1. Introduction

Spontaneous emission from quantum sources to vacuum is one of the most crucial topics in the field of quantum optics. One prominent feature of spontaneous emission is its emission rate can be modulated by the surrounding electromagnetic fields, which is predicted by the Purcell effect [1]. From the beginning, most of the research focus on the spontaneous emission from electric dipole (ED) transitions, while that from magnetic dipole (MD) transitions has always been neglected due to the weak interaction between magnetic component of light and natural materials [2,3]. In the year of 2011, a famous plasmonic nanoantenna named “diabolo” [4], a counterpart of bowtie, was proposed for enhancing the optical magnetic field as well as its interaction with matter. From then on, tremendous plasmonic structures have been proposed for enhancing magnetic field, including bowtie apertures [5,6], fractal diabolo [7], graphene diabolo [8] and the derivative structure of split ring [9,10]. In the past five years, thanks to the development of nanofabrication technology, dielectric nanoantennas supporting both ED and MD resonances [11] become feasible to be fabricated [12]. Different from plasmonic nanoantenna, where the concentrated magnetic field outside the antenna originates from the oscillating surface current, the dielectric nanoantenna produces magnetic hot spot by its internal MD resonance, which is induced by an electric displacement current loop inside the particle. Recently, it has been demonstrated both theoretically [13–15] and experimentally [16,17] that the magnetic hot spot can be utilized for enhancing MD emissions from an emitter inside or around the structure.

However, all nanophotonic structures mentioned above are designed for single-wavelength operation, as they depend on a single plasmonic or MD resonance and thus applicable for narrowband applications. For different MD emitters, such design becomes insufficient, which is because most MD transitions operate in different wavelengths across the visible and near-IR range [18]. On the other hand, structure engineering of broadband photonic devices for near-IR applications has made a significant progress in the past few years, including absorbers [19–21], light emitters [22] and waveguides [23]. Especially in the field of broadband absorbers, one elegant design is the tapered hyperbolic metamaterials (THM), which is composed of alternating metal−dielectric plates and capable of supporting high-efficiency and ultra-broadband absorption in near-IR range. This fascinating property originates from the multiple resonant MD modes excited along the structure axis [19]. Different from dielectric particles, the MD modes inside THM involves along its hyperbolic isofrequency contour (IFC) [21]. Consequently, multiple coaxial magnetic hot spots can be produced under different wavelengths. Besides, it has also been demonstrated that hyperbolic metamaterials can be employed for enhancing ED emissions [24–27]. By embedding an ED emitter inside hyperbolic metamaterials, a 500-fold Purcell factor can be achieved in the visible range [25]. Yet due to the absorption loss in THM, the radiative decay rate enhancement of the emission is only around 10-fold.

In this paper, we unprecedentedly utilize the THM to improve MD emissions from an optical emitter. By introducing an air hole through the structure, we construct the tapered hollow hyperbolic metamaterial (THHM), which makes the multiple internal MD modes become accessible for emitters. We numerically demonstrate a multiband absorption property in THHM. Due to this exclusive multiband property, the proposed structure could enhance MD emissions in five discrete bands, which means such a design could support different kinds of MD emitters. Furthermore, we demonstrate the identity of the five bands in the spectra of absorption, field enhancement and MD emissions. At last, we show that THHM holds higher quantum efficiencies than THM. The result of this work may bring new possibilities to maximize MD emission over several bands simultaneously, which can find applications in the magnetic light−matter manipulations, as well as the novel light-emitting devices.

2. Multiband absorption property and multiband near-field enhancement

The schematic diagram of a standalone THHM is depicted in Fig. 1(a), where W_t = 150 nm, W_b, = 600 nm, W_h = 90 nm, and h = 1 μm stand for the widths of top, bottom, the central square hole (fixed at 90 nm if not specially clarified) and the total structure height, respectively. W_h is kept constant along y direction, which means cross section of hole keeps its shape invariable along the structure axis. The hyperbolic metamaterial is composed of silver and germanium multilayer films, which is one of the commonly used combinations in hyperbolic metamaterial [28]. The thicknesses of silver and germanium films are t_m = 15 nm and t_d = 35 nm, which makes the period of multilayered structure much smaller than the interested wavelengths. Based on the effective medium approach [29], the multilayered structure is considered to be homogeneous but extremely anisotropic with diagonal permittivity tensor $