Abstract

We present an approach for achieving large Kerr χ(3)–mediated thermal energy transfer at the nanoscale that exploits a general coupled-mode description of triply resonant, four-wave mixing processes. We analyze the efficiency of thermal upconversion and energy transfer from mid- to near-infrared wavelengths in planar geometries involving two slabs supporting far-apart surface plasmon polaritons and separated by a nonlinear χ(3) medium that is irradiated by externally incident light. We study multiple geometric and material configurations and different classes of intervening mediums—either bulk or nanostructured lattices of nanoparticles embedded in nonlinear materials—designed to resonantly enhance the interaction of the incident light with thermal slab resonances. We find that even when the entire system is in thermodynamic equilibrium (at room temperature) and under typical drive intensities ~ W/μm2, the resulting upconversion rates can approach and even exceed thermal flux rates achieved in typical symmetric and non-equilibrium configurations of vacuum-separated slabs. The proposed nonlinear scheme could potentially be exploited to achieve thermal cooling and refrigeration at the nanoscale, and to actively control heat transfer between materials with dramatically different resonant responses.

© 2017 Optical Society of America

1. Introduction

The field of nonlinear optics has experienced unprecedented growth in the last several decades, leading to advances in a wide range of optical technologies with applications for signal processing [1, 2], detectors [3, 4], spectroscopy [5], among others. Due to the inherently weak nature of bulk optical nonlinearities [6], most nonlinear devices rely on resonant systems e.g. large-etalon mirrors [7], photonic-crystal defects [8] and plasmonic resonators [9], that confine light into small mode volumes and over long timescales [10], thereby reducing power requirements. As these power requirements are scaled down, even relatively small effects stemming from thermal fluctuations can be altered by material nonlinearities [11, 12], but such phenomena have only just begun to be explored [13, 14]. For instance, we recently showed in [13] that at high temperatures, optical nonlinearities can alter the emission spectrum of photonic resonators, leading to asymmetric lineshapes and greater-than-blackbody emission under passive (purely thermodynamic) operating conditions. In subsequent work [15], we showed that an optically driven photonic resonator can exhibit nonlinear thermal effects at a lower “temperature scale” as well as lead to new phenomena, including the appearance of Stokes/anti-Stokes emission lines and thermally activated transitions [11]. In this paper, we extend our earlier work to show that the Kerr χ(3) nonlinear response of a passive medium can be exploited to efficiently upconvert thermal radiation from mid-infrared to near-infrared or visible wavelengths.

Thermal radiation from hot bodies has been studied for over a century now [16–18]. The recent development and application of powerful numerical techniques capable of describing thermal radiation from complex, structured materials has paved the way for the design of emitters with unique and spectrally selective properties [19–21]. A significant body of work has focused on the study of radiation in the near field (short gap sizes ≪ thermal wavelength λT ≈ 10μm near room temperature), where bound surface resonances can cause two objects, held at different temperatures and separated by sub-micron gaps, to exchange heat at rates that are orders of magnitude larger than those predicted by the Stefan–Boltzmann law (applicable only in the far field) [22–25]. This in turn has created opportunities for potential advances in the areas of photovoltaics [26], imaging [27], and more generally, thermal devices operating at the nano scale [28–30]. More recently, there has been interest in achieving active control of near-field heat transfer [31], such as through gain media [32,33] or via chemical potentials [34], with potential applications to nanoscale thermal refrigeration [35–37]. In this paper, we propose a different active mechanism for tailoring radiative heat transfer that exploits parametric optical nonlinearities mediated by externally incident light to extract and upconvert “thermal energy” trapped in the near field of a planar body unto another. As a proof of concept, we analyze heat exchange in planar geometries supporting surface plasmon polaritonic (SPP) resonances at far-apart wavelengths. In particular, we study planar configurations of silicon carbide (SiC) slabs separated from either silver (Ag), potassium (K), or indium tin oxide (ITO) slabs by a nonlinear (bulk or nanostructured) chalcogenide (ChG) film. Although the large SPP mismatch of the slabs leads to negligible heat exchange in the absence of a pump—SiC supports mid-infrared SPPs while the absorber slabs can support either near-infrared or visible SPPs—we show that under external illumination, energy transfer across the gap can be significant. In particular, we consider multiple intervening-gap designs—either bulk nonlinear thin films or nanostructured media consisting of lattices of nanoparticles embedded in a nonlinear medium—and find that even when the entire system is in thermodynamic equilibrium (at room temperature), the energy flux rate of mid-infrared photons which get upconverted and subsequently absorbed at near-infrared or visible wavelengths can be as large as 104 W/m2 for relatively low pump intensities ~ W/μm2, approaching and even exceeding typical flux rates observed in more commonly studied, non-equilibrium (passive) scenarios in which the slabs are held at a large temperature differential. Hence, this scheme allows significant thermal extraction and absorption of thermal radiation at short wavelengths (otherwise inaccessible under purely passive scenarios) and between very different (such as non-resonant) materials. Our calculations suggest that with additional design optimizations, it may be possible to exploit this approach for radiative cooling and refrigeration.

The paper is organized as follows. In Sec. 2, we consider a representative and generic system consisting of three resonators supporting modes at far-apart frequencies ω1, ω2 and ω3 = ω1 + 2ω2. The resonator modes are coupled by a Kerr χ(3) medium which is driven at ω2 by externally incident light, leading to upconversion of thermal radiation from ω1 to ω3. Our analysis is based on a coupled mode theory framework [38] that describes the underlying resonant four-wave mixing process and lays out general geometric and operating conditions required to maximize upconversion. In Sec. 3, we consider concrete, physical examples of this scheme based on planar geometries and explore different possible material choices of emitter and absorber slabs separated by a nonlinear medium. Section 3.1.1 and Sec. 3.1.2 describe situations where the upconversion is mediated by an intermediate medium consisting of a lattice of nanoparticles (composite core/shell nanospheres or graphene nanodisks) embedded in ChG, whereas Sec. 3.2 focuses on upconversion through a bulk ChG film. Finally, in Sec. 4 we provide a summary of our main results and possible directions for future work.

2. Coupled-mode theory

We first illustrate the basic thermal upconversion mechanism by considering a representative system, depicted in the top inset of Fig. 1(a), involving resonators that support modes at ω1, ω2 and ω3 = ω1 + 2ω2, the second of which is coupled to an external channel. The modes are assumed to have widely different frequency and therefore do not couple linearly to one another. They can however interact nonlinearly through a four-wave mixing process [6] mediated by a χ(3) medium via their field Ej (j = 1, 2, 3) overlaps and initiated by externally incident light at ω2 from the channel. Such a system is well described by the following temporal coupled-mode equations in terms of a few key geometric parameters [38]:

da1dt=(iω1γ1)a1iβω1a3(a2*)2+2γ1dξ1
da2dt=(iω2γ2)a2iβω2a3a1*a2*+2γ2dξ2+2γ2csin
da3dt=(iω3γ3)a3iβ*ω3a1a22+2γ3dξ3
where aj denotes the mode amplitude of mode j ∈ [1, 3], normalized so that |aj|2 is the mode energy, and γj = γjd + γjc denotes its decay rate, resulting from either material dissipation γjd or coupling to external channels γjc (e.g. radiation or a waveguide). Each mode is assumed to be in local thermodynamic equilibrium [13] and hence subject to thermal sources ξj satisfying ξj*(ω)ξj(ω)=Θ(ω,Tj)δ(ωω), where Θ(ω, Tj) = ħω/[exp(ħω/kBTj) − 1] is the Planck distribution associated with the local resonator temperature Tj and 〈⋯〉 denotes a thermodynamic, ensemble average [39]. A monochromatic coherent drive incident from the channel, sin = s0exp(2t) normalized such that P = |s0|2 denotes the power, facilitates the nonlinear interaction captured by the following nonlinear coupling coefficient:
β=dVχijkl(3)E1iE2jE2kE3l*2ε0dV(ωε)ωE1i*E1idV(ωε)ωE3i*E3idV(ωε)ωE2i*E2i,
that depends on a spatial overlap of the linear cavity fields Eαi over the (generally anisotropic) susceptibility χijkl(3) of the nonlinear medium in space. The indices α ∈ [1, 2, 3] and {i, j, k, l} ∈ [x, y, z] run over the mode number and cartesian components of the mode profiles, while ε0 and ε denote the vacuum and relative permittivity of the system.

 figure: Fig. 1

Fig. 1 (a) Schematic of three equal-temperature T resonators supporting modes at ωj, each with decay rate γj stemming from internal dissipation γjd and/or radiation into an external channel γjc, with j = {1, 2, 3}. The thermal modes 1 and 3 are strongly coupled to one another through a four-wave mixing process involving a Kerr χ(3) medium excited by externally incident light of power P from a waveguide that resonantly couples to the mode at ω2 = 1/2(ω3ω1). Nonlinear mixing between modes 1 and 3 can be described by an effective, linear but power-dependent coupling κ [Eq. (7)] obtained via a spatial overlap between the three modes, given in Eq. (4). (b) Thermal emission spectra Pj (ω) (blue and red curves) normalized by kBT, for a choice of far-apart frequencies ω1 = kBT/ħ and ω3 = 10ω1, and decay rates γjd = γjc = 0.01ω1, as a function of the dimensionless frequency ħω/kBT. Emission at any ω is bounded above by the Planck distribution Θ(ω, T) (black curve) in the absence of the pump κ = 0 (blue curves), but is exponentially enhanced under finite κ > 0 (red curves). (c) Thermal emission P1(ω) (solid lines) and heat-transfer Pex(ω) (dotted lines) spectra near ω1 as a function of the dimensionless frequency (ωω1)/γ1, demonstrating splitting of the resonances into Stokes (red-shifted) and anti-Stokes (blue-shifted) peaks, which grow apart with increasing κ and further enhance emission.

