g = \mathrm {det^{-2} A}\) is the determinant of the matrix with components \(g_{ij} = \boldsymbol{g}_u \cdot \boldsymbol{g}_v\), A is the Jacobian transformation matrix and \(\Gamma ^w_{uv}=\partial \boldsymbol{g}_u/\partial x^v \cdot \boldsymbol{g}^w\) is the Christoffel symbol resulting in \(\Gamma ^w_{wu} = (\partial _u\sqrt{g})/\sqrt{g}\). Similarly, we can rewrite the conductive term as

$$\begin{aligned} \nabla \cdot [\boldsymbol{\kappa }\cdot \nabla T] = \frac{1}{\sqrt{g}}\partial _u[\sqrt{g}\kappa ^{uv}\partial _v T] \end{aligned}$$
(12.11)

Combining the two aforementioned expressions, the dominant equation can be reformulated as follows

$$\begin{aligned} \sqrt{g}\rho C \partial _t T - \partial _u [\sqrt{g}(\tau ^{uv}T^3+\kappa ^{uv})\partial _v T] = 0 \end{aligned}$$
(12.12)

which is expressed in the curvilinear space. Then, rephrase it within the physical space to obtain the transformation rules for material properties,

$$\begin{aligned} \sqrt{g}\rho C \partial _t T - \partial _{u^\prime }\frac{\partial x^{u^\prime }}{\partial x^u} [\sqrt{g}(\tau ^{uv}T^3+\kappa ^{uv})\frac{\partial x^{v^\prime }}{\partial x^v}\partial _v T] = 0, \end{aligned}$$
(12.13)

where \(\partial x^{u^\prime /\partial x^u}\) and \(\partial x^{v^\prime /\partial x^v}\) are the components of the Jacobian transformation matrix A.

Finally, we substitute space transformation with material transformation,

$$\begin{aligned} \frac{\rho C}{\textrm{det A}}\partial _t T - \partial _{u^\prime }[(\frac{A^{u^\prime }_u \tau ^{uv}A^{v^\prime }_v}{\textrm{det A}}T^3 + \frac{A^{u^\prime }_u \kappa ^{uv}A^{\nu ^\prime }_v}{\textrm{det A}})\partial _{v^\prime }T] = 0. \end{aligned}$$
(12.14)

We can notice that this equation maintains the initial structure of the diffusion equation, provided that the parameters adhere to the transformation rules outlined below:

$$\begin{aligned} \begin{aligned} (\rho C)^\prime & =\frac{ \rho _0 C_0}{\textrm{det A}}, \\ \boldsymbol{\tau }^\prime & =\frac{\textrm{A} \boldsymbol{\tau }_0 \textrm{A}^{\textrm{T}}}{\textrm{det A}}, \\ \boldsymbol{\kappa }^\prime & =\frac{\textrm{A} \boldsymbol{\kappa }_0 \textrm{A}^{\textrm{T}}}{\textrm{det A}}, \end{aligned} \end{aligned}$$
(12.15)

where \(\rho _0\), \(C_0\), \(\tau _0\), and \(\kappa _0\) are the density, heat capacity, radiative coefficient, and thermal conductivity of the background respectively.

Now we have the capacity to engineer thermal metamaterials [17, 18] for use in radiative and conductive environments [19]. Figure 12.1 depicts the conceptual designs of the cloak, concentrator, and rotator, respectively.

Fig. 12.1
4 illustrations. a, 3 concentric circles with radii r 1, r m, and r 2 titled background. J rad + J con equals J tot. b, an aircraft enclosed in a cloak stopping I Rs. c, a solar energy concentrator with incident solar energy. d, incident heat waves on a rotator for direction control.

(from Ref. [16])

Figure a illustrates different thermal flux representations through the use of unique arrow styles: the wavy, dashed, and solid arrows signify radiative flux, conductive flux, and total flux, respectively, in the domain of multithermotics. In Fig. b, the notion of cloaking is demonstrated, providing protection for internal objects, such as aircraft, against infrared detection. Figure c showcases a concentrator, highlighting its capacity to greatly improve energy utilization efficiency. Finally, Fig. d presents rotators, which enable controlled directional alterations in local heat flux.

Full size image

Since the concept of transformation multithermotics is not limited by spatial dimensions, we opt for a two-dimensional system when creating these components.

