Photonic refrigeration from time-modulated thermal emission

TL;DR

This paper introduces a novel active photonic cooling mechanism based on time-modulated thermal emission, which can approach the Carnot limit without relying on luminescence, opening new possibilities for cooling and energy harvesting.

Contribution

It proposes a new photonic cooling method utilizing temporal modulation of thermal emission, bypassing the need for luminescence efficiency constraints.

Findings

High thermodynamic performance approaching Carnot limit

Does not rely on luminescence efficiency

Enables new active control of thermal emission for cooling

Abstract

Active photonic cooling is of significant importance to realize robust, compact, vibration-free, all-solid-state refrigeration. Currently proposed photonic cooling approaches are based on luminescence and impose stringent requirements on luminescence efficiency. We propose a new photonic cooling mechanism arising from temporal modulation of thermal emission. We show that this mechanism has a high thermodynamic performance that can approach the Carnot limit and yet does not rely on luminescence. Further, our work opens exciting new avenues in active, time-modulated control of thermal emission for cooling and energy harvesting applications.

Equations

$$-i\frac{d}{dt}\text{a}$$

$$M(t)=\left(\begin{array}{cc}0& \sqrt{\frac{{\omega }_1}{{\omega }_2}}V{e}^{-i\Omega t}\\ \sqrt{\frac{{\omega }_2}{{\omega }_1}}{V}^{\ast }{e}^{i\Omega t}& 0\end{array}\right),$$

$$P_{out}=4{\gamma }^2|V{|}^2\frac{{\omega }_2}{{\omega }_1}\int d\omega \frac{1}{|Z_2(\omega ){|}^2}\Theta (\omega -\Omega ,T_c),$$

$$P_{in}=4{\gamma }^2|V{|}^2\int d\omega \frac{1}{|Z_1(\omega ){|}^2}\Theta (\omega +\Omega ,T_h),$$

$$\dot{W}=4{\gamma }^2|V{|}^2\int d\omega \left[\frac{\Omega }{{\omega }_1}\frac{\Theta (\omega ,T_c)}{|Z_1(\omega ){|}^2}-\frac{\Omega }{{\omega }_2}\frac{\Theta (\omega ,T_h)}{|Z_2(\omega ){|}^2}\right]$$

$$\frac{{\omega }_2}{{\omega }_1}\ge \frac{T_h}{T_c}.$$

$$COP=\frac{{\omega }_1}{{\omega }_2-{\omega }_1}\le \frac{T_c}{T_h-T_c},$$

$$⟨J_{\alpha }(\text{r},\omega )J_{\beta }^{\ast }({\text{r}}^{\prime },$$

$$\frac{{ϵ}_{\alpha \beta }(\text{r},\omega )-{ϵ}_{\beta \alpha }^{\ast }(\text{r},\omega )}{2i}\delta (\text{r}-{\text{r}}^{\prime })\delta (\omega -{\omega }^{\prime })$$

$$\displaystyle\Delta P=\int_{0}^{\infty}d\omega\int_{-\infty}^{\infty}\frac{dk_{x}}{2\pi}\int_{-\infty}^{\infty}\frac{dk_{y}}{2\pi}\Big{[}\Phi_{f}(\omega,k_{x},k_{y})\times$$

$$\displaystyle\Theta(\omega,T_{c})-\Phi_{b}(\omega,-k_{x},-k_{y})\Theta(\omega,T_{h})\Big{]},$$

$$\displaystyle\dot{W}=\int_{0}^{\infty}d\omega\frac{2}{\pi}\omega\Theta(\omega,T)\int d\textbf{r}\int d\textbf{r}^{\prime}\operatorname{Im}\operatorname{Tr}\big{[}\mathcal{W}\hat{\epsilon}(\textbf{r})\times$$

$$\displaystyle\mathcal{G}(\textbf{r},\textbf{r}^{\prime})\delta_{\omega,\omega^{\prime}}\operatorname{Im}\left[\epsilon(\textbf{r}^{\prime})\right]\mathcal{G}^{\dagger}(\textbf{r},\textbf{r}^{\prime})\big{]},$$

$${\Phi }_{f,b}(\omega )={\int }_{-\infty }^{\infty }\frac{d{k}_x}{2\pi }{\int }_{-\infty }^{\infty }\frac{d{k}_y}{2\pi }{\Phi }_{f,b}(\omega ,k_x,k_y),$$