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Since typical pump energies tend to be much greater than the available thermal energy in the system, i.e. PγkBT, it is safe to ignore the down-conversion term iβω2a3a1*a2* in Eq. (2), in which case the equation for the pump decouples and is linear in the incident drive field:

da2dt=(iω2γ2)a2+2γ2csin.
Such an undepleted-pump approximation [40] greatly simplifies the description of the four-wave mixing process, which is thence described by the following coupled linear equations:
da1dt=(iω1γ1)a1iκe2iω2ta3+2γ1dξ1
da3dt=(iω3γ3)a3iω3ω1κ*e2iω2ta1+2γ3dξ3,
Here, the role of the mediator mode at ω2 is captured by the effectively linear coupling coefficient,
κ=2βω1γ2cPγ22.
Generally, in addition to frequency mixing, the Kerr nonlinearity also leads to cross-phase modulation [6], which acts to shift the resonator frequencies and hence disturbs frequency matching, ω3 = ω1+2ω2 [40,41]. Since the corresponding modulation terms are temporally and hence dynamically decoupled from the effective equations under the undepleted approximation, in the following we account for such as well as other possible sources of frequency mismatch, e.g. material dispersion or even fabrication imperfections, by introducing a time-independent frequency offset Δω = ω3ω1 − 2ω2 into one of the cavity frequencies.

To analyze energy transfer in this system, it suffices to consider the rate of energy loss associated with each mode, d|aj|2dt, obtained through Eq. (5) and Eq. (6). Collecting terms proportional to the coupling coefficient κ, one finds that the rate of energy extraction from a1 is given by Pex=2Im[κexp(2iω2t)a3*a1], while the rate at which energy is upconverted and gained by a3 is given by Pup=2Im[ω3ω1κexp(2iω2t)a3*a1]=ω3ω1Pex. Note that in contrast to the case of two resonantly and linearly coupled modes [31], energy exchange within this four-wave mixing process is not symmetric, i.e. PexPup, but instead satisfies a photon-number conservation condition [42], i.e. Pexω1=Pupω3, which ensures that the number of photons lost by a1 is equal to that gained by a3. Finally, the linearity of the coupled-mode equations allows the extraction/upconversion rates and emission rates Pj = 2γjc〈|aj|2〉 to be expressed in closed form, leading to the following power spectral densities:

Pex(ω)=4|κ|2D1(ω)[γ1γ3dΘ(ω+2ω2,T3)+γ1dγ3(ω3/ω1)Θ(ω,T1)]
P1(ω)=4γ1cD1(ω)[γ1d|i(ωω1Δω)+γ3|2Θ(ω,T1)+γ3d|κ|2Θ(ω+2ω2,T3)]
P3(ω)=4γ3cD3(ω)[γ3d|i(ωω3Δω)+γ1|2Θ(ω,T3)+γ1d(ω3/ω1)|κ|2Θ(ω2ω2,T1)]
where D1(ω)=|(i(ωω1)+γ1)(i(ωω1Δω)+γ3)+ω3ω1|κ|2|2, D3 = D1(1 ↔ 3, Δω → −Δω), and as noted above, the upconverted power Pup=ω3ω1Pex.

The expressions above capture the most important features of this four-wave mixing scheme. As an example, we consider the particular scenario of equal-temperature T cavities with ω1 = kBT/ħ, ω3 = 10ω1, and zero frequency mismatch Δω = 0. We focus on the special situation of resonances having equal dissipative and radiation rates, γjd = γjc = 0.01ω1, in which case both resonators exhibit perfect thermal emissivities φjPj/Θ(ωj, T) (blue curves) in the absence of the pump, i.e. κ = 0. Note that we have chosen ω3kBT/ħ = ω1, in which case there is negligible radiation at ω3 despite the near-unity emissivity. Figure 1(b) shows the thermal radiation spectrum near the two resonances for both κ = 0 (blue curve) and κ = γ1 (red curves), revealing giant enhancements in φ3 ≫1 in the presence of the pump due to significant energy transfer from ω1 to frequencies ω3kBT/ħ at which fluctuations are otherwise exponentially suppressed by the Planck distribution (black curve). Figure 1(c) shows P1(ω) and Pex(ω) near ω1, illustrating that the pump also causes the mode resonances to split into Stokes (−) and anti-Stokes (+) peaks which grow apart with increasing κ and have center frequencies:

ω1±=ω1+Δω±Δω2+4(γ1γ3+ω3/ω1|κ|2)2ω3±=ω1±(13,ΔωΔω)
As a consequence, in addition to mediating energy transfer, the pump-induced red shift allows the resonator to effectively draw additional energy available at longer wavelengths from the Planckian reservoir, thus increasing its emission rate. Specifically, owing to the red shift, the largest emissivity associated with the Stokes mode is found to be φ3max=kBT/Θ(ω3,T) rather than Θ(ω1, T)/Θ(ω3, T). Such enhancements, however, are achieved only in the unrealistic limit of strong coupling κγ1,3, perfect frequency- and rate-matching, Δω = 0 and γ1 = γ3, respectively, and negligible spurious losses, γ1c = γ3d = 0. The largest possible extraction rate in this system, Pexmax=2γ1dkBT, occurs under similar conditions, except in a regime wherein thermal excitations at ω1 are upconverted and reabsorbed at ω3 at a faster rate than they decay in resonator 1, achieved in the limit of |κ| ≫ γ3γ1. While such a strong-coupling limit typically renders the coupled-mode framework invalid [38], as we show below, it is still possible to observe significant splitting in situations κγj where coupled-mode theory is still accurate. Moreover, while perfect frequency matching Δω = 0 is desirable in order to guarantee optimal flux rates, in practice one can still achieve significant transfer rates so long as the frequency mismatch is smaller than either the resonator bandwidths, Δωγj, or the coupling rate, κγj. Finally, as expected, it follows from Eq. (7) that in order to achieve strong coupling |κ|, one requires large nonlinear overlaps β, input power P, and long lifetimes 1/γ2.

The coupled-mode equations above describe a wide variety of triply resonant systems, with the choice of implementation affecting only the corresponding coupled-mode parameters. Aside from allowing different temperature reservoirs, one reason to consider a system of three physically distinct resonators is that it allows extraction of energy from one of the resonators while mitigating additional heating introduced by the coherent drive, thus paving the way for an active near-field cooling mechanism analogous to a recently proposed scheme based on gain media [32]. In what follows, we examine a set of possible, physical implementations of this upconversion scheme based on a configuration of two planar slabs supporting a plurality of resonances and which allow significant extraction of thermal energy from one slab to another.

3. Near-field thermal upconversion and energy transfer between slabs

Consider the planar geometry depicted schematically in Fig. 2(a) and comprising two semi-infinite materials of relative permittivities ϵ1 and ϵ3 which are held at temperatures T1 and T3, respectively. The slabs are separated by a gap of size d that is filled with a χ(3) nonlinear medium of permittivity ϵ2. They support a large number of SPPs localized around their respective interfaces and characterized by their conserved in-plane momenta, k1 and k3, with resonant frequencies ω1(k1) and ω3(k3) satisfying,

e2iκz2d=(ϵ1κz2ϵ2κz1)(ϵ3κz2ϵ2κz3)(ϵ2κz3+ϵ3κz2)(ϵ1κz2+ϵ2κz1)
where kzj=ϵjω2/c2k2 and k = |k|. Such a configuration is known to result in large thermal energy transfer when the two slabs are identical and hence support equal-frequency resonances, ω1 = ω3 [43]. In what follows, we consider the atypical situation of two different materials with dissimilar dispersions and far-apart SPP resonance frequencies, where there is negligible heat exchange under passive, non-equilibrium conditions but where the presence of the nonlinear medium and a (mediator) mode at ω2ω3ω12 that is excited by externally incident light can cause a large amount of thermal energy in slab 1 (emitter) to be upconverted and absorbed in slab 3 (absorber). A schematic of the frequency dispersions of such a system is shown in Fig. 2(a). As discussed above, in order for any two modes (in principle characterized by different k1,3) to couple efficiently, their frequencies must satisfy the frequency-matching condition Δωγ1,3, κ described above. While material dispersion and intrinsic fabrication imperfections generally preclude this from occurring for all modes, such a condition can however be enforced by appropriate nanostructuring and/or dispersion engineering [44, 45]. In principle, while it is possible to achieve frequency matching for a plurality of thermal modes by introducing and exciting multiple resonances near ω2, such a situation would require a more complicated analysis and is therefore out of the scope of this work.

 figure: Fig. 2

Fig. 2 (a) Schematic of two plates consisting of different materials held at temperatures T1 and T3 and which support surface plasmon polaritons (SPPs) at far-apart resonant frequencies ω1 and ω3. Also shown is a schematic of the corresponding SPP dispersions ωj (k) (blue and red). Thermal upconversion and transfer of energy from ω1 to ω3 is facilitated by a mediator mode at ω2~(ω3ω1)2 (green). (b) Table summarizing three different geometric configurations that result in significant thermal upconversion, along with various possible choices of emitting, absorbing, and nonlinear materials. The main difference between configurations is the choice of intervening medium, which consist of either a (i) lattice of nanoparticles embedded in the nonlinear medium or (ii) bulk nonlinear thin film, and serves as a tunable means to enhance the incident light.

