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Degeneracy mediated thermal emission from nanoscale optical resonators with high-order symmetry

  • Zexiao Wang ORCID logo , Jiayu Li , Zhuo Li , Xiu Liu , Yibai Zhong , Tianyi Huang and Sheng Shen EMAIL logo
Published/Copyright: November 27, 2024
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Nanophotonics
From the journal Nanophotonics Volume 13 Issue 26

Abstract

Conventional thermal emitters, such as a blackbody or the incandescent filament of a light bulb, lack the directionality or narrow linewidth required in many applications such as thermophotovoltaics and infrared sensing. Although thermal emission from bulk materials is well understood based on phenomenological heat transfer concepts like emissivity and the framework of classical electrodynamics, there still remains a significant gap in our understanding of thermal emission at the nanoscale. In this work, by leveraging the quasi-normal mode theory, we derive a general and self-consistent formalism to describe the thermal radiation from nanoscale resonant thermal emitters with high-order symmetric geometries, which are the basic building blocks of metasurfaces and metamaterials. The complex symmetrical geometries of the emitters yield degeneracy of quasi-normal modes. The introduction of the degeneracy can strongly mediate far-field thermal emission from nanoscale resonators, which is closely correlated to the number of degenerate modes and the coupling among the degenerate modes. Our formalism from the quasi-normal mode theory serves as a general guideline to design the complex metastructures with high-ordered degeneracy to achieve optimized absorption or emission capabilities.

Keywords: thermal radiation; quasi-normal mode theory; group theory; nanoscale thermal emitter; fluctuational electrodynamics

1 Introduction

The manipulation of thermal radiation from a nanoscale thermal emitter plays a vital role in energy-, and sensing-related areas, including thermophotovoltaics [1], [2], [3], [4], radiative cooling [5], [6], [7], sensing [8], [9], [10] and infrared imaging [11], [12]. Since a resonant thermal emitter functions as a thermal heat source and an optical cavity simultaneously at the nanoscale, its thermal radiation can be dramatically modulated by its electromagnetic resonance mode according to the Purcell effect [13]. By engineering the localized density of states (LDOS) of a nanoscale thermal emitter [14], its emission spectrum can be extremely narrow-band in both far- and near-fields compared to the broad-band spectrum of a blackbody. One paradigm for quantitatively designing such thermal optical resonators is to employ the quasi-normal modes (QNM) which can exactly depict the non-Hermitian nature of these thermal emitters [15], [16], [17], [18]. Under the assumption of a single predominant mode based on the QNM theory, the fundamental limit of the spectral thermal emission power from an optical resonator is derived by matching the resistive and radiative fractional mode losses of the optical resonator. However, as the geometry of a nanoscale thermal emitter becomes non-trivial, degenerate modes emerge when the symmetry group of the thermal emitter is promoted to a higher order, as shown in Figure 1 [18], [19]. In addition, multiple resonant modes can be spectrally bundled around the same frequency, leading to highly complex physical behaviors. In terms of degeneracy due to the non-trivial symmetry group, coupled-mode theory (CMT) [19] has been applied to account for the coupling among degenerate modes by a phenomenological coupling parameter. A comprehensive CMT-based theoretical framework has been established [20] on a group of tightly bundled emitters, which predicted and explained the unique enhancing or suppressing effect of thermal emission due to degenerate coupling. Yet, regarding the more complicated multi-mode emitter structures, the upper limit of the degree of degeneracy, the physical mechanism of the coupling among the degenerate modes, and their influence on far-field thermal radiation still need further exploration.

Figure 1: Degenerate resonant modes of thermal emitters with high-order symmetry.
Figure 1:

Degenerate resonant modes of thermal emitters with high-order symmetry.

In this work, we develop a general and self-consistent theoretical framework from the QNM theory and fluctuational electrodynamics to describe the thermal radiation from resonant thermal emitters with higher order symmetric geometries. By applying group theory [21], we identify the upper limit to the degree of degeneracy of a resonant thermal emitter given its geometry. The far-field thermal radiation from emitters supporting a set of degenerate QNMs is explicitly formulated with a revised expansion of the dyadic Green’s function [22]. We find that the existence of degeneracy strongly mediates far-field thermal radiation of a resonant thermal emitter, yet the non-zero coupling among degenerate modes turns out to be detrimental to far-field thermal emission. When quasi-static approximation is valid for the QNMs in a resonant thermal emitter, we show that multiple spectrally close QNMs contribute to far-field thermal radiation independently. When the size of the thermal emitter is no longer in the deep-subwavelength scale, the coupling between spectrally close QNMs compromises the far-field thermal radiation. By designing thermal emitters with higher-order symmetry groups, the far-field thermal radiation intensity at certain resonance frequencies can be significantly enhanced as compared to the single-mode emitter when normalized to the emitting volume. This degeneracy based enhancement effect thus provides a unique way of boosting thermal radiation from a resonant emitter. Following the theoretical framework proposed here, narrow-band thermal emitters with more compact size and higher emission power can be designed, which is beneficial for increasing the efficiency of thermophotovoltaic emitters and sensitivity of infrared sensors [23], [24].