Use the coordinate transformations for cloaks [20, 21]

$$\begin{aligned} {\left\{ \begin{array}{ll} r^\prime = (r_2-r_1)r/r_2+r_1 \\ \theta ^\prime = \theta \end{array}\right. } \end{aligned}$$
(12.16)

concentrators

$$\begin{aligned} {\left\{ \begin{array}{ll} r^\prime = r_1r/r_m, r<r_m \\ r^\prime = [(r_2-r_1)r+(r_1-r_m)r_2]/(r_2-r_m), r_m<r<r_2 \\ \theta ^\prime = \theta \end{array}\right. } \end{aligned}$$
(12.17)

and rotators

$$\begin{aligned} {\left\{ \begin{array}{ll} r^\prime = r \\ \theta ^\prime = \theta +\theta _0, r<r_1 \\ \theta ^\prime = \theta +\theta _0(r-r_2)/(r_1-r_2), r_1<r<r_2 \end{array}\right. } \end{aligned}$$
(12.18)

The Jacobian transformation matrix can be calculated by

$$\begin{aligned} \textrm{A} = \begin{pmatrix} \partial r^\prime /\partial r &{} \partial r^\prime /(r\partial \theta ) \\ r^\prime \partial \theta ^\prime /\partial r &{} r^\prime \partial /(r\partial \theta ) \end{pmatrix}. \end{aligned}$$
(12.19)

In a broader context, thermal radiation, thermal conduction, and thermal convection may coexist simultaneously. By introducing terms representing convective flux into the equations, a method has been developed to manipulate all three fundamental heat transfer modes through the transformation of total heat [22]. It’s worth noting that the parameters acquired through transformation multithermotics exhibit anisotropic characteristics, a feature rarely observed in natural materials. These material properties can be realized through the design of layered structures [23] and effective medium theories [24]. Additionally, achieving a similar effect is possible through the theories of scattering cancellation [24] and conformal mapping [25].

2.3 Thermal Camouflage with Transformation Theory

Thermal camouflage, often referred to as thermal illusion [26] serves the purpose of concealing the genuine thermal signal while projecting an altered signal to external detectors, resulting in deceptive effects. For instance, by modifying the thermal properties of an object, it can be camouflaged as an entirely different object during detection [27]. In another study, regionalization transformation techniques were employed to craft thermal illusions, successfully achieving encrypted thermal printing that reveals concealed information exclusively when exposed to the correct heat source [28]. This concept was further extended into the realm of three dimensions to overcome limitations inherent in two-dimensional approaches [29].

The control over thermal radiation patterns can also be attained through the regulation of thermal conduction, enabling the concurrent management of both thermal radiation and conduction, as indicated by Dede et al. [30]. Li and his team introduced an inventive method to achieve conductive transformation thermotics specifically for radiative applications [31]. Their investigation delved into existing camouflage techniques, emphasizing the direct options of adjusting surface temperatures or emissivities for customizing typical infrared imaging. However, engineering emissivity methods rely on the background, which could limit their practicality. In this particular case, the camouflage surface was devised through a dual transformation approach, revealing a discreet concealed area on the device’s surface, thus allowing objects to remain hidden from thermal radiation detection. By structuring layers of copper and polydimethylsiloxane in accordance with transformation theory, they successfully experimentally verified the anisotropic thermal conductivity. This advancement facilitated the integration of conductive transformations within radiative fields, expanding the scope of transformation applications. Notably, the concept of thermal camouflage surfaces was subsequently applied to three-dimensional scenarios [32]. In another strand of research, the application of regionalization transformation theory was explored for constructing thermal metamaterial structures that function as distinct infrared signatures [28]. By manipulating the thermal conductivity within distinct rectangular areas, one could harness heat from specific locations to create the thermal representation of a basic stroke. These individual strokes were then combined to construct the complete alphabet. This method allowed for the thermal encoding of information, exclusively visible in infrared imagery.

Expanding beyond individual physical domains, approaches for multifaceted physical camouflage have been investigated. An innovative multispectral camouflage apparatus, designed to cover the infrared, visible light, laser, and microwave spectra, has been put forth [33]. It is foreseeable that the convergence of diverse disciplines and the thinning of device profiles will likely emerge as predominant trends in the field of thermal camouflage.

3 Metamaterial Design of Far-Field and Near-Field Thermal Radiation Beyond Transformation Theory

The Rosseland diffusion approximation represents one variation of radiation diffusion approximations, primarily applicable to optically thick media. Frank Graziani presented an exposition on the fundamental principles of radiation transport, elucidating the genesis of the diffusion approximation and its accompanying transport corrections. He provided an overview of three radiation diffusion approximations and various numerical simulation techniques [2]. These diffusion approximation techniques are often linked with far-field phenomena. In addition to the aforementioned concepts, progress in manipulating far-field thermal radiation has resulted in significant applications like daytime radiative cooling and thermophotovoltaic systems. On the other hand, as systems shrink to micro and nanoscale dimensions, the influence of evanescent waves becomes more pronounced. Near-field radiation can transcend the constraints of the Stefan-Boltzmann law in the far-field, thus significantly impacting various thermal technologies. In the forthcoming sections, we will present a concise overview of notable designs pertaining to radiative cooling and thermophotovoltaic devices, encompassing both far-field and near-field thermal radiation domains.