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For the sake of generality, we consider two classes of possible geometric configurations, depicted in Fig. 2(b) and differing primarily in the choice of material and intervening medium, which involve either (i) nanostructured or (ii) bulk nonlinear materials supporting SPP resonances at ω2. In (i), laterally incident light couples to a periodic lattice of composite nanospheres or nanodisks (supporting dipolar resonances) while in (ii), vertically incident light couples to a low-frequency SPP through a grating. As described below, we ensure that regardless of implementation, an emitter mode at k1 couples to a unique absorber mode at k3, which greatly simplifies the calculation of flux transfer by avoiding otherwise cumbersome analysis of multiply interacting degenerate modes. Such a simplification also allows us to easily compute the nonlinear coupling coefficient corresponding to each pair of modes β(k1, k3) and to exploit the analytical expression for Pex(ω, k1, k3) given in Eq. (8). In either configuration, the sparsity (low filling fractions) and off-resonant nature of the nanoparticles and grating allow us to ignore their impact on the translational symmetry and dispersion relations of the slab modes.

The table in Fig. 2(b) summarizes a number of potential configurations and material choices. For the sake of comparison and consistency, we choose the emitter to be SiC, whose permittivity is obtained from [46], and the intervening nonlinear medium to be ChG, whose permittivity ϵ2 = 6.25 and χ(3) ~ 1 × 10−17m2/V2 are taken from various references [47–50]. For computational and conceptual convenience, we assume an isotropic χ(3) tensor with elements χxxxx = 3χxxyy = 3χxyxy = χ(3) [6, 48]. While there are many choices of possible absorber materials, e.g. K, Ag, ITO, Au, and AZO, here we focus on a select few, depending on the choice of implementation; their permittivities are taken from several references [51–53]. Finally, we fix the gap size to be d = 60nm and consider only the situation in which the entire system is at thermodynamic equilibrium, with T1 = T3 = 300K, for which there is zero heat exchange in the absence of the pump.

3.1. Nanoparticles lattice

In this section, we examine situations in which the intervening medium consists of a two-dimensional lattice of nanoparticles embedded in ChG. The purpose of the lattice is threefold: First, it acts as a grating which allows externally incident light to excite a given (mediator) Bloch mode at ω2, described by the coupling coefficient of Eq. (7), which couples SPPs at ω1 and ω3. Second, the particle shapes, sizes, and materials can be exploited to engineer the resonant frequency ω2 and associated decay rates, and thereby the pump field and coupling coefficient κ, as needed. Third, the lattice provides many degrees of freedom with which to enforce “quasi-phase matching” [54] over a broad range of k, allowing the β coefficient corresponding to multiple mode pairs (k1, k3) to be optimized.

Owing to the subwavelength size of the nanoparticles, their optical response can be treated within a quasi-static, dipolar approximation [55]. They are also placed far apart from one another, mitigating many-body scattering effects and allowing the field induced by the incident drive to be conveniently expressed as a linear superposition ∑p E2(xxp, z) of the isolated particle resonances, where x and xp denote the transverse co-ordinates and center of each particle while E2(xxp, z) denotes its mode profile. The p-polarized field profiles of the planar resonances are given by Ejl(z)eikj.x, with j = 1, 3 and lx, y, z (nonzero components along k^ and z^ directions), in which case the nonlinear coupling coefficient β for a given mode pair is given by:

β(k1,k3)=pdVχijk(3)ei(k3k1).xE1i(z)E2j(xxp,z)E2k(xxp,z)E3*(z)2ϵ0(pdVϵωω|E2(xxp,z)|2)(dVϵωω|E1(z)|2)1/2(dVϵωω|E3(z)|2)1/2
Here, the sum is taken with respect to the index p of each particle, {i, j, k, } denote cartesian field components, and χijk(3) is the Kerr tensor of the background medium. The SPP dispersions ωj (k) and corresponding (purely dissipative) decay rates γjd (k) are obtained by the complex-frequency solutions of Eq. (11). Because of the relatively low in-plane and volume filling fractions as well as the off-resonant response of the nanoparticle lattice, they are assumed to have a negligible effect on the planar dispersion and mode profiles.

In order to take advantage of the large density of states available in the near field, we choose geometric and material parameters that ensure large β for as wide a range of modes as possible. We restrict our analysis to modes having equal momentum k = k1 = k3, in which case the integral in Eq. (12) no longer exhibits a phase factor that depends on k1,3. Such a simplifying assumption is further justified by the fact that although technically the nonlinearity can couple modes propagating in different directions (k1 ≠ k3), a situation that arises in the configuration of Sec. 3.2, these typically suffer from diminished mode overlaps and hence reduce the overall upconversion efficiency. Our analysis is further simplified by ensuring that the induced particle fields exhibit cylindrical symmetry about the z^ axis, in which case the field E2(xxp) and hence the coupling between each mode pair is independent of the k^ direction. The chosen implementations involve either a lattice of spherical nanoparticles or graphene nanodisks, respectively, which act like z^-oriented dipoles under illumination by light incident along either the parallel or z^ directions, respectively. These considerations allow us to write Pex(ω, k1, k3) as Pex(ω, k), and thus to express the net heat extraction rate per unit area H across the gap as:

H=0dω2π0dω2πkPex(ω,k)Φex(ω)
where we further define the quantity Pex(k)=dω2πPex(ω,k) as the frequency-integrated flux at each k. Similarly, one can define the associated flux spectral density, Φex(ω)=0dk2πPex(ω,k), by integrating instead over all wavevectors. Finally, in addition to the spatial profile of each mode pair, it is equally important to enforce the frequency-matching condition Δω(k) = ω3(k) − ω1(k) − 2ω2γj, κ, which is generally not guaranteed and as illustrated below, depends sensitively on the choice of materials.

3.1.1. Nanospheres

Figure 3(a) shows a rectangular lattice of nanospheres of equal radii R and unit-cell size Λx × Λy that is illuminated by a z-polarized incident wave ∝ z^ei(k2y+ω2t), with k2=ε2ω2/c. The incident field excites dipolar modes with field profiles of the form E2j(xxp,z)eik2y. Since for any given mode pair the nonlinear overlap integral in Eq. (12) involves a sum over all particle positions, vanishing unless the individual unit-cell contributions add constructively, we make an appropriate choice of period Λy = π/k2. Furthermore, although the lack of a “Bloch phase” in the x^ direction means that at least in principle Λx can be much smaller than Λy, here we choose a large enough Λx = 5R in order to ignore many-body effects on the induced field. Taking advantage of the equal contribution of each unit cell to Eq. (12) and restricting our analysis to momentum-matched mode pairs k1 = k3, the coupling κ(k) is given by:

κ(k)=ω1σabsIcellχijk(3)E1i(z)E2j(xxp,z)E2k(xxp,z)E3*(z)4ϵ0γ2dΛxΛy(cellϵωω|E2(xxp,z)|2)(dzϵωω|E1(z)|2)1/2(dzϵωω|E3(z)|2)1/2
where ω2, σabs, and γ2d are the resonance frequency, absorption cross-section, and dissipation rate of the nanoparticle resonances, and I denotes the incident-field intensity. To ensure frequency matching, we employ core-shell nanoparticles [56, 57] with carefully chosen core/shell radii and material dispersions [58]. In particular, we choose particles made of silica [59] core and Au [52] shells having inner and outer radii of 15nm and 20nm, respectively. These parameters yield a resonance frequency ω2 = 8.74 × 1014rad/s, dissipative and radiative decay rates, γ2d = 4.31 × 1012rad/s and γ2c = 8.29 × 1012rad/s, respectively, and lattice parameters Λx = 5R = 100nm and Λy = πc/ω2 = 431nm, resulting in volume and in-plane filling fractions of 0.01 and 0.03, respectively. The corresponding absorption cross-section σabs = 1.2 × 10−13m2 can also be obtained via a well-known dipolar analysis [55,58]. Choosing SiC as the emitter and K as the absorber means that the two interacting slab resonances have frequencies ω1 ~ 1.68 × 1014rad/s (λ1 ~ 11.22μm) and ω3 ~ 2 × 1015rad/s (λ3 ~ 0.94μm), respectively.

 figure: Fig. 3

Fig. 3 (a) Schematic of a planar system of SiC and K slabs at thermal equilibrium (room temperature) separated by a gap of size d = 60nm that is filled with a rectangular lattice of nanospheres with unit-cell size Λx × Λy embedded in a χ(3) nonlinear medium (ChG). The SiO2/Au core/shell nanoparticles have core/shell radii of 15nm and 20nm, respectively, and the dimensions of the unit cell are Λx = 5R and Λy = πc/ω2. The SiC and K slabs support SPPs at frequencies ω1 and ω3, respectively, which couple nonlinearly to dipolar particle resonances at ω2~12(ω3ω1) excited by a laterally incident, monochromatic, z^-polarized planewave of frequency ω2 and intensity I. (b) xy and yz cross sections of the particle resonances |E2|2, with yellow/black denoting maximum/zero amplitude. (c) Normalized slab mode-profiles at a representative wavenumber kd = 2, with the shaded region indicating the intervening medium. (d) Variations in the nonlinear coupling κ, frequency mismatch Δω, and frequency ω1 with respect to kd, normalized by the corresponding dissipation rate γ3 of the K slab. (e) Frequency-integrated heat-extraction spectrum Pex(k), normalized by P0 = 2γ1Θ(ω1, T), as a function of kd and for multiple incident intensities I. Also shown are the (f) associated spectral density Φex(ω) and (g) net extracted and upconverted flux rates, Hex and Hup, as a function of I. For comparison, (e–g) also show the heat-transfer rates associated with two vacuum-separated SiC slabs held at either 300K (light blue) or 1K (dark blue) temperature differences.