2 Theory and methods

2.1 QNM theory for far-field thermal radiation from degenerate resonant thermal emitters

For an optical resonator in vacuum, its resonant mode profile and frequency are described through the eigen-solutions of Maxwell equation:

(1) ∇ × E n r = i ω n μ r , ω H n r , ∇ × H n r = − i ω n ε r , ω E n r .

Given the geometry and optical properties of a resonant structure, the resonant modes can then be obtained by solving the eigen-value problem M ̂ ϕ n r = ω n ϕ n r . , where ϕ n is the nth eigen-mode of the corresponding structure with resonant frequency ω n , and the Maxwell operator M ̂ can be defined as

(2) M ̂ = 0 i ε − 1 r , ω ∇ × − i μ − 1 r , ω ∇ × 0 .

In the regime of linear optics, the Maxwell operator is a linear operator. Finding a set of eigen-modes can be useful in expanding the Green’s function of the Maxwell equation and therefore acquiring the electromagnetic response of the given structure. In most cases, such eigen-modes are presumed to be non-degenerate: one resonant frequency ω n only corresponds to one mode configuration ϕ n .

The introduction of symmetry groups to the Maxwell operator leads to degeneracy in resonance modes. Assume that the symmetry group of the resonator structure’s Maxwell operator has the order of n, and the Maxwell operator commutes with the symmetry operator O ̂ , that is, M ̂ , O ̂ = M ̂ O ̂ − O ̂ M ̂ = 0 . For a certain eigen-mode of the Maxwell operator ϕ n with the corresponding eigenvalue ω n , the new mode ϕ n g produced by the symmetry group operation g ̂ is also the solution to the eigenvalue problem, M ̂ g ̂ ϕ n = g ̂ M ̂ ϕ n = g ̂ ω n ϕ n = ω n g ̂ ϕ n = ω n ϕ n g .

As proven by group theory, the upper limit of the degenerate modes for a certain eigen-value is given by the dimension of the largest irreducible representation of the symmetry group [21], [22]. It is worth noting that the upper limit of degeneracy predicted by group theory explicitly excludes accidental degeneracy. Although the largest number of the degenerate modes can be dictated based on the symmetry group that the Maxwell operator has, it is not guaranteed that all the degenerate modes can be found at a certain resonance frequency. The exact spectrum of the eigen-values still relies on the details of a certain physical system.

In the scenario of lossy resonators, the entire optical system is non-Hermitian. The resonant modes usually have higher losses and therefore reduced quality factors. The corresponding eigen-frequency of the resonant mode becomes a complex value, where the real part indicates the position of the resonant frequency, and the imaginary part represents the loss of the mode. Such a set of modes are called QNMs E n r , H n r . The Dyadic Green’s function of the resonator G ̄ ω , r , r ′ , which describes the impulse response of the Maxwell equation, can be expanded onto the QNM modes, G ̄ ω , r , r ′ = ∑ n E n r ⋅ E n † r ′ ω μ 0 ω n − ω N n n . Here, E n is the electric field expressed as a 3-by-1 column vector, †denotes the conjugate-transpose, r and r′ are the spatial position of the observation and source points, respectively. N nm is the normalization factor corresponding to a QNM with resonant frequency ω n [13]:

(3) N n m = ∫ V ∞ d r 3 ∂ ω ε r , ω ∂ ω E n T ⋅ E m − ∂ ω μ ∂ ω H n T ⋅ H m .

Further taking degeneracy into consideration, the QNM expansion of the Green’s function can be revised as:

(4) G ̄ ω , r , r ′ = ∑ n ∑ g E n g r ⋅ E n g , † r ′ ω μ 0 ω n − ω N n n g ,

where g denotes the degenerate number and the expansion of G ̄ ω , r , r ′ needs to go through all possible degenerate modes. It can be proved that the normalization factor N n n g is same for all degenerate modes if the symmetry operator is unitary. The validity of Eq. (4) has been justified in our previous work [18].