Full text

Active photonic cooling is of significant importance with the potential to realize robust, compact, vibration-free, all-solid-state refrigeration. To date, the proposed photonic cooling approaches, including laser cooling Epstein et al. (1995); Seletskiy et al. (2010); Zhang et al. (2013) and electroluminescent cooling Tauc (1957); Chen et al. (2015); Xiao et al. (2018); Zhu et al. (2019), are all based on luminescence and impose stringent requirements on luminescence efficiency. Recent advances in nanophotonic control of thermal emission Lenert et al. (2014); Raman et al. (2014); Ilic et al. (2016); Greffet et al. (2002); Liu et al. (2011); Khandekar et al. (2015); Li and Fan (2018) have offered a promising approach to manipulate photon emission without the luminescence process. In this Letter, we propose a photonic refrigeration technique based on time-modulation of thermal emission. This technique results in a mechanism of refrigeration with a significantly higher performance than laser cooling of solids, while also overcoming the stringent quantum efficiency requirements of electroluminescent cooling. Our work opens exciting new avenues in active, time-modulated control of thermal emission for active cooling and energy harvesting applications.

Laser cooling and electroluminescent cooling are the two main existing approaches to active photonic refrigeration. The mechanism of laser cooling is based on an anti-Stokes luminescence up-conversion process Pringsheim (1929), where the energy difference between a luminescence photon and a pump photon is supplied by the lattice phonon, resulting in cooling. Net laser cooling has been demonstrated in rare earth metal doped crystals Epstein et al. (1995); Seletskiy et al. (2010) and more recently in II-VI semiconductors Zhang et al. (2013). However, the performance of laser cooling is inherently limited to be several orders of magnitude below the Carnot bound due to the small energy difference between the luminescence photon and the pump photon Seletskiy et al. (2016). Alternatively, electroluminescent cooling has been suggested to realize photonic cooling Tauc (1957); Berdahl (1985); Chen et al. (2015, 2016); Zhu et al. (2019) with the potential to overcome such limitations on performance. In positive electroluminescent cooling, a positive chemical potential for photons enables thermal emission at a radiant temperature that is much higher than the thermodynamic temperature. However, no positive electroluminescent cooling has been demonstrated to date due to the stringent requirements on the luminescence efficiency. More recently, negative electroluminescent cooling was proposed Chen et al. (2016) and demonstrated Zhu et al. (2019), but it suffers from the limitation of low power density.

As another photon emission process, thermal radiation results from a direct conversion of heat into photon emission due to the thermally induced fluctuations of particles or quasi-particles. Particularly, recent advances in using sub-wavelength nanophotonic structures to control fundamental properties of thermal radiation has offered a tremendous number of opportunities Li and Fan (2018); Greffet et al. (2002); Liu et al. (2011); Khandekar et al. (2015) and enabled new applications such as passive radiative cooling Raman et al. (2014). However, so far, almost all of the existing work on thermal radiation control has focused on static systems, which can only perform passive cooling Raman et al. (2014), where heat flows from a high-temperature to a low-temperature object.

In this paper we consider thermal emission from temporally modulated systems. In recent years, temporal modulation of the refractive index has offered exciting opportunities to manipulate photons. Such modulation, via the conversion of photon frequencies, can achieve optical isolation and circulation Yu and Fan (2009); Fang et al. (2012); Sounas and Alù (2017) and the breaking of symmetry between emission and absorption Hadad et al. (2016). While temporally modulated systems such as electro-optic modulators have been widely used in optical information processing and communication systems, the thermodynamic implications of temporal modulation have not been previously explored. In this work, we develop a statistical temporal coupled mode theory to show that temporal modulation of a thermal photonic system achieves refrigeration. Further, by a rigorous fluctuational electrodynamics approach, we verify the predictions of our theory and numerically demonstrate photonic refrigeration by computing the heat transfer from a temporally modulated body maintained at a certain temperature to a passive thermal emitter at a higher temperature.