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Figure 3(b) shows the xy and yz cross-sections of the nanosphere mode profiles |E|2 within a unit cell, where black/yellow represent zero/maximum values. The spatial mode profiles of the slab resonances are illustrated in (c) at a representative kd = 2, where the shaded region indicates the intervening medium. Figure 3(d) shows the variation in the coupling coefficient |κ| for increasing intensities I = {0.5, 1, 5, 10} W/μm2 (from black to red), frequency mismatch Δω, and resonance frequency ω1, as a function of the dimensionless wavenumber kd. All quantities are normalized by the corresponding dissipation rate γ3 at each k, which is the largest of the loss rates in the system. Figure 3(e) shows the ratio of Pex(k) to the thermal radiation rate of an isolated thermal resonance P0 = 2γ1Θ(ω1, T1) as a function of kd, while Fig. 3(f) shows the associated spectrum Φex(ω), in units of Wm−2rad−1s, for multiple I. Finally, the integrated extraction Hex and upconversion Hup rates are plotted in Fig. 3(g) as a function of I.

We now explain the most salient and important features associated with heat exchange in this system. As shown in (e), Pex(k) exhibits a small peak at kd ≈ 2 that grows and widens with increasing I, causing the overall heat transfer per unit area Hex to monotonically increase and eventually saturate as I → ∞. The associated spectrum Φex(ω) also shows a corresponding increase along with linewidth broadening with increasing I. These features are explained as follows: First, Eq. (8) shows that in the weak coupling regime |κ|/γ3 ≪ 1, the flux rate Hex ~ |κ|2 ~ I2, a consequence of the fact that to lowest order in the nonlinearity, the coupling scales linearly with incident power. Second, as evident from Fig. 3(d), the bandwidth of the spectrum is primarily determined by the range of modes satisfying the frequency matching constraint, Δωγj,|κ|, with the peak occurring at the value of k which minimizes Δω. At small I or equivalently, κ/γ3 ≪ 1, the range of modes satisfying Δωκ, γ3 is narrow, but higher intensities and hence increasing κ allow frequency matching to be satisfied over a wider range of k. Third, the rapid decrease in the flux rate at large kd ≫ 1 happens because β and hence κ depend on the spatial decay of the planar mode profiles, which become increasingly localized to the surface with increasing k. The precise value of k at which Δω(k) = 0 depends primarily on the choice of materials and geometric parameters: although its occurrence at larger k would facilitate increased bandwidths due to the lack of dispersion at large kω/c, the resulting exponential suppression in β suggests instead an optimal choice of particle parameters for a given d, here chosen at an intermediate kd ≈ 2. Note also that we only consider Δω arising from material dispersion and ignore power-dependent shifts due to cross-phase modulation, [40] which tend to be small and in any case can be compensated by suitable lattice parameters. Finally, as discussed above, the ratio Pex/P0 can exceed one when the system enters the strong coupling regime γ3 ≲ |κ| ≪ ω1, in which case the Stokes peak can draw additional thermal energy available at longer wavelengths.

Figure 3 also shows the flux rate associated with a symmetric configuration of two identical SiC plates maintained at T1 and T3 and separated by the same gap size (albeit in vacuum). The net heat-transfer rate in this more typical scenario can be computed via the well-known fluctuational electrodynamics framework [23] and is given by:

H(0)=0dω2π[Θ(ω,T1)Θ(ω,T3)]0dkk2Im{r21p}Im{r23p}e2Im{kz2}dπ|1r21pr23pe2Im{kz2}d|2Φ(0)ω)
where rjp is the p-polarized Fresnel reflection coefficient between mediums j and and where, as before, one can define and compute the frequency-integrated flux P(0) (k) and flux spectral density Φ(0) (ω). For the sake of comparison, we consider two different operating conditions corresponding to either (i) large (T1 = 300 K, T3 = 0 K) (light blue curves) or (ii) small (T1 = 301 K, T3 = 300 K) (dark blue curves) temperature differentials.

From Fig. 3(e), one immediately observes that compared to the nonlinear scheme, the exponential decay in Pex(0)(k) occurs at smaller kd ≈ 1. The reasons are twofold: First, the increased proximity of the nanoparticle resonances to the slab interfaces results in slightly larger mode overlaps. Second and most importantly, increasing I and hence κ allows thermal energy in the SiC to be upconverted and then absorbed at a much faster rate than it is dissipated in the SiC. Consequently, the range of participating modes and hence the bandwidth of Pex(k) increases with increasing I. When combined with the aforementioned Stokes enhancement, the net effect is a significant increase in the nonlinear flux rates, which exceed the corresponding linear flux rates ≈ 104 W/m2 at a temperature-dependent threshold intensity Ic, which is approximately Ic ≈ 3W/μm2 in scenario (i) and Ic ≈ 0.1W/μm2 in scenario (ii). Finally, we note that in contrast to the typically investigated system of two vacuum-separated identical slabs, the frequency-matching wavenumber κ and hence peak value of k in the nonlinear scheme can be tuned by engineering the core/shell nanoparticle geometry or by an appropriate choice of emitter/absorber materials. Nevertheless, we find that the threshold intensities do not vary much with respect to the choice of absorber material. As shown in Fig. 3(f) (green curves), fixing the parameters of the emitter and core/shell nanoparticles but replacing K with Ag also leads to significant energy transfer (albeit slightly smaller owing to the smaller frequency-matching wavenumber).

Finally, we remark that our choice of geometry and operating conditions is by no means optimal. For instance, one could further increase the mode overlaps and frequency-matching wavenumbers by exploiting more complicated and diverse particle shapes (e.g. nanorods or even asymmetric particles) or by employing other emitter materials (e.g. hBN, Au), leading to greater efficiencies and lower power requirements. Moreover, while the choice of lattice period in this example is primarily motivated by the need to enforce quasi-phase matching for a laterally incident pump, as we show in the next section, that and other geometric considerations are also highly dependent on the choice of incident direction and/or particle shape.

3.1.2. Nanodisks

Figure 4(a) shows a square lattice of doped graphene nanodisks of unit-cell size Λ × Λ that is illuminated by a normally incident, circularly polarized wave ∝ (x^+iy^)ei(k2z+ω2t), with k2=ε2ω2/c. Owing to the normal incidence, the induced fields E2j (xxp, z) at each lattice site have the same phase and consequently, each individual unit cell contributes equally irrespective of the lattice period. For simplicity, we choose Λ = 3R which allows us to ignore many-body scattering. As before, the cylindrical symmetry of the disks and the circular polarization of the incident light imply that β(k1, k2) → β(k) is independent of the k^ direction. Our choice of graphene as opposed to other potential polaritonic/plasmonic materials (e.g. Au, Ag, etc) is motivated by the large degree of tunability in the resonance frequency ω2 and hence frequency-matching wavenumber with respect to the Fermi energy or doping concentration of graphene, as well as by recent experiments exploring related structures [60, 61]. In what follows, we choose graphene nanodisks of radius R = 20nm, fermi energy Ef = 0.7eV, and intrinsic lifetime τ = 6 × 10−13, consistent with recent experimental measurements [61], which results in resonance frequencies ω2 = 3.04 × 1014rad/s and dissipation rates γ2d = 3.3 × 1012rad/s. For an incident light of intensity I, the coupling κ(k) is given by Eq. (14), where σabs = 2 × 10−14m2 is calculated from the effective dipolar susceptibility [60]. The corresponding field profiles are obtained via numerical simulation of Maxwell’s equations using the boundary element method (BEM) [62], where the graphene nanodisk is assumed to have an effective thickness h = 0.5nm and dielectric permittivity ϵG=14πiσ(ω)ωh, with σ(ω) denoting the sheet conductivity of graphene [60]. Assuming SiC and ITO as the emitter and absorber in this configuration, respectively, one finds ω1 ~ 1.68 × 1014rad/s (λ1 ~ 11.22μm) and ω3 ~ 8.4 × 1014rad/s (λ3 ~ 2.24μm).

 figure: Fig. 4

Fig. 4 (a) Schematic of a planar system of SiC and ITO slabs at thermal equilibrium (room temperature) separated by a gap of size d = 60nm that is filled with a square lattice of doped graphene nanodisks with unit-cell size Λ × Λ embedded in a χ(3) nonlinear medium (ChG). The graphene nanodisks have radius R = 20nm, Fermi energy EF = 0.7eV and are placed at a distance of 10nm from ITO slab. The SiC and ITO slabs support SPPs at frequencies ω1 and ω3, respectively, which couple nonlinearly to nanodisk resonances at ω2~12(ω3ω1) excited by a normally incident, monochromatic, x^+iy^-polarized planewave of frequency ω2 and intensity I. (b) xy and yz cross sections of the particle resonances |E2|2, with yellow/black denoting maximum/zero amplitude. (c) Normalized slab mode-profiles at a representative wavenumber kd = 1, with the shaded region indicating the intervening medium. (d) Variations in the nonlinear coupling κ, frequency mismatch Δω, and frequency ω1 with respect to kd, normalized by the corresponding dissipation rate γ3 of the ITO slab. (e) Frequency-integrated heat-extraction spectrum Pex(k), normalized by P0 = 2γ1Θ(ω1, T), as a function of kd and for multiple incident intensities I. Also shown are the (f) associated spectral density Φex(ω) and (g) net extracted and upconverted flux rates, Hex and Hup, as a function of I. For comparison, (e–g) also show the heat-transfer rates associated with two vacuum-separated SiC slabs held at either 300K (light blue) or 1K (dark blue) temperature differences.