Consider a thermal emitter V E at temperature T E and a closely separated object V A that are both placed in a vacuum. Our previous work [18] has discussed the QNM formulation for near-field thermal radiation between V E and V A , but here the second object V A is still preserved for the generalization of the derivation. By neglecting the terms related to V A , the far-field thermal radiation from V E can be accounted for within this framework. Here, the materials are assumed to be nonmagnetic with isotropic electrical response. Since the thermal radiation from V E is physically the emission of electromagnetic waves E r , ω , H r , ω generated by the thermally induced random currents j r , ω , T E inside V E , the spectral thermal energy transfer from V E to the far-field, ϕ ∞ ω equals the integration of the averaged Poynting vector over an enclosure surface ∂V. It can be further expressed by the total power from the current sources minus the near-field power absorption in both V E and V A according to energy conservation,

(5) ϕ ∞ ω = ∫ ∂ V d r 2 n ̂ 1 2 Re E × H † , ϕ ∞ ω = − ∫ V E d r 3 1 2 Re j † × E − ∫ V E d r 3 1 2 σ E E 2 − ∫ V A d r 3 1 2 σ A E 2 ,

where the first term on the right-hand side (RHS) is the total emission energy from the emitter; the second and third terms correspond to the resistive energy loss inside the emitter and receiver, respectively.

Based on the Dyadic Green’s function, the electric field emitted by the random currents can be represented as E r , ω = i ω μ ∫ V E d r ′ 3 G ̄ ω , r , r ′ ⋅ j r ′ , ω , T E . Meanwhile, the autocorrelation of the random currents j r ′ , ω , T E can be characterized by the fluctuation-dissipation theorem as j r ′ , ω , T E ⋅ j † r ′ , ω , T E = 4 π σ E Θ ω , T E δ r − r ′ I ̄ , where Θ ω , T E is the Planck distribution Θ ω , T E = ℏ ω / exp ℏ ω / k T − 1 and I ̄ is the 3-by-3 entity matrix. Therefore, the reduced spectral thermal emission ψ ∞ ω = ϕ ∞ / Θ / 2 π into the far-field can be further expressed as:

(6) ψ ∞ ω = 4 ω μ 0 Tr ∫ V E d r 3 σ E Im G ̄ ω , r , r ′ − 4 ω 2 μ 0 2 T r ∫ V E d r 3 ∫ V E d r ′ 3 σ E 2 G ̄ ω , r , r ′ † ⋅ G ̄ ω , r , r ′   − 4 ω 2 μ 0 2 T r ∫ V A d r 3 ∫ V E d r ′ 3 σ A σ E G ̄ ω , r , r ′ † ⋅ G ̄ ω , r , r ′ .

To classify the effect of degeneracy on far-field thermal radiation, we consider a set of degenerate resonant modes whose resonant frequency Re ω is well-separated from other modes. Thus, the Green’s function around this frequency can be approximated by Eq. (4), in which the first term on the RHS of Eq. (6) becomes:

(7) 4 ω μ 0 Tr ∫ V E d r 3 σ E Im G ̄ ω , r , r ′ = 4 ω μ 0 ∑ g ∫ V E d r 3 σ E Im E n g r 2 ω μ 0 ω n − ω N n n g ≈ 4 ∑ g L ω − 1 Im ω n × Re ∫ V E d r 3 σ E E n g r 2 N n n g .

According to the divergence theorem, it has ∫ ∂ V d r 2 E n g × H n g † + E n g † × H n g = − 2 ∫ V d r 3 H n g 2 Im ω n μ 0 + E n g 2 Im ω n ε ω n and Im ω n ε ω n = σ Re ω n + Im ω n Re ∂ ω ε ∂ ω ω = Re ω n + o Im ω n 2 . As a result,

(8) − 1 Im ω n = 1 2 ∫ V d r 3 Re ∂ ω ε ∂ ω ω n E n g 2 + μ 0 H n g 2 ∫ ∂ V d r 2 1 2 Re E n g × H n g † + ∫ V E + V A d r 3 1 2 σ Re ω n E n g 2

Then, the first term on the RHS of Eq. (6), which describes the total emission power from the emitter, can be finally expressed as 4 L ω ∑ g D n , E g D n , E g + D n , A g + D n , ∞ g F n g P n g , where L ω is the Lorentzian line shape function with the peak at degenerate resonant frequency Re ω n , and quality factor of Q n = Re ω n / 2 Im ω n . D n , x g = 1 2 ∫ V x d r 3 σ x Re ω n E n g 2 , where x ∈ E , A , represents the resistive energy loss inside the emitter and the receiver. The factors due to the non-Hermitian imperfection are given as P n g = R e 1 N n n g / 1 N n n g and F n g = 1 2 ∫ V d r 3 Re ∂ ω ε ∂ ω ω n E n g 2 + μ 0 H n g 2 / N n n g .