3.1 Far-Field Thermal Radiation

3.1.1 Radiative Cooling

Daytime radiative cooling, a passive process that demands no supplementary energy input, represents a promising frontier in renewable energy research. In the prevailing environmental temperatures of the universe, blackbody radiation extensively aligns with the Earth’s atmospheric transparency window, wavelength range is 8–13 \(\mathrm {{\mu }m}\). This characteristic permits a substantial portion of objects on Earth to emit thermal energy into space, thereby achieving radiative cooling.

The utilization of titanium dioxide to create white paint for enhancing radiative cooling was first introduced as far back as 1978 [34]. However, this concept didn’t receive significant attention at the time. It wasn’t until 2014 that a photonic approach for developing radiative coolers was proposed, and it was subsequently experimentally realized [35]. This approach revolves around the design of a device that selectively reflects solar radiation while emitting thermal radiation in the mid-infrared spectrum. This unique property was achieved through a multilayer structure consisting of \(\mathrm {HfO_2}\) and \(\mathrm {SiO_2}\). Since then, substantial advancements have been made toward the widespread adoption of this technology [36]. Radiative cooling leverages the chill of the cosmos, carrying profound implications for the realm of energy technology and establishing a pioneering frontier in renewable energy research [37].

3.1.2 Thermophotovoltaic Systems

In the realm of solar energy harvesting, single-junction solar cells confront an efficiency limitation recognized as the Shockley-Queisser limit, which is capped at 41%  [38]. The root cause of this reduced conversion efficiency lies in the broad spectrum of solar radiation, primarily spanning from 200 nm to 2500 nm. Photons with energy levels lower than the solar cell’s bandgap cannot be absorbed, while photons with energy levels surpassing the bandgap lead to the descent of high-energy electrons to the band’s edge, resulting in additional energy loss in the form of heat. Solar thermophotovoltaic (STPV) systems represent a pivotal approach for transcending the Shockley-Queisser limit. The concept of utilizing STPV technology to capture thermal radiation energy was initially introduced in 1979 [39]. STPV operates as a solar-driven heat engine, extracting electrical energy from thermal radiation. The overarching objective is to absorb and convert the wide-ranging solar radiation spectrum into a narrower thermal emission spectrum, specifically tailored to match the spectral response of the photovoltaic (PV) cell [40].

Within the domain of STPV systems, an intermediate element absorbs incoming sunlight, undergoes heating, and subsequently emits thermal radiation, aligned with the bandgap of the solar cell, thus surpassing the theoretical limit. Although the theoretical efficiency ceiling can theoretically extend to as high as 85.4%, practical experimental results have encountered impediments owing to a range of optical and thermal losses. There is optimism for enhancing STPV efficiency through various approaches, including material selection, structural design, and elevating the operating temperatures of emitters and absorbers [42]. In STPV, the power source for PV cells encompasses not only thermal radiation but also various other forms of solar emissions. Furthermore, a notable advantage of STPV lies in its ability to function continuously, incorporating alternative heat sources or thermal storage [42].

Thermophotovoltaic (TPV) systems are closely related to STPV systems, harnessing thermal radiation from localized heat sources to generate electricity in photovoltaic cells [40]. In TPV systems, a key approach is to harness photovoltaic cells with minimal bandgap absorption and high-quality back reflectors for light recovery below the bandgap [40].

3.2 Near-Field Thermal Radiation

3.2.1 Radiative Cooling

Besides macroscopic radiative cooling, microscopic near-field scenarios, as discussed in [43], also enable the control of heat through radiative cooling.

Guha et al. showcased effective cooling through near-field radiative heat transfer in an innovative setup, as reported in [44]. This setup featured the direct integration of temperature sensors into thermally isolated hotspots. Bringing an oxidized probe in proximity to a \(\mathrm {SiO_2}\) membrane, measuring 100 \(\times \) 50 \(\mathrm {\mu m^2}\) and acting as a localized hotspot, resulted in a substantial near-field radiative heat flux that led to cooling by several tens of degrees. This cooling mechanism was accomplished without any physical contact, circumventing conductive heat transfer and introducing an innovative cooling capability. The observed trends in radiative cooling corresponded with theoretical predictions, primarily influenced by the geometry of the utilized probe and the minimum achievable separation within the experimental setup.

In another study by Zhou et al., the technique of nanoparticle doping was introduced into the near-field radiative negative luminescence cooling domain. By capitalizing on its light-modulating capability, cooling between the host material and the semiconductor was effectively realized [45].

Furthermore,Liu and colleagues introduced an intelligent radiative thermostatic system, driven by near-field radiative thermal diodes. This system integrates passive radiative cooling at the top layer and internal heat modulation. Remarkably, it has the capability to autonomously monitor its own temperature and regulate it to remain close to the desired set point, as discussed in [46].