Consider a cavity with modes 1 and 2 at frequencies $\omega_{1,2}$, respectively, with $\omega_{1}<\omega_{2}$, as shown schematically in Fig. 1(a). The amplitudes in the two modes $a_{1,2}$ are normalized such that $|a_{1,2}|^{2}$ represent the energy in the modes. The modes have internal loss rates $\gamma_{1,2}^{i}$ due to absorption and are in thermal equilibrium with a heat bath at temperature $T_{c}$. The modes also radiatively couple to an external heat bath at temperature $T_{h}(\geq T_{c})$ via coupling rates $\gamma_{1,2}^{e}$. By the fluctuation-dissipation theorem, there are associated compensating noise sources Otey et al. (2010); Zhu et al. (2013) $n_{1,2}^{i}$ for internal loss and $n_{1,2}^{e}$ for external radiative coupling, respectively. The strength of these noise sources are defined by $\langle n_{1,2}^{i*}(\omega)n^{i}_{1,2}(\omega^{\prime})\rangle=2\pi\delta(\omega-\omega^{\prime})\Theta(\omega,T_{c})$ and $\langle n_{1,2}^{e*}(\omega)n^{e}_{1,2}(\omega^{\prime})\rangle=2\pi\delta(\omega-\omega^{\prime})\Theta(\omega,T_{h})$, where $\langle\cdot\rangle$ denotes a thermal ensemble average and $\Theta(\omega,T)=\hbar\omega/(\exp(\hbar\omega/k_{B}T)-1)$ is the Planck distribution at frequency $\omega$ and temperature $T$. Defining $\textbf{a}=(a_{1},a_{2})^{T}$, $\textbf{n}_{i}=(n_{1}^{i},n_{2}^{i})^{T}$ and $\textbf{n}_{e}=(n_{1}^{e},n_{2}^{e})^{T}$, the time evolution of the two modes is described by

[TABLE]

Here, $H_{0}=\operatorname{diag}(\omega_{1},\omega_{2})$ is the Hamiltonian of the unmodulated, closed system. $D_{i}=\operatorname{diag}(\sqrt{2\gamma_{1}^{i}},\sqrt{2\gamma_{2}^{i}})$, $D_{e}=\operatorname{diag}(\sqrt{2\gamma_{1}^{e}},\sqrt{2\gamma_{2}^{e}})$, $\Gamma_{i}=D_{i}D_{i}^{\dagger}/2$ and $\Gamma_{e}=D_{e}D_{e}^{\dagger}/2$. The operator $M(t)$ describes the modulation-induced coupling between the modes. Here, we assume that the cavity is modulated by an index modulation proportional to $\cos(\Omega t)$, where $\Omega=\omega_{2}-\omega_{1}$. Then, in the simplest approximation that includes the rotating-wave approximation, $M(t)$ is given by (see Supplementary Information (SI), Section I)

[TABLE]

where $V$ is related to the strength of the index modulation. We emphasize that $M(t)$ is not Hermitian. In our formalism, the modal amplitudes $a_{1,2}(t)$ are normalized with respect to energy, and such time modulation preserves the total number of photons Yu and Fan (2009); Fang et al. (2012) and not the total energy.

We now show that the system shown in Fig. 1(a), which is described by Eqs. (1)-(2), can achieve photonic refrigeration. For illustration purposes, we first consider the simplest case: in the setup of Fig. 1(a), we assume that mode 1 has no radiative coupling to the high temperature heat bath, i.e., $\gamma_{1}^{e}=0$. Mode 2 is assumed to have a nonzero external radiative coupling rate, but has no internal loss, i.e., $\gamma_{2}^{i}=0$. Therefore, the thermal emission and absorption of the unmodulated system in this ideal limit is zero. For simplicity, we take the remaining rates to be equal, i.e., $\gamma_{1}^{i}=\gamma_{2}^{e}=\gamma$. For this ideal system, the flux of the thermal emission from the cold side is (see SI Sec. IV, based on Secs. II-III)