The second term on the RHS of Eq. (6), which describes the energy absorption inside the emitter, hence becomes:

(9)   4 ω 2 μ 0 2 T r ∫ V E d r 3 ∫ V E d r ′ 3 σ E 2 G ̄ ω , r , r ′ † ⋅ G ̄ ω , r , r ′ = L ω 4 Im ω n 2 ∑ g ∫ V E d r 3 ∫ V E d r ′ 3 σ E 2 E n g r ′ 2 E n g r 2 N n n g 2 + ∑ k ≠ g ∫ V A d r 3 ∫ V E d r ′ 3 σ E 2 E n k † r ⋅ E n g r ′ E n g † r ⋅ E n k r ′ N n n g * N n n k = 4 L ω ∑ g D n , E g D n , E g + D n , A g + D n , ∞ g F n g 2   + ∑ k ≠ g C n , E g k D n , E g + D n , A g + D n , ∞ g C n , E k g D n , E k + D n , A k + D n , ∞ k ,

where the index k iterates through all the degenerate modes and k ≠ g. The coupling term between the kth and gth degenerate modes is given as C n , E g k = ∫ V E d r ′ 3 σ E E n k † r ′ ⋅ E n g r ′ .

Similarly, the third term on the RHS of Eq. (6) describing the absorption by the receiver becomes:

(10) 4 ω 2 μ 0 2 T r ∫ V A d r 3 ∫ V E d r ′ 3 σ A σ E G ̄ ω , r , r ′ † ⋅ G ̄ ω , r , r ′ = 4 L ω ∑ g D n , E g D n , E g + D n , A g + D n , ∞ g F n g × D n , A g D n , E g + D n , A g + D n , ∞ g F n g + ∑ k ≠ g C n , A g k D n , E g + D n , A g + D n , ∞ g F n g × C n , E k g D n , E k + D n , A k + D n , ∞ k F n k .

If the symmetry operator that generates the degenerate modes is a real unitary operator, for example, the rotation matrix in a two-dimensional space, it can be proved that D n , γ g = D n , γ k , F n g = F n k , P n g = P n k and C n , E g k = C n , E k g * , where γ represents emitter V E , absorber V A or far-field ∞. Then, through substituting Eqs. (7), (9) and (10) into Eq. (6), the far-field thermal radiation from the emitter can be simplified as:

(11) ψ ∞ ω = 4 L ω ∑ g η n , E g η n , ∞ g − ∑ k ≠ g γ n , A g k + γ n , E g k γ n , E k g ,

where η n , E g and η n , ∞ g are the fractional mode losses in the emitter and to the far-field given by η n , E = F n D n , E / D n , E + D n , A + D n , ∞ and η n , ∞ = P n − F n D n , E + D n , A / D n , E + D n , A + D n , ∞ , γ n , A g k and γ n , E g k are the fractional mode coupling terms in the absorber and emitter given by γ n , A g k = F n C n , A g k / D n , E + D n , A + D n , ∞ and γ n , E g k = F n C n , E g k / D n , E + D n , A + D n , ∞ . In the absence of the near-field absorber, since γ n , E g k ⋅ γ n , E k g ≥ 0 always hold, the overall far-field thermal radiation from a set of degenerate modes is compromised due to the coupling term among them.

Recall that C n , E g k = ∫ V E d r ′ 3 σ E E n k † r ′ ⋅ E n g r ′ . Since we only consider the degenerate modes that result from the symmetry operation g ̂ of the Maxwell operator, E g r can be written as E g r = g ̂ E k r = g ̃ E k g ̃ − 1 r , where the first g ̃ shuffles the components of the electric field and the second g ̃ − 1 operates on the coordinates of the electric field. The coupling between different degenerate modes now becomes C n , E g k = ∫ V E d r ′ 3 σ E E n k † r ′ ⋅ g ̃ E n k g ̃ − 1 r ′ . As long as the mode configuration stays the same, we can claim that ∫ V E d r ′ 3 σ E E n k † r ′ ⋅ g ̃ E n k g ̃ − 1 r ′ = β ∫ V E d r 3 σ E E n k † r ′ ⋅ E n k r ′ , where β ≥ 0 denotes the coupling coefficient. Consequently, the far-field thermal radiation from a set of degenerate QNMs in the absence of the near-field absorber are expressed as:

(12) ψ ∞ ω = 4 L ω ∑ g η n , E g η n , ∞ g − ∑ j β j 2 η n , E g 2 ,

where the index of coupling coefficient β j ranges from one to g g − 1 / 2 . Furthermore, we can obtain the upper limit for far-field thermal radiation given the constraint that η n,E + η n,∞ = P. When η n , E = P / 2 1 + ∑ j β j 2 , η n , ∞ = 1 + 2 ∑ j β j 2 P / 2 1 + ∑ j β j 2 , the maximum far-field thermal radiation from g degenerate QNMs is achieved as:

(13) ψ ∞ ω = L ω g P 2 1 + ∑ j β j 2 .