3.2.2 Thermophotovoltaic Systems

Thermal objects emit electromagnetic radiation, and their far-field spectral features are delineated by Planck’s thermal radiation theory. However, this emission is subject to limitations imposed by the blackbody effect, which arises because the gap between the emitter and the PV cell is significantly larger than the characteristic wavelength. When this radiation reaches a PV cell, it stimulates the generation of electron-hole pairs, which are separated by the built-in electric field of the PV cell, thus yielding an electric current, as described in [47].

The analysis of far-field thermophotovoltaic systems relies on classical thermal radiation theory for modeling the thermal properties of the emitter. The PV cell, on the other hand, is typically modeled using traditional PV cell device models. In contrast, near-field thermophotovoltaic systems integrate fluctuational electrodynamics (FED) with conventional PV cell device models. This approach involves coupling Maxwell’s equations with stochastic current sources, in accordance with the fluctuation-dissipation theorem [48]. The radiative energy transfer is then computed by assessing the time-averaged Poynting flux. As depicted in Fig. 2 in [47], multiple electromagnetic modes are thermally excited within the thermal body, extending to the emitter-vacuum interface. Only modes with incident angles below the critical angle can propagate through the vacuum and reach the PV cell. These modes, sustained in the vacuum, are termed propagating waves and contribute to far-field radiative energy transfer. Modes with incident angles exceeding the critical angle experience total internal reflection (frustrated modes) and cannot traverse the vacuum gap. This phenomenon is responsible for the limiting constraint observed in blackbody radiative processes.

To significantly improve the power conversion efficiency of thermophotovoltaic devices, Whale and colleagues introduced the innovative concept of utilizing evanescent waves to enhance radiative power, as discussed in [49]. When the separation between objects is comparable to or much smaller than the dominant wavelength, employing surface plasmon resonances or surface phonon resonances can substantially boost the efficiency of near-field tunneling and expand the spatial spectrum range of evanescent wave thermal photons. This approach enables radiative energy transmission power that exceeds the Planck limit by a considerable margin. In the near-field regime, power densities up to ten times higher than those of standard devices can be achieved. In 2011, Francoeur and colleagues further investigated the influence of PV cell heating on the performance of near-field photovoltaic systems, as documented in [50]. Given the broadband nature of near-field enhancement, heat transfer to the PV cell is also intensified, potentially leading to challenges in thermal management [47]. It is foreseeable that subsequent research endeavors have been dedicated to enhancing thermoelectric conversion possibilities in near-field thermophotovoltaic systems, with a focus on factors like emitter materials and spectral matching.

4 Conclusion and Outlook

Thermal radiation, a fundamental mode of heat transfer, coexists with thermal conduction in objects. The Stefan-Boltzmann law dictates that any object at a non-zero temperature emits thermal radiation, a phenomenon of paramount importance in various fields. This chapter commences by introducing the concept of transformation multithermotics theory, demonstrating how the Rosseland approximation is integrated within this framework to simultaneously account for both thermal conduction and radiation. Subsequently, we provide a concise overview of the design of thermal control devices related to thermal radiation using transformation theory, with a particular focus on the field of thermal camouflage. Additionally, we highlight the potential of utilizing thermal radiation in the designs of metamaterial devices that extend beyond the scope of transformation theory. Given the distinct behavior of radiation in far-field and near-field scenarios, we offer brief overviews of applications in both contexts, encompassing radiative cooling and thermophotovoltaics.

Beyond the applications mentioned above, thermal radiation plays a crucial role in various fields, including infrared thermal sensing [51], thermal diodes [52], thermal rectification [53], biosensors, and chemical sensors [54]. The interplay of thermal radiation and conduction [55] has led to innovative approaches in thermal control. Strategies like spatiotemporal modulation have significantly impacted the fields of thermal conduction and convection [56]. Emerging phenomena, including topological effects [57], thermal hall effects [58], and nonreciprocity [59], are also becoming pivotal in thermal control strategies, and the role of thermal radiation in these domains is poised for future exploration.

While thermal conduction follows diffusion equations, which are inherently distinct from electromagnetic wave radiation governed by wave equations, the study of non-Fourier heat conduction [60] highlights that thermal conduction also exhibits wave-like properties, particularly in micro- and nanoscale systems. On the other hand, considering that thermal radiation, under the diffusion approximation, is equivalent to diffusion heat transfer with nonlinear thermal conductivity, this equivalence provides a valuable tool for investigating nonlinear thermal phenomena [61]. Mathematically, the analogy between the electric field intensity in nonlinear optics and temperature gradient in nonlinear thermal physics is evident. However, in nonlinear thermal physics, thermal conductivity depends on temperature rather than temperature gradient. This analogy suggests the potential for uncovering surprising new physics.