[TABLE]

while the flux received from the hot side at $T_{h}$ is

[TABLE]

for a work input of

[TABLE]

where $Z_{1,2}(\omega)=(\omega-\omega_{1,2}-i\gamma_{1})(\omega-\omega_{1,2}-i\gamma_{2})-|V|^{2}$. As seen from Eqs. (3)-(5), when the modulation is turned on, i.e., $V\neq 0$, a fraction of the thermally generated photons from mode 1 are up-converted to mode 2 and emitted. These photons carry power $P_{out}$ away from the low-temperature reservoir and therefore their emission constitutes a cooling mechanism. Similarly, a fraction of the photons received by mode 2 are down-converted to mode 1, where they are absorbed. These photons carry power $P_{in}$ into the low-temperature heat bath and thus their absorption constitutes a heating mechanism. In Fig. 1(b), we plot the net cooling given by $P_{out}-P_{in}-\dot{W}$ (blue curve) and the work input $\dot{W}$ (red curve) as a function of the ratio $\omega_{2}/\omega_{1}$ for $V=2\gamma$. We note that net cooling starts to occur when

[TABLE]

As $\omega_{2}/\omega_{1}$ increases beyond the threshold value of $T_{h}/T_{c}$, the cooling power also increases. In Fig. 1(c), we plot the coefficient of performance (COP), defined as the ratio between the cooling power and the work input, as a function of $\omega_{2}/\omega_{1}$. We observe that the COP reaches the Carnot bound of $T_{c}/(T_{h}-T_{c})$ at the threshold condition of Eq. (6), and decreases as $\omega_{2}/\omega_{1}$ increases beyond the threshold.

The threshold condition for $\omega_{2}/\omega_{1}$ in Eq. (6) can also be derived analytically from Eqs.(3)-(5) (SI Section IV). Here, we provide an intuitive argument. For simplicity, we assume the classical limit of $k_{B}T_{c,h}\gg\ \hbar\omega_{1,2}$. In the unmodulated cavity, the number of thermal photons in mode 1 is $k_{B}T_{c}/\hbar\omega_{1}$, while the number of thermal photons in mode 2 is $k_{B}T_{h}/\hbar\omega_{2}$, due to its radiative coupling to the high-temperature heat bath and the lack of internal loss in mode 2. When modulation is turned on, since the rate of up- and down-conversion for an individual photon is equal Yu and Fan (2009); Fang et al. (2012), net cooling will be observed when $k_{B}T_{c}/\hbar\omega_{1}\geq k_{B}T_{h}/\hbar\omega_{2}$, which leads to the threshold condition of Eq. (6). When the condition of Eq. (6) is met, for each photon that is emitted by the modulated system, the system at $T_{c}$ experiences cooling by $\hbar\omega_{1}$. On the other hand, the work input per emitted photon is the energy difference of the two modes, $\hbar\omega_{2}-\hbar\omega_{1}$. Therefore, the COP of such a refrigerator is given by

[TABLE]

where the inequality follows from Eq. (6). This upper bound indicates that modulation-induced refrigeration obeys the Carnot limit on performance. Interestingly, the value of COP for this ideal refrigerator is independent of the modulation strength $V$. A rigorous derivation of the Carnot bound on the COP is included in the SI (Section V).