2.2 QNM ‘bundling’ effect

In the previous section, the QNMs are assumed to be well-isolated from each other. However, the QNMs tend to locate or bundle around the same frequency when the geometry of the emitter has higher order symmetry groups. Suppose that there are two different but closely separated non-degenerate QNMs with resonant frequencies ω 1 and ω 2, respectively. The Green’s function at frequency ω which is close to ω 1 and ω 2 needs to be written as the sum of contributions from two QNMs, namely, G ̄ ω , r , r ′ = ∑ p = 1,2 E n ω p , r ⋅ E n † ω p , r ′ ω μ 0 ω p − ω N p p . After a similar mathematical manipulation, the far-field thermal radiation in this case can be expressed by two terms. The first term is simply the contribution from each QNM alone, while the second term indicates the coupling of the two closely separated QNMs. The coupling term is directly proportional to the integral ∫ V E d r 3 E 1 † r ⋅ E 2 † r evaluated at frequency ω within the emitter. Recall that the normalization factor for the QNMs is defined as N n m = ∫ V ∞ d r 3 ∂ ω ε r , ω ∂ ω E n T ⋅ E m − ∂ ω μ ∂ ω H n T ⋅ H m . Under the quasi-static condition, one can choose the integral volume V to only enclose the near-field components of the resonant mode. The portion of the resonant mode outside V behaves like propagating wave and therefore satisfies ∂ ω ε ∂ ω E n T ⋅ E m = ε E n T ⋅ E m ≈ μ H n T ⋅ H m . The portion of the resonant mode inside V behaves to be quasi-static, where H n = 1 i ω μ ∇ × E n ≈ 0 . Thus, N n m ≈ ∫ V d r 3 ∂ ω ε r , ω ∂ ω E n T ⋅ E m ≈ δ n m . Since the quasi-static electric field has a real value, the coupling term satisfies that ∫ V d r 3 E n T ⋅ E m ≈ ∫ V d r 3 E n † ⋅ E m . Finally, under the quasi-static approximation of the QNMs, the coupling term can be neglected, and the far-field thermal radiation from a bundle of closely-separated QNMs is a sum of the contribution from each QNM. It should be noted that if the quasi-static approximation no longer holds, the coupling of the bundled QNMs will compromise the far-field thermal emission. In this scenario, one can still explicitly evaluate the coupling between the QNMs in terms of their mode profiles.

3 Results and discussion

Here, we give three examples of degeneracy mediated far-field thermal radiation from nanoscale thermal emitters. The first example is the nanocross structure composed of two single nanorod emitters. As a building block, each nanorod has a length of 2.5 μm with a 40 nm × 40 nm square shape cross-section. The Drude model of ε ω = 1 − ω p 2 / ω 2 − i ω ⋅ ω c is selected as the dielectric function, with ω p = 1.37 × 1016 rad/s and ω c = 5.32 × 1013 rad/s corresponding to those of gold [17]. In this architecture, the cross structure has the D4h symmetry group where the largest dimension of its irreducible representation is 2. Therefore, there are up to two degenerate modes at the same resonant frequency. Here, a direct numerical approach based on fluctuating surface current (FSC) [25], [26] is utilized to calculate the far-field thermal emission spectrum of the nano-cross [27]. The resulting spectrum is plotted in Figure 2(a), with the spectrum of a single nanorod shown as a reference. Note that at the first resonance from the left, we observe an emission peak from the cross structure that is two times larger than that from the single nanorod emitter. For the third resonance from the left, there is a strong emission peak with an intensity almost three times larger than that of the single nanorod emitter.

Figure 2: Far-field thermal emission spectrum from a nanocross structure with D4h symmetry. (a) Calculated far-field thermal emission spectrum of the nanocross emitter compared with a single nanorod from FSC; (b) predicted thermal radiation spectrum with the contribution of each mode from our QNM theory, versus calculated spectrum for the nanocross structure from FSC.
Figure 2:

Far-field thermal emission spectrum from a nanocross structure with D4h symmetry. (a) Calculated far-field thermal emission spectrum of the nanocross emitter compared with a single nanorod from FSC; (b) predicted thermal radiation spectrum with the contribution of each mode from our QNM theory, versus calculated spectrum for the nanocross structure from FSC.