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Figure 4(b) shows the xy and yz cross-sections of the nanodisk mode profiles |E|2 within a unit cell. We remark that since the modes are largely confined to the nanodisk circumference, further close packing or potentially smaller lattice periods could potentially be employed to enhance κ. Figure 4(c) shows profiles of the planar resonances at a representative kd = 1, with the shaded region indicating the intervening medium. While most of the features observed in this configuration and illustrated in Fig. 4(d–g) are qualitatively similar to the previous implementation in Sec. 3.1.1 based on nanospheres, we emphasize some of the main differences. First, the possibility of exciting cylindrically symmetric nanoparticle resonances with normally incident light allows a reduction of the lattice period and leads to slightly reduced threshold intensities, with Ic ≈ 0.5W/μm2 in the scenario corresponding to SiC slabs held at a 300K temperature difference (light blue curves). Second, graphene offers additional (potentially dynamic) tunability with respect to the frequency-matching wavenumber, leading to large (order of magnitude) differences in the heat flux for relatively small changes in Ef (not shown). Third, the two-dimensional nature of graphene nanodisks allows greater choice in their vertical placement; one could for instance consider multiple sets of nanodisks distributed vertically throughout the intervening medium, which would further increase κ. One important constraint to consider when choosing the shapes of the nanoparticles is the choice of pump and wavelengths; for instance, while graphene supports polaritons in the mid-infrared, other metallic nanoparticles can support resonances at higher (e.g. near-infrared or visible) wavelengths. Finally, we note that our choice of direction for the incident pump in either scenario is motivated by the desire to restrict our analysis to cylindrically symmetric nonlinear coupling coefficients (i.e. independent of the SPP propagation direction). One could also tune the lattice parameters and particle shapes so as to allow obliquely incident pumps, though this would likely lead to much more complicated β(k1, k3). We analyze such a situation in the following section, in which we study upconversion in the case of a bulk nonlinear thin film as the intervening medium.

3.2. Bulk media

Figure 5(a) shows a 1d y-periodic grating of period Λ resting at the interface of the ChG and K slabs. The grating is assumed to have a negligible effect on the SPPs at ω1 and ω3, and chosen so that incident light of wavevector k2y along the y direction can excite a SPP of frequency ω2=ω3(k2y) localized around the ChG–K interface. The angle of incidence θinc with respect to the z^ axis is chosen so as to satisfy,

ϵ2ω2sin(θinc)c+2πΛ=k2y,
thus ensuring that only the first diffracted order is excited by the pump [63,64]. The p-polarized field profiles of the planar resonances are given by Ej(z)eikj.x, with j ∈ [1, 3] and x, y, z (nonzero components along k^ and z^), in which case the nonlinear coupling β is given by:
β(k1,k3)=dVχijk(3)ei(k3k12k2yy^).xE1i(z)E2j(ω2,z)E2k(ω2,z)E3*(z)2ϵ0(dVϵωω|E2(ω2,z)|2)(dVϵωω|E1(z)|2)1/2(dVϵωω|E3(z)|2)1/2
Notably, one finds that nonzero coupling is only possible under the momentum-matching condition, k1+2k2yy^=k3, represented schematically in Fig. 5(b). Such a condition ensures that a wave at k1 couples to a unique wave at k3, in which case β(k1, k3) → β(k, θ), where k = |k1| and θ is the angle extended by k1 with respect to the y axis. Employing the translational invariance of the overlap integrals for such a process, one finds that coupling κ(k, θ) is given by:
κ(k,θ)=ω1γ2cIdzχijk(3)E1i(ω1,z)E2j(ω2,z)E2k(ω2,z)E3*(ω3,z)ϵ0γ22dzϵωω|E2(ω2,z)|2(dzϵωω|E1(ω1,z)|2)1/2(dzϵωω|E3(ω3,z)|2)1/2
where as before, I is the intensity of the incident light and γ2c is obtained by solving the full scattering problem [63, 64]. The net heat-transfer rate per unit area across the gap is then given by Hex(k)=02πdθ2π0dω2πPex(ω,k,θ), where Pex follows from Eq. (13) and the associated flux spectral density is given by Φex(ω)=02πdθ2π0dk2πkPex(ω,k,θ). In what follows, we choose a grating period Λ = 205nm in order to couple incident light at an angle θinc = π/4 to a SPP of frequency ω2 = 9 × 1014rad/s, wavenumber k2y = 2.8ω2/c, and dissipation rate γ2d = 2.3 × 1012rad/s. For convenience and expediency, we ignore the impact of the grating on the dispersions at ω1,3 and assume critical coupling, γ2c = γ2d. Of course, any deviation from this condition would result in decreased efficiency [10]. Choosing SiC and K as the emitter and absorber materials, respectively, one finds that ω1 ~ 1.68 × 1014rad/s (λ1 ~ 11.22μm) and ω3 ~ 2 × 1015rad/s (λ2 ~ 0.94μm).

 figure: Fig. 5

Fig. 5 (a) Schematic of a planar system of SiC and K slabs at thermal equilibrium (room temperature) separated by a gap of size d = 60nm that is filled with χ(3) nonlinear medium (ChG). The SiC and K slabs support SPPs at frequencies ω1 and ω3, respectively, which couple nonlinearly to a SPP resonance at ω2~12(ω3ω1) that is excited via a grating (of period Λ) by monochromatic light of frequency ω2, intensity I, and wavevector k2y incident at an angle θinc with respect to the z^ direction. (b) Schematic illustration of the coupling of SPP resonances of wavevectors k1 and k3; the polar plot shows the directional dependence of the nonlinear coupling coefficient κ(k, θ), where k1 = (k, θ) is expressed in polar coordinates, with θ denoting the angle extended by k1 with respect to y^. (c) Normalized slab mode-profiles at a representative wavenumber kd = 2 and angle θ = 0, with the shaded region indicating the intervening medium. (d) Variations in the nonlinear coupling κ, frequency mismatch Δω, and frequency ω1 with respect to kd, normalized by the corresponding dissipation rate γ3 of the K slab. (e) Frequency-integrated, angle-averaged heat-extraction spectrum Pex(k), normalized by P0 = 2γ1Θ(ω1, T), as a function of kd and for multiple incident intensities I. Also shown are the (f) associated spectral density Φex(ω) and (g) net extracted and upconverted flux rates, Hex and Hup, as a function of I. For comparison, (e–g) show the heat-transfer rates associated with two vacuum-separated SiC slabs held at either 300K (light blue) or 1K (dark blue) temperature differences.

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Figure 5(c) shows the various mode profiles |E|2 at a representative kd = 2 and for θ = 0, illustrating the larger spatial extent of the mediator mode owing to its effectively smaller wavenumber k2y d ≈ 0.5. The polar plot in Fig. 5(b) shows the normalized |κ(k, θ)|/γ3 for increasing values of kd, illustrating the directional dependence of the coupling coefficient and the lack of frequency matching stemming from material dispersion and the absence of a ω3(k3) mode in certain directions.

Figure 5(d,f,g) shows that heat exchange in this configuration is qualitatively similar to what is observed in the case of nanoparticle lattices. Despite the similar frequency spectrum and net flux rate dependence on I, the complicated dependence of κ on both k and θ does lead to noticeably different Pex(k), shown in (e). Moreover, while this configuration has the advantage of simplicity and ease of realization, as expected the significantly decreased modal overlaps lead to much larger power requirements. For instance, the threshold intensity needed to surpass the flux rate between two SiC slabs held at a 300K temperature difference (light blue curve) is roughly an order of magnitude larger, with I ≈ 10W/μm2. Finally, we remark that Fig. 5(g) also show the net flux rate achieved by replacing K with Ag under the same lattice parameters and excitation conditions, indicating significant robustness with respect to the choice of materials.