Motivated by the results of our coupled-mode theory, we proceed to consider a physical structure whose radiative thermal properties can be tailored by temporal modulation of its dielectric function, shown in Fig. 2(a). The structure consists of a one-dimensional photonic crystal constructed using two materials with dielectric constants $\epsilon_{1}=14$ (blue layers) and $\epsilon_{2}=4$ (yellow layers). Such a 1D photonic crystal with a large index contrast possesses a bandgap for waves that can propagate in vacuum. To introduce two modes in the bandgap of the photonic crystal, we create two defect layers with thicknesses greater than those of the remaining layers, indicated by ‘Defect 1’ and ‘Defect 2’ in Fig. 2(a). The material constituting the Defect 1 (orange layer) is assumed to be a narrowband absorber. Such narrowband absorbers help suppress parasitic heating arising from frequencies away from the modulated modes under consideration. As an example, assume Defect 1 comprises a medium that is a random mixing of silicon carbide and a lossless high-index medium of $\epsilon_{1}=14$ in a 1:9 ratio. Using the Maxwell-Garnett approximation, the dielectric constant of such a medium is then $\epsilon(\omega)=0.1\times\epsilon_{\infty}(\omega_{LO}^{2}-\omega^{2}-i\omega\gamma)/(\omega_{TO}^{2}-\omega^{2}-i\omega\gamma)+0.9\times 14$, where $\epsilon_{\infty}=6.7$, $\omega_{LO}=1.83\times 10^{14}$ rad/s, $\omega_{TO}=1.49\times 10^{14}$ rad/s and $\gamma=8.97\times 10^{11}$ rad/s. Further, a layer in between the defects, marked in green, experiences a temporal modulation given by $\epsilon(t)=\epsilon_{2}+\delta\cos(\Omega t)$, where $\delta$ is the modulation strength and $\Omega$ is the modulation frequency. This structure is maintained at $T_{c}=290$ K and faces a narrowband emitter in the far-field, composed of the same material as Defect 1 and at a temperature of $T_{h}=300$ K.

To perform calculations of thermal emission and absorption, we extend the formalism of radiative heat transfer Polder and Van Hove (1971); Chen et al. (2018) to include time-varying dielectric functions. This formalism combines rigorous coupled wave analysis Whittaker and Culshaw (1999); Inampudi et al. (2019) with the fluctuation-dissipation theorem Henry and Kazarinov (1996) to compute thermal emission for spatio-temporally modulated layered structures. In our system, the modulated layer does not have loss and hence by itself does not generate thermal radiation. For the lossy layer, the fluctuation-dissipation theorem has the form Chen et al. (2018)

[TABLE]

where $\alpha,\beta=\{x,y,z\}$, $\textbf{J}(\textbf{r},\omega)$ is the current source at position r and frequency $\omega$ that produces thermal fluctuations, and $\epsilon(\textbf{r},\omega)$ is the dielectric tensor of the structure. Within this formalism, the net heat transfer between two bodies at temperatures $T_{h}$ and $T_{c}$ separated by a vacuum gap is given by

[TABLE]

where $(k_{x},k_{y})$ are the wavevector components parallel to the layers and $\Phi_{f,b}(\omega,k_{x},k_{y})$ are the Poynting flux spectra in the vacuum gap generated by sources in the cold and hot sides, respectively. For passive reciprocal structures, $\Phi_{f}(\omega,k_{x},k_{y})=\Phi_{b}(\omega,-k_{x},-k_{y})$. In this system, $\Phi_{f}(\omega,k_{x},k_{y})\neq\Phi_{b}(\omega,-k_{x},-k_{y})$ due to the presence of an actively modulated region. In addition to the flux of thermally generated photons, we compute the work done by the modulation directly from Maxwell’s equations, given by (see SI, Section VI)

[TABLE]

where $\mathcal{W}$, $\delta_{\omega,\omega^{\prime}}$ and $\hat{\epsilon}$ are matrices defined by $\mathcal{W}_{nm}=(\omega+m\Omega)\delta_{nm}$, $\delta_{\omega,\omega^{\prime}}=\delta_{n=0,m=0}$ and $\hat{\epsilon}_{nm}(\textbf{r})=\frac{\Omega}{2\pi}\int_{0}^{2\pi/\Omega}\epsilon(\textbf{r},t)e^{-i(n-m)\Omega t}dt$, with $\hat{\epsilon}^{\dagger}=\hat{\epsilon}$ in the modulation layer. $\mathcal{G}(\textbf{r},\textbf{r}^{\prime})$ is the Green’s function for the electric field at r in the modulated layer originating from a source at $\textbf{r}^{\prime}$ in the lossy layers. The operator $\delta_{\omega,\omega^{\prime}}$ ensures that thermal photons are generated only at $\omega$ but not at the sideband frequencies, since the lossy layer at $\textbf{r}^{\prime}$ is unmodulated. We also note that the expression for work in Eq. (5) is a coupled-mode theory version of the general formula given by Eq. (10).

As a first numerical demonstration, we fit our coupled-mode theory to direct numerical calculations of thermal emission into vacuum, for the structure shown in Fig. 2(a) without the hot side. In Fig. 2(b)-(c), we plot in blue the emissivity of the two modes in the unmodulated structure in the $(k_{x},k_{y})=(0,0)$ channel. We extract the parameters $\omega_{1,2}$, $\gamma_{1,2}^{i}$ and $\gamma_{1,2}^{e}$ by fitting the emissivity profiles, shown in red dotted lines (parameter values listed in SI, Section VII). In this structure, the lossy Defect 1 layer is further away from the top surface as compared to the lossless Defect 2 layer. Thus, $\gamma_{1}^{e}$ and $\gamma_{2}^{i}$ are much smaller than the other two rates, and therefore the thermal emission of the unmodulated system is very low. We then introduce a modulation of $\delta\cos(\Omega t)$ in the green layer in Fig. 2(a), where $\delta=0.4$ and $\Omega=\omega_{2}-\omega_{1}=2\pi\cdot 1.64$ THz. In Fig. 2(d)-(e), we plot the emissivities of the two modes from the numerical calculation in blue lines and fit them using our coupled-mode theory in red dotted lines, exhibiting a very good agreement. With modulation, the emissivity near $\omega_{1}$ is suppressed, whereas the emissivity near $\omega_{2}$ is enhanced as compared with the unmodulated system. In fact, for a larger modulation of $\delta=2.0$, shown in Fig. 2(f)-(g), the emission near $\omega_{2}$ becomes super-Planckian: the emissivity, which is defined as the emitted power density normalized against a blackbody at the same temperature, begins to exceed unity. The results here demonstrate that there is significant up-conversion induced by the temporal modulation. In addition, we observe modulation-induced Rabi splitting Shi et al. (2018) of the modes for $\delta=2.0$, resulting in dips in thermal emission near the frequencies where emission was maximum in the unmodulated system. We note that the emissivity under any modulation strength can be numerically computed accurately. On the other hand, coupled-mode theory, which is a first-order perturbation theory, is not a physically accurate model of the underlying dynamics for large modulation strengths and a coupled-mode theory fit should only be used to guide intuition. The parameters of the coupled mode theory fit for Figs. 2(b)-(g) are provided in the SI (Section VII).

To demonstrate cooling for this single channel, in the presence of the narrowband emitter on the hot side, in Fig. 2(f), we plot the net cooling of the cold side (blue curve) and the work input to the modulated region (red curve) as a function of the modulation strength $\delta$. In Fig. 2(g), we plot the corresponding COP. It is seen that the system of Fig. 2(a) does achieve cooling for the single channel under consideration with a large COP, reaching a maximum value of about 11. For reference, the Carnot limit on performance for the temperatures used in our setup is $T_{c}/(T_{h}-T_{c})=29$, although this limit is attained only at net zero cooling power.

Now, we proceed to demonstrate that the system of Fig. 2(a) exhibits refrigeration even after integration over all propagating channels $(k_{x},k_{y})$ in Eq. (9) and all frequencies. Defining

[TABLE]

in Fig. 3(a), we plot the spectral heat flux $\Phi_{f}(\omega)\Theta(\omega,T)$ (blue curve) and $\Phi_{b}(\omega)\Theta(\omega,T)$ (red curve) for the passive, unmodulated structure when the two sides are at the same temperature of $300$ K. It is seen that $\Phi_{f}(\omega)=\Phi_{b}(\omega)$ for all frequencies, as dictated by electromagnetic reciprocity. On the other hand, in the presence of modulation, $\Phi_{f}(\omega)\neq\Phi_{b}(\omega)$ due to the presence of the active region in the structure on the cold side, where power is either consumed or generated. This is seen in Fig. 3(b) for a modulation of $\delta=1.5$ and $\Omega=2\pi\cdot 1.64$ THz, where $\Phi_{f}(\omega)\Theta(\omega,T_{c})$ and $\Phi_{b}(\omega)\Theta(\omega,T_{h})$ differ significantly in their spectral shape. Strikingly different from passive structures, $\Phi_{b}(\omega)$ can be negative at some frequencies in such modulated structures. This is because a current source in the hot emitter at frequency $\omega$ generates photons that cross the vacuum gap and generate sideband photons at $\omega+n\Omega$, which in turn experience partial reflection back into the vacuum gap, resulting in negative values of Poynting flux at the sideband frequencies.

By integrating the spectral heat flux in Fig. 3(b), we obtain $\Delta P=+0.2033\ \mathrm{W/m^{2}}$, indicating that heat flows against the temperature gradient. Further, we obtain a total power input of $64.8\ \mathrm{mW/m^{2}}$ using Eq. (10). This gives us a thermodynamic COP of $(\Delta P/\dot{W})-1=2.1373$. Therefore, the structure shown in Fig. 2(a) indeed achieves photonic refrigeration after integration over all frequencies and wavevectors. In Fig. 3(c), we plot the COP obtained from this system as a function of the modulation strength $\delta$ for a fixed $\Omega=2\pi\cdot 1.64$ THz. We see that the system begins to exhibit cooling for $\delta>0.87$, saturating at a COP of around 2.3 for large modulation strengths. The performance of the full system is below the ideal limit of Fig. 1 and the single-channel case of Fig. 2 since the modes in the photonic crystal have varying frequency separations $\Omega$ and linewidths as the wavevectors (channels) are varied. In Fig. 3(d), we plot the frequencies of modes 1 and 2 as a function of the angle of emission from the system. It is seen that the frequency separation between the modes varies from 1.64 THz to about 1.94 THz as the angle is varied. Due to this mismatch between the modulation frequency and the modal frequency separation, not all channels contribute equally to the cooling. Further improvements to the performance are possible by detailed optimization of the modal shapes involved in cooling, engineering the bands to be parallel over a larger angular range, considering non-planar geometries, as well as reducing the gap distance between the hot and cold sides to the near field regime.

We now comment on a few important aspects of our cooling approach. First, as compared to laser cooling, our cooling scheme enables a much higher COP. For a laser cooling system which up-converts a pump photon at energy $\hbar\omega$ to a luminescence photon at energy $\hbar\omega_{f}$, the COP can be expressed as Seletskiy et al. (2016) $(\hbar\omega_{f}-\hbar\omega)/\hbar\omega\approx\ k_{B}T/\hbar\omega$, which inherently limits its COP to be on the order of 0.025 at room temperature for $\hbar\omega\simeq 1$ eV. By contrast, our approach can achieve a COP that is several orders of magnitude larger than laser cooling, reaching values close to the Carnot limit. In addition, our approach, being based on thermal emission, does not involve luminescence. Therefore, it can potentially overcome the stringent requirement on luminescence quantum efficiency in laser cooling and electroluminescent cooling.

To summarize, we introduce a new mechanism of active photonic refrigeration that is induced by temporal modulation of the refractive index in a thermal emission system. This mechanism has the potential to overcome the inherent performance limitations of laser cooling as well as the stringent luminescence requirements of electroluminescent cooling. In addition, our work also points to exciting new avenues for tailoring thermal emission. For example, the observed modulation-induced Rabi splitting and super-Planckian thermal emission provide a novel pathway for tuning thermal emission spectra for sensing applications. The combination of thermal photonics with temporal modulation opens up new avenues to tailor thermal emission for active cooling and energy harvesting applications.

This work was supported by Lockheed Martin, the U.S. Department of Energy Grant DE-FG-07ER46426, and the U.S. Department of Energy “Photonics at Thermodynamic Limits” Energy Frontier Research Center under Grant DE-SC-0019140. S.B. acknowledges the support of a Stanford Graduate Fellowship. Valuable discussions with Avik Dutt and Professor David A.B. Miller are gratefully acknowledged.