To provide a detailed analysis on the contributing modes, eigenmode analysis is conducted on the cross emitter through COMSOL Multiphysics with the wave optics module and the built-in eigenfrequency solver. Due to the relatively large mesh and memory requirement for eigenfrequency calculation, the computation is executed with the Bridges-2 supercomputer at Pittsburgh Supercomputing Center [28]. The extracted mode complex frequencies corresponding to the second and third emission peaks are summarized in Table 1, together with the mode fractional losses, degeneracy, and coupling terms evaluated from the mode profiles based on Eq. (11). The result indicates that, the third peak is the combination of one doubly degenerate asymmetric mode and one non-degenerate symmetric mode, with their resonant frequencies around 8.066 × 1014 rad/s and 8.157 × 1014 rad/s, respectively. The calculation on the mode parameters shows that, the doubly degenerate modes emerging at the third resonant peak have negligible coupling between the degenerate modes, with the coupling term γ = 0.014. The existence of three non-coupling modes for the third peak explains its three-fold enhancement versus the single nanorod structure. Plugging the mode parameters into Eq. (12) gives the thermal emission spectrum predicted by our QNM theory, as shown in Figure 2(b), which indicates excellent agreement with the direct FSC calculation.

Table 1:

Mode parameters of resonant modes around the 2nd and 3rd peak of the nanocross emitter.

Mode # ω (1014 rad/s) η E η ∞ g γ
2nd-1 6.042 + 0.212i 0.484 0.515 1 –
3rd-1 8.066 + 0.220i 0.475 0.523 2 0.014
3rd-2 8.157 + 0.412i 0.310 0.669 1 –

The second example consists of three nanorods arranged in a 3-dimensional (3-D) cross structure with the symmetry group of Th, as shown in Figure 3(a). With the increased dimension of irreducible representation, this structure supports degeneracy up to 3. Figure 3(a) shows the calculated far field emission spectrum for the 3-D cross structure by FSC, compared with that of a single nanorod emitter. The third peak of reduced heat flux for the 3-D cross structure reaches 0.75, almost 5 times as large as the intensity of the single nanorod emitter. On the contrary, the intensity of the second mode is largely suppressed to below 0.1.

Figure 3: Far-field thermal emission spectrum from a 3-D nanocross structure with Th symmetry. (a) Calculated far-field thermal emission spectrum of the Th 3-D nanocross emitter compared with a single nanorod from FSC; (b) predicted thermal radiation spectrum with the contribution of each mode from our QNM theory, versus calculated spectrum for the 3-D nanocross structure from FSC.
Figure 3:

Far-field thermal emission spectrum from a 3-D nanocross structure with Th symmetry. (a) Calculated far-field thermal emission spectrum of the Th 3-D nanocross emitter compared with a single nanorod from FSC; (b) predicted thermal radiation spectrum with the contribution of each mode from our QNM theory, versus calculated spectrum for the 3-D nanocross structure from FSC.

The eigenmode analysis from COMSOL Multiphysics reveals the QNM composition for the second and third peaks. The calculated mode parameters are listed in Table 2, where the third peak consists of one triply degenerate mode around 8.077 × 1014 rad/s, and a doubly degenerate mode at 8.159 × 1014 rad/s. Both the degenerate modes have negligible coupling intensity around 0.01. Thus, the mode coupling does not play a significant role in the far field emission, resulting in the emission enhancement approaching 5-folds for the third peak. Besides, the second mode for the 3-D cross structure is a dark mode with radiation loss only η ∞ = 0.083, indicating that the introduction of the extra nanorod suppresses the electromagnetic emission perpendicular to the cross structure. This explains the reduced emission intensity for the second peak compared to the 2-D cross and single nanorod.

Table 2:

Mode parameters of resonant modes around the 2nd and 3rd peak of the 3-D nanocross emitter.

Mode # ω (1014 rad/s) η E η ∞ g γ
2nd-1 6.559 + 0.115i 0.902 0.083 1 –
3rd-1 8.077 + 0.221i 0.474 0.523 3 0.008
3rd-2 8.159 + 0.413i 0.310 0.669 2 0.014

Finally, a star-shape emitter consisting of three co-planar single nanorods is further discussed in Figure 4. For this degenerate structure consisting of 3 equal-length nanorods arranged in the same plane with 120° rotation interval, it has the D6h symmetry group where the largest dimension of its irreducible representation is still 2. FSC is used to calculate the far field emission spectrum, as shown in Figure 4(a). The third emission peak around 8.0 × 1014 rad/s shows around 4-fold enhancement compared with the single nanorod structure, which is smaller than the 3-D cross structure with the Th symmetry.

Figure 4: Far-field thermal emission spectrum from a star-shape structure with D6h symmetry. (a) Calculated far-field thermal emission spectrum of the star-shape emitter compared with a single nanorod from FSC; (b) predicted thermal radiation spectrum with the contribution of each mode from our QNM theory, versus calculated spectrum for the star-shape structure from FSC.
Figure 4:

Far-field thermal emission spectrum from a star-shape structure with D6h symmetry. (a) Calculated far-field thermal emission spectrum of the star-shape emitter compared with a single nanorod from FSC; (b) predicted thermal radiation spectrum with the contribution of each mode from our QNM theory, versus calculated spectrum for the star-shape structure from FSC.

Here, we introduce non-zero coupling terms γ E g k to fully understand the emission behavior of the D6h structure. By calculating the QNMs with COMSOL Multiphysics, we identify one single QNM for the second peak, and three bundles of QNMs for the third peak, which includes one non-degenerate mode and two groups of doubly degenerate modes. Their corresponding mode parameters are summarized in Table 3. The predicted spectrum from our QNM theory is then plotted and compared with the direct FSC calculation in Figure 4(b), which shows good agreement. The results indicate that the second and third modes for the third peak show non-negligible coupling, with the coupling strength of 0.053 and 0.134, respectively.

Table 3:

Mode parameters of resonant modes around the 2nd and 3rd peak of the star-shape three nanorod emitter.

Mode # ω (1014 rad/s) η E η ∞ g γ
2nd-1 6.355 + 0.290i 0.357 0.642 1 –
3rd-1 7.820 + 0.255i 0.432 0.566 1 –
3rd-2 7.948 + 0.350i 0.362 0.623 2 0.053
3rd-3 8.086 + 0.192i 0.530 0.455 2 0.134

The induced current density profiles of the third peak QNMs are shown in Figure 5. For the resonant mode at 7.820 × 1014 rad/s in Figure 5(b), it is a non-degenerate mode whose total dipole moment is zero. For the modes at 7.948 × 1014 rad/s in Figure 5(c) and (d), they are doubly-degenerate modes whose total dipole moment is also zero. For the modes at 8.086 × 1014 rad/s in Figure 5(e) and (f), they are doubly degenerate modes with non-zero dipole moment. The mode profiles in Figure 5D–F also revealed a non-negligible overlap among degenerate modes, which explains the relatively stronger coupling term in Table 3. Although there are in total 5 different QNMs around the third resonance, the overall far-field thermal radiation is only enhanced by around 4 times, which is attributed to the coupling between each pair of doubly degenerate modes. The underlying physics behind this behavior can be understood as the ‘mode switching’ in analog to the energy diagram of excitons, as shown in Figure 5(a). When the emitter is thermally pumped, the photon mode will be elevated to the unstable high energy state, where it will quickly decay to the meta-stable states, as dictated by the QNMs of the emitter. When degeneracy arises in the thermal emitter, there exist multiple such meta-stable states at the same energy level. There are two mechanisms that the photon mode can return to the ground state. One is that the photon mode directly transits from those meta-stable states to the ground state, releasing a thermal photon with the same energy but different polarizations. The other is that the photon mode will experience a state transition from one degenerate state to the other, due to the non-zero overlapping of the QNMs. During this transition process, extra resistive loss is introduced to the system and eventually causes a suppression in far-field thermal radiation.

Figure 5: Induced current density profiles for the 3rd peak of the star-shpe emitter. (a) Energy diagram of the exciton in analog to the case when there are degenerate QNMs in the emitter. (b)–(f) Induced current density amplitude inside the emitter volume at different modes for the third peak: (b) 3rd-1 mode at 7.820 × 1014 rad/s; (c), (d) doubly-degenerated 3rd-2 modes at 7.948 × 1014 rad/s; (e), (f) doubly-degenerated 3rd-3 modes at 8.086 × 1014 rad/s.
Figure 5:

Induced current density profiles for the 3rd peak of the star-shpe emitter. (a) Energy diagram of the exciton in analog to the case when there are degenerate QNMs in the emitter. (b)–(f) Induced current density amplitude inside the emitter volume at different modes for the third peak: (b) 3rd-1 mode at 7.820 × 1014 rad/s; (c), (d) doubly-degenerated 3rd-2 modes at 7.948 × 1014 rad/s; (e), (f) doubly-degenerated 3rd-3 modes at 8.086 × 1014 rad/s.

It is worth noting that although the far-field thermal radiation from each set of degenerate QNMs is not optimized, the total far-field thermal radiation can far exceed the simple sum of the contribution from three single nanorod emitters. While it is not guaranteed that every set of degenerate or non-degenerate modes can be optimized simultaneously, a global optimization can be performed to search for the maximized far-field thermal radiation. The star-shape nanoemitter also illustrates the trade-off between the complexity of the emitter geometry and its far-field thermal emission. Increasing the complexity of the emitter geometry, e.g. promoting its symmetry group to a higher order one, can bring up a set of degenerate QNMs bundled around the frequency of interest, therefore boosting the total far-field thermal radiation. On the other hand, the inevitable coupling between the degenerate modes and the spectrally close modes may compromise the total far-field thermal radiation. A balance between these two factors needs to be carefully optimized to maximize thermal emission.

4 Conclusions

In this work, we formulate a general and self-consistent theory based upon fluctuational electrodynamics and QNMs to describe the thermal radiation from nanoscale resonant thermal emitters with higher order symmetric geometries. The far-field thermal radiation from a set of degenerate QNMs is explicitly described under the expansion of the dyadic Green’s function. The introduction of the degeneracy in nanoscale resonators can result in significant enhancement of their far-field thermal radiation which is closely related to the number of degenerate modes, yet the coupling among the degenerate modes can further compromise the thermal radiation. The upper limit of far-field thermal radiation is derived in terms of the coupling strength between degenerate modes. We also show that multiple spectrally close QNMs can contribute to the far-field thermal radiation independently when quasi-static approximation is valid for the QNMs in a thermal emitter. By utilizing thermal emitters with higher-order symmetry groups, the far-field thermal radiation intensity at certain resonance frequencies can be dramatically boosted as compared to single-mode emitters when normalized to the emitting volume. This new theoretical framework thus serves as a general guideline to design the meta-structures with perfect absorption/emission, which has important implications for thermophotovoltaic power generation, thermal infrared sources, radiative cooling, and infrared sensing.

Corresponding author: Sheng Shen, Department of Mechanical Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA, E-mail: 

Zexiao Wang, Jiayu Li and Zhuo Li contributed equally to this work.

Funding source: Defense Threat Reduction Agency

Award Identifier / Grant number: HDTRA1-19-1-0028

Funding source: National Science Foundation

Award Identifier / Grant number: 2137603

Award Identifier / Grant number: 2138259

Award Identifier / Grant number: 2138286

Award Identifier / Grant number: 2138296

Award Identifier / Grant number: 2138307

Award Identifier / Grant number: CBET-1931964

Funding source: Office of Navy Research

Award Identifier / Grant number: N00014-23-1-2173

Acknowledgments

This work used Bridges-2 at Pittsburgh Supercomputing Center through allocation MCH2400043 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) program, which is supported by National Science Foundation grants #2138259, #2138286, #2138307, #2137603, and #2138296.

  1. Research funding: This work was mainly supported by the National Science Foundation (Grant No. CBET-1931964) and Defense Threat Reduction Agency (Grant No. HDTRA1-19-1-0028). This work was also partially supported by the Office of Navy Research (Grant No. N00014-23-1-2173).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. SS and JL came up with the idea. JL, ZL, and ZW derived the theory. ZW and XL developed the simulation based on the theory; ZW, YZ, and TH implemented the simulation on the supercomputing platforms. SS supervised the work and provided funding supports. ZW and JL prepared the manuscript with contributions from all co-authors.

  3. Conflict of interest: Authors state no conflict of interest.

  4. Informed consent: Informed consent was obtained from all individuals included in this study.

  5. Ethical approval: The conducted research is not related to either human or animals use.

  6. Data availability: The datasets generated and/or analysed during the current study are available from the corresponding author upon reasonable request.

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Received: 2024-10-10
Accepted: 2024-11-15
Published Online: 2024-11-27

© 2024 the author(s), published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/nanoph-2024-0534
Keywords for this article
thermal radiation; quasi-normal mode theory; group theory; nanoscale thermal emitter; fluctuational electrodynamics
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Articles in the same Issue

  1. Frontmatter
  2. Review
  3. Plasmon-driven molecular scission
  4. Research Articles
  5. Ultra-compact thin-film-lithium-niobate photonic chip for dispersion compensation
  6. Optimization of NC-LSPR coupled MoS2 phototransistors for high-performance broad-spectrum detection
  7. Impact of temperature on the brightening of neutral and charged dark excitons in WSe2 monolayer
  8. Designing rotational motion of charge densities on plasmonic nanostructures excited by circularly polarized light
  9. Twist-tunable in-plane anisotropic polaritonic crystals
  10. An overview on plasmon-enhanced photoluminescence via metallic nanoantennas
  11. Unique features of plasmonic absorption in ultrafine metal nanoparticles: unity and rivalry of volumetric compression and spill-out effect
  12. Constant-force photonic projectile for long-distance targeting delivery
  13. Emission dynamics and spectrum of a nanoshell-based plasmonic nanolaser spaser
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  15. Erratum
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