4. Concluding remarks

We have presented a scheme for achieving large near-field thermal energy transfer at the nanoscale based on the nonlinear χ(3)-mediated interaction of thermal modes with externally incident light. We employed a general coupled-mode framework to predict power requirements and thermal flux rates in planar configurations consisting of different (emitter and absorber) materials separated by either a bulk or nanostructured nonlinear medium, and in which resonantly enhanced incident light at mid-infrared wavelengths upconverts mid-infrared thermal energy to either near-infrared or visible wavelengths. We find that even when the entire system is held at room temperature and at relatively low pump intensities ~ W/μm2, the rate of energy upconversion can approach and even exceed 104W/m2, which compares with typical flux rates expected in the more common situation of identical slabs separated by vacuum and held up to large ~ 300K temperature differences. This scheme therefore not only provides a means to achieve significant energy transfer at typically inaccessible near-infrared or visible wavelengths, but also facilitates heat exchange between dissimilar materials even when these are originally in thermodynamic equilibrium. Finally, we remark that our specific choices of planar geometries and materials are by no means optimal and represent only a proof of concept. More importantly, while this scheme could potentially be exploited in nanoscale radiative cooling [32, 34, 37] and power generation [23,29] applications, the efficiency and utility of the nonlinear process in those situations will necessarily be degraded by heating introduced by the pump and neglected in this work. Such additional heating stems from the presence of SPP resonances at the incident (mid-infrared) wavelength, which necessarily cause conductive heating of the slab but can potentially be mitigated by the introduction of a small vacuum gap separating the nonlinear material from either or both of the slabs. Similarly, instead of exploiting SPP resonant enhancement of the incident light, one might also consider dielectric resonances [65, 66] in semi-transparent materials. A detailed analysis of the merits of this scheme for radiative cooling or related applications that incorporates pump-induced heating and related considerations will be the subject of future work.

Funding

National Science Foundation (DMR-1454836) Princeton Center for Complex Materials, a MRSEC supported by NSF (1420541).

Acknowledgments

We are thankful to Weiliang Jin and especially grateful to Riccardo Messina for helpful comments and suggestions.

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57. S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. 288, 243 (1998). [CrossRef]  

58. A. H. Sihvola, “Peculiarities in the dielectric response of negative-permittivity scatterers,” Prog. Electromagn. Res. 66191 (2006). [CrossRef]  

59. R. Kitamura, L. Pilon, and M. Jonasz, “Optical constants of silica glass from extreme ultraviolet to far infrared at near room temperature,” Appl. Opt. 46, 8118 (2007). [CrossRef]   [PubMed]  

60. Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. G. de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7, 2388 (2013). [CrossRef]   [PubMed]  

61. F. H. Koppens, D. E. Chang, and F. J. G. de Abajo, “Graphene plasmonics: a platform for strong light–matter interactions,” Nano Lett. 11, 3370 (2011). [CrossRef]   [PubMed]  

62. U. Hohenester and A. Trügler, “MNPBEM-A Matlab toolbox for the simulation of plasmonic nanoparticles,” Comput. Phys. Comm. 183, 370 (2012). [CrossRef]  

63. J. R. Sambles, G. W. Bradbery, and F. Yang, “Optical excitation of surface plasmons: an introduction,” Contemp. Phys. 32, 173 (1991). [CrossRef]  

64. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845 (1999). [CrossRef]  

65. Q. Quan and M. Lončar, “Deterministic design of wavelength scale, ultra-high Q photonic crystal nanobeam cavities,” Opt. Exp. 19, 18529 (2011). [CrossRef]  

66. I. W. Frank, P. B. Deotare, M. W. McCutcheon, and M. Lončar, “Programmable photonic crystal nanobeam cavities,” Opt. Express 18, 8705 (2010). [CrossRef]   [PubMed]  

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    [Crossref] [PubMed]
  61. F. H. Koppens, D. E. Chang, and F. J. G. de Abajo, “Graphene plasmonics: a platform for strong light–matter interactions,” Nano Lett. 11, 3370 (2011).
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  62. U. Hohenester and A. Trügler, “MNPBEM-A Matlab toolbox for the simulation of plasmonic nanoparticles,” Comput. Phys. Comm. 183, 370 (2012).
    [Crossref]
  63. J. R. Sambles, G. W. Bradbery, and F. Yang, “Optical excitation of surface plasmons: an introduction,” Contemp. Phys. 32, 173 (1991).
    [Crossref]
  64. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845 (1999).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]

2016 (3)

H. Soo and M. Krüger, “Fluctuational electrodynamics for nonlinear media,” EPL 115, 41002 (2016).
[Crossref]

D. Ding, T. Kim, and A. J. Minnich, “Active Thermal Extraction and Temperature Sensing of Near-field Thermal Radiation,” Sci. Rep. 6, 32744 (2016).
[Crossref] [PubMed]

C. Khandekar, W. Jin, O. D. Miller, A. Pick, and A. W. Rodriguez, “Giant frequency-selective near-field energy transfer in active-passive structures,” Phys. Rev. B 94, 115402 (2016).
[Crossref]

2015 (4)

K. Chen, P. Santhanam, S. Sandhu, L. Zhu, and S. Fan, “Heat-flux control and solid-state cooling by regulating chemical potential of photons in near-field electromagnetic heat transfer,” Phys. Rev. B 91, 134301 (2015).
[Crossref]

B. Song, A. Fiorino, E. Meyhofer, and P. Reddy, “Near-field radiative thermal transport: From theory to experiment,” AIP Adv. 5, 053503 (2015).
[Crossref]

C. Khandekar, Z. Lin, and A. W. Rodriguez, “Thermal radiation from optically driven Kerr χ(3) photonic cavities,” Appl. Phys. Lett. 106, 151109 (2015).
[Crossref]

C. Khandekar, A. Pick, S. G. Johnson, and A. W. Rodriguez, “Radiative heat transfer in nonlinear Kerr media,” Phys. Rev. B 91, 115406 (2015).
[Crossref]

2014 (3)

C. R. Otey, L. Zhu, S. Sandhu, and S. Fan, “Fluctuational electrodynamics calculations of near-field heat transfer in non-planar geometries: A brief overview,” J. Quant. Spectrosc. Radiat. Transfer 1323 (2014).
[Crossref]

P. Ben-Abdallah and S. Biehs, “Near-field thermal transistor,” Phys. Rev. Lett. 112, 044301 (2014).
[Crossref] [PubMed]

Z. Lin, T. Alcorn, M. Loncar, S. G. Johnson, and A. W. Rodriguez, “High-efficiency degenerate four-wave mixing in triply resonant nanobeam cavities,” Phys. Rev. A 89, 05389 (2014).
[Crossref]

2013 (3)

R. Messina and P. Ben-Abdallahm, “Graphene-based photovoltaic cells for near-field thermal energy conversion,” Sci. Rep. 3, 1383 (2013).
[Crossref]

A. C. Jones, B. T. O’Callahan, H. U. Yang, and M. B. Raschke, “The thermal near-field: coherence, spectroscopy, heat-transfer, and optical forces,” Prog. Surf. Sci. 88, 349 (2013).
[Crossref]

Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. G. de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7, 2388 (2013).
[Crossref] [PubMed]

2012 (3)

U. Hohenester and A. Trügler, “MNPBEM-A Matlab toolbox for the simulation of plasmonic nanoparticles,” Comput. Phys. Comm. 183, 370 (2012).
[Crossref]

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photon. 6, 737 (2012).
[Crossref]

B. Guha, C. Otey, C. B. Poitras, S. Fan, and M. Lipson, “Near-field radiative cooling of nanostructures,” Nano Lett. 12, 4546 (2012).
[Crossref] [PubMed]

2011 (7)

D. M. Ramirez, A. W. Rodriguez, H. Hashemi, J. D. Joannopoulos, M. Soljačič, and S. G. Johnson, “Degenerate four-wave mixing in triply resonant Kerr cavities,” Phys. Rev. A 83, 033834 (2011).
[Crossref]

H. Zhao, J. Zhang, G. Liu, and N. Tansu, “Surface plasmon dispersion engineering via double-metallic Au/Ag layers for III-nitride based light-emitting diodes,” Appl. Phys. Lett. 98, 151115 (2011).
[Crossref]

X. Yan, X. Zhang, S. Shi, Z. Liu, and J. Tian, “Third-order nonlinear susceptibility tensor elements of CS 2 at femtosecond time scale,” Opt. Express 19, 5559 (2011).
[Crossref] [PubMed]

F. Kheirandish, E. Amooghorban, and M. Soltani, “Finite-temperature Casimir effect in the presence of nonlinear dielectrics,” Phys. Rev. A 83, 032507 (2011).
[Crossref]

Q. Quan and M. Lončar, “Deterministic design of wavelength scale, ultra-high Q photonic crystal nanobeam cavities,” Opt. Exp. 19, 18529 (2011).
[Crossref]

F. H. Koppens, D. E. Chang, and F. J. G. de Abajo, “Graphene plasmonics: a platform for strong light–matter interactions,” Nano Lett. 11, 3370 (2011).
[Crossref] [PubMed]

R. G. Chaudhari and S. Paria, “Core/shell nanoparticles: classes, properties, synthesis mechanisms, characterization, and applications,” Chem. Rev. 112, 2373 (2011).
[Crossref]

2010 (4)

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4, 795 (2010).
[Crossref]

A. Bahabad, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase-matching of momentum and energy in nonlinear optical processes,” Nat. Photon. 4, 570 (2010).
[Crossref]

I. W. Frank, P. B. Deotare, M. W. McCutcheon, and M. Lončar, “Programmable photonic crystal nanobeam cavities,” Opt. Express 18, 8705 (2010).
[Crossref] [PubMed]

C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104, 154301 (2010).
[Crossref] [PubMed]

2009 (5)

I. Chowdhury, R. Prasher, K. Lofgreen, G. Chrysler, S. Narasimhan, R. Mahajan, D. Koester, R. Alley, and R. Venkatasubramanian, “On-chip cooling by superlattice-based thin-film thermoelectrics,” Nat. Nanotechnol. 4, 235 (2009).
[Crossref] [PubMed]

S. Basu, Z. M. Zhang, and C. J. Fu, “Review of near-field thermal radiation and its application to energy conversion,” Int. J. Energy Res. 33, 1203 (2009).
[Crossref]

M. G. Blaber, M. D. Arnold, and M. J. Ford, “Search for the ideal plasmonic nanoshell: the effects of surface scattering and alternatives to gold and silver,” J. Phys. Chem. C 113, 3041 (2009).
[Crossref]

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photon. 3, 216 (2009).
[Crossref]

R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photon. 3, 696 (2009).
[Crossref]

2008 (2)

M. Francoeur, M. P. Mengüc, and R. Vaillon, “Near-field radiative heat transfer enhancement via surface phonon polaritons coupling in thin films,” Appl. Phys. Lett. 93, 043109 (2008).
[Crossref]

S. Franzen, “Surface plasmon polaritons and screened plasma absorption in indium tin oxide compared to silver and gold,” J. Phys. Chem. C 112, 6027 (2008).
[Crossref]

2007 (3)

R. Kitamura, L. Pilon, and M. Jonasz, “Optical constants of silica glass from extreme ultraviolet to far infrared at near room temperature,” Appl. Opt. 46, 8118 (2007).
[Crossref] [PubMed]

A. W. Rodriguez, M. Soljačič, J. D. Joannopoulos, and S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Exp. 15, 7303 (2007).
[Crossref]

J. Greffet and C. Henkel, “Coherent thermal radiation,” Contemp. Phys. 48, 183 (2007).
[Crossref]

2006 (2)

Y. de Wilde, F. Formanek, R. Carminati, B. Gralak, P. Lemoine, K. Joulain, J. Mulet, Y. Chen, and J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nat. 444, 740 (2006).
[Crossref]

A. H. Sihvola, “Peculiarities in the dielectric response of negative-permittivity scatterers,” Prog. Electromagn. Res. 66191 (2006).
[Crossref]

2005 (1)

K. Joulain, J. Mulet, F. Marquier, R. Carminati, and J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57, 59 (2005).
[Crossref]

2004 (2)

M. Soljačič and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211 (2004).
[Crossref]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847 (2004).
[Crossref] [PubMed]

2003 (1)

A. Zakery and S. R. Elliott, “Optical properties and applications of chalcogenide glasses: a review,” J. Non-Cryst. Solids 330, 1 (2003).
[Crossref]

2002 (1)

1999 (2)

D. Cotter, R. J. Manning, K. J. Blow, A. D. Ellis, A. E. Kelly, D. Nesset, I. D. Phillips, A. J. Poustie, and D. C. Rogers, “Nonlinear optics for high-speed digital information processing,” Science 286, 1523 (1999).
[Crossref] [PubMed]

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845 (1999).
[Crossref]

1998 (1)

S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. 288, 243 (1998).
[Crossref]

1991 (1)

J. R. Sambles, G. W. Bradbery, and F. Yang, “Optical excitation of surface plasmons: an introduction,” Contemp. Phys. 32, 173 (1991).
[Crossref]

1985 (1)

C. G. Granqvist, “Spectrally selective coatings for energy efficiency and solar applications,” Phys. Scripta 32, 401 (1985).
[Crossref]

1984 (1)

W. Eckhardt, “Macroscopic theory of electromagnetic fluctuations and stationary radiative heat transfer,” Phys. Rev. A 29, 1991 (1984).
[Crossref]

1978 (1)

L. G. Aio, A. M. Efimov, and V. F. Kokorina, “Refractive index of chalcogenide glasses over a wide range of compositions,” J. Non-Cryst. Solids 27, 299 (1978).
[Crossref]

1975 (1)

M. I. Dykman, “Theory of nonlinear nonequilibrium oscillators interacting with a medium,” Zh. Eksp. Theor. Fiz 68, 2082 (1975).

1971 (1)

D. Polder and M. Van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4, 3303 (1971).
[Crossref]

Aggarwal, I. D.

Aio, L. G.

L. G. Aio, A. M. Efimov, and V. F. Kokorina, “Refractive index of chalcogenide glasses over a wide range of compositions,” J. Non-Cryst. Solids 27, 299 (1978).
[Crossref]

Ajayan, P. M.

Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. G. de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7, 2388 (2013).
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K. Chen, P. Santhanam, S. Sandhu, L. Zhu, and S. Fan, “Heat-flux control and solid-state cooling by regulating chemical potential of photons in near-field electromagnetic heat transfer,” Phys. Rev. B 91, 134301 (2015).
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ACS Nano (1)

Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. G. de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7, 2388 (2013).
[Crossref] [PubMed]

AIP Adv. (1)

B. Song, A. Fiorino, E. Meyhofer, and P. Reddy, “Near-field radiative thermal transport: From theory to experiment,” AIP Adv. 5, 053503 (2015).
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Appl. Opt. (1)

Appl. Phys. Lett. (3)

M. Francoeur, M. P. Mengüc, and R. Vaillon, “Near-field radiative heat transfer enhancement via surface phonon polaritons coupling in thin films,” Appl. Phys. Lett. 93, 043109 (2008).
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C. Khandekar, Z. Lin, and A. W. Rodriguez, “Thermal radiation from optically driven Kerr χ(3) photonic cavities,” Appl. Phys. Lett. 106, 151109 (2015).
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Chem. Phys. Lett. (1)

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Chem. Rev. (1)

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Comput. Phys. Comm. (1)

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Int. J. Energy Res. (1)

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Figures (5)

Fig. 1
Fig. 1 (a) Schematic of three equal-temperature T resonators supporting modes at ωj, each with decay rate γj stemming from internal dissipation γjd and/or radiation into an external channel γjc, with j = {1, 2, 3}. The thermal modes 1 and 3 are strongly coupled to one another through a four-wave mixing process involving a Kerr χ(3) medium excited by externally incident light of power P from a waveguide that resonantly couples to the mode at ω2 = 1/2(ω3ω1). Nonlinear mixing between modes 1 and 3 can be described by an effective, linear but power-dependent coupling κ [Eq. (7)] obtained via a spatial overlap between the three modes, given in Eq. (4). (b) Thermal emission spectra Pj (ω) (blue and red curves) normalized by kBT, for a choice of far-apart frequencies ω1 = kBT/ħ and ω3 = 10ω1, and decay rates γjd = γjc = 0.01ω1, as a function of the dimensionless frequency ħω/kBT. Emission at any ω is bounded above by the Planck distribution Θ(ω, T) (black curve) in the absence of the pump κ = 0 (blue curves), but is exponentially enhanced under finite κ > 0 (red curves). (c) Thermal emission P1(ω) (solid lines) and heat-transfer Pex(ω) (dotted lines) spectra near ω1 as a function of the dimensionless frequency (ωω1)/γ1, demonstrating splitting of the resonances into Stokes (red-shifted) and anti-Stokes (blue-shifted) peaks, which grow apart with increasing κ and further enhance emission.
Fig. 2
Fig. 2 (a) Schematic of two plates consisting of different materials held at temperatures T1 and T3 and which support surface plasmon polaritons (SPPs) at far-apart resonant frequencies ω1 and ω3. Also shown is a schematic of the corresponding SPP dispersions ωj (k) (blue and red). Thermal upconversion and transfer of energy from ω1 to ω3 is facilitated by a mediator mode at ω 2 ~ ( ω 3 ω 1 ) 2 (green). (b) Table summarizing three different geometric configurations that result in significant thermal upconversion, along with various possible choices of emitting, absorbing, and nonlinear materials. The main difference between configurations is the choice of intervening medium, which consist of either a (i) lattice of nanoparticles embedded in the nonlinear medium or (ii) bulk nonlinear thin film, and serves as a tunable means to enhance the incident light.
Fig. 3
Fig. 3 (a) Schematic of a planar system of SiC and K slabs at thermal equilibrium (room temperature) separated by a gap of size d = 60nm that is filled with a rectangular lattice of nanospheres with unit-cell size Λx × Λy embedded in a χ(3) nonlinear medium (ChG). The SiO2/Au core/shell nanoparticles have core/shell radii of 15nm and 20nm, respectively, and the dimensions of the unit cell are Λx = 5R and Λy = πc/ω2. The SiC and K slabs support SPPs at frequencies ω1 and ω3, respectively, which couple nonlinearly to dipolar particle resonances at ω 2 ~ 1 2 ( ω 3 ω 1 ) excited by a laterally incident, monochromatic, z ^ -polarized planewave of frequency ω2 and intensity I. (b) xy and yz cross sections of the particle resonances |E2|2, with yellow/black denoting maximum/zero amplitude. (c) Normalized slab mode-profiles at a representative wavenumber kd = 2, with the shaded region indicating the intervening medium. (d) Variations in the nonlinear coupling κ, frequency mismatch Δω, and frequency ω1 with respect to kd, normalized by the corresponding dissipation rate γ3 of the K slab. (e) Frequency-integrated heat-extraction spectrum Pex(k), normalized by P0 = 2γ1Θ(ω1, T), as a function of kd and for multiple incident intensities I. Also shown are the (f) associated spectral density Φex(ω) and (g) net extracted and upconverted flux rates, Hex and Hup, as a function of I. For comparison, (e–g) also show the heat-transfer rates associated with two vacuum-separated SiC slabs held at either 300K (light blue) or 1K (dark blue) temperature differences.
Fig. 4
Fig. 4 (a) Schematic of a planar system of SiC and ITO slabs at thermal equilibrium (room temperature) separated by a gap of size d = 60nm that is filled with a square lattice of doped graphene nanodisks with unit-cell size Λ × Λ embedded in a χ(3) nonlinear medium (ChG). The graphene nanodisks have radius R = 20nm, Fermi energy EF = 0.7eV and are placed at a distance of 10nm from ITO slab. The SiC and ITO slabs support SPPs at frequencies ω1 and ω3, respectively, which couple nonlinearly to nanodisk resonances at ω 2 ~ 1 2 ( ω 3 ω 1 ) excited by a normally incident, monochromatic, x ^ + i y ^ -polarized planewave of frequency ω2 and intensity I. (b) xy and yz cross sections of the particle resonances |E2|2, with yellow/black denoting maximum/zero amplitude. (c) Normalized slab mode-profiles at a representative wavenumber kd = 1, with the shaded region indicating the intervening medium. (d) Variations in the nonlinear coupling κ, frequency mismatch Δω, and frequency ω1 with respect to kd, normalized by the corresponding dissipation rate γ3 of the ITO slab. (e) Frequency-integrated heat-extraction spectrum Pex(k), normalized by P0 = 2γ1Θ(ω1, T), as a function of kd and for multiple incident intensities I. Also shown are the (f) associated spectral density Φex(ω) and (g) net extracted and upconverted flux rates, Hex and Hup, as a function of I. For comparison, (e–g) also show the heat-transfer rates associated with two vacuum-separated SiC slabs held at either 300K (light blue) or 1K (dark blue) temperature differences.
Fig. 5
Fig. 5 (a) Schematic of a planar system of SiC and K slabs at thermal equilibrium (room temperature) separated by a gap of size d = 60nm that is filled with χ(3) nonlinear medium (ChG). The SiC and K slabs support SPPs at frequencies ω1 and ω3, respectively, which couple nonlinearly to a SPP resonance at ω 2 ~ 1 2 ( ω 3 ω 1 ) that is excited via a grating (of period Λ) by monochromatic light of frequency ω2, intensity I, and wavevector k2y incident at an angle θinc with respect to the z ^ direction. (b) Schematic illustration of the coupling of SPP resonances of wavevectors k1 and k3; the polar plot shows the directional dependence of the nonlinear coupling coefficient κ(k, θ), where k1 = (k, θ) is expressed in polar coordinates, with θ denoting the angle extended by k1 with respect to y ^. (c) Normalized slab mode-profiles at a representative wavenumber kd = 2 and angle θ = 0, with the shaded region indicating the intervening medium. (d) Variations in the nonlinear coupling κ, frequency mismatch Δω, and frequency ω1 with respect to kd, normalized by the corresponding dissipation rate γ3 of the K slab. (e) Frequency-integrated, angle-averaged heat-extraction spectrum Pex(k), normalized by P0 = 2γ1Θ(ω1, T), as a function of kd and for multiple incident intensities I. Also shown are the (f) associated spectral density Φex(ω) and (g) net extracted and upconverted flux rates, Hex and Hup, as a function of I. For comparison, (e–g) show the heat-transfer rates associated with two vacuum-separated SiC slabs held at either 300K (light blue) or 1K (dark blue) temperature differences.

Equations (20)

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d a 1 d t = ( i ω 1 γ 1 ) a 1 i β ω 1 a 3 ( a 2 * ) 2 + 2 γ 1 d ξ 1
d a 2 d t = ( i ω 2 γ 2 ) a 2 i β ω 2 a 3 a 1 * a 2 * + 2 γ 2 d ξ 2 + 2 γ 2 c s in
d a 3 d t = ( i ω 3 γ 3 ) a 3 i β * ω 3 a 1 a 2 2 + 2 γ 3 d ξ 3
β = d V χ i j k l ( 3 ) E 1 i E 2 j E 2 k E 3 l * 2 ε 0 d V ( ω ε ) ω E 1 i * E 1 i d V ( ω ε ) ω E 3 i * E 3 i d V ( ω ε ) ω E 2 i * E 2 i ,
d a 2 d t = ( i ω 2 γ 2 ) a 2 + 2 γ 2 c s in .
d a 1 d t = ( i ω 1 γ 1 ) a 1 i κ e 2 i ω 2 t a 3 + 2 γ 1 d ξ 1
d a 3 d t = ( i ω 3 γ 3 ) a 3 i ω 3 ω 1 κ * e 2 i ω 2 t a 1 + 2 γ 3 d ξ 3 ,
κ = 2 β ω 1 γ 2 c P γ 2 2 .
P ex ( ω ) = 4 | κ | 2 D 1 ( ω ) [ γ 1 γ 3 d Θ ( ω + 2 ω 2 , T 3 ) + γ 1 d γ 3 ( ω 3 / ω 1 ) Θ ( ω , T 1 ) ]
P 1 ( ω ) = 4 γ 1 c D 1 ( ω ) [ γ 1 d | i ( ω ω 1 Δ ω ) + γ 3 | 2 Θ ( ω , T 1 ) + γ 3 d | κ | 2 Θ ( ω + 2 ω 2 , T 3 ) ]
P 3 ( ω ) = 4 γ 3 c D 3 ( ω ) [ γ 3 d | i ( ω ω 3 Δ ω ) + γ 1 | 2 Θ ( ω , T 3 ) + γ 1 d ( ω 3 / ω 1 ) | κ | 2 Θ ( ω 2 ω 2 , T 1 ) ]
ω 1 ± = ω 1 + Δ ω ± Δ ω 2 + 4 ( γ 1 γ 3 + ω 3 / ω 1 | κ | 2 ) 2 ω 3 ± = ω 1 ± ( 1 3 , Δ ω Δ ω )
e 2 i κ z 2 d = ( ϵ 1 κ z 2 ϵ 2 κ z 1 ) ( ϵ 3 κ z 2 ϵ 2 κ z 3 ) ( ϵ 2 κ z 3 + ϵ 3 κ z 2 ) ( ϵ 1 κ z 2 + ϵ 2 κ z 1 )
β ( k 1 , k 3 ) = p d V χ i j k ( 3 ) e i ( k 3 k 1 ) . x E 1 i ( z ) E 2 j ( x x p , z ) E 2 k ( x x p , z ) E 3 * ( z ) 2 ϵ 0 ( p d V ϵ ω ω | E 2 ( x x p , z ) | 2 ) ( d V ϵ ω ω | E 1 ( z ) | 2 ) 1 / 2 ( d V ϵ ω ω | E 3 ( z ) | 2 ) 1 / 2
H = 0 d ω 2 π 0 d ω 2 π k P ex ( ω , k ) Φ ex ( ω )
κ ( k ) = ω 1 σ abs I cell χ i j k ( 3 ) E 1 i ( z ) E 2 j ( x x p , z ) E 2 k ( x x p , z ) E 3 * ( z ) 4 ϵ 0 γ 2 d Λ x Λ y ( cell ϵ ω ω | E 2 ( x x p , z ) | 2 ) ( d z ϵ ω ω | E 1 ( z ) | 2 ) 1 / 2 ( d z ϵ ω ω | E 3 ( z ) | 2 ) 1 / 2
H ( 0 ) = 0 d ω 2 π [ Θ ( ω , T 1 ) Θ ( ω , T 3 ) ] 0 d k k 2 Im { r 21 p } Im { r 23 p } e 2 Im { k z 2 } d π | 1 r 21 p r 23 p e 2 Im { k z 2 } d | 2 Φ ( 0 ) ω )
ϵ 2 ω 2 sin ( θ inc ) c + 2 π Λ = k 2 y ,
β ( k 1 , k 3 ) = d V χ i j k ( 3 ) e i ( k 3 k 1 2 k 2 y y ^ ) . x E 1 i ( z ) E 2 j ( ω 2 , z ) E 2 k ( ω 2 , z ) E 3 * ( z ) 2 ϵ 0 ( d V ϵ ω ω | E 2 ( ω 2 , z ) | 2 ) ( d V ϵ ω ω | E 1 ( z ) | 2 ) 1 / 2 ( d V ϵ ω ω | E 3 ( z ) | 2 ) 1 / 2
κ ( k , θ ) = ω 1 γ 2 c I d z χ i j k ( 3 ) E 1 i ( ω 1 , z ) E 2 j ( ω 2 , z ) E 2 k ( ω 2 , z ) E 3 * ( ω 3 , z ) ϵ 0 γ 2 2 d z ϵ ω ω | E 2 ( ω 2 , z ) | 2 ( d z ϵ ω ω | E 1 ( ω 1 , z ) | 2 ) 1 / 2 ( d z ϵ ω ω | E 3 ( ω 3 , z ) | 2 ) 1 / 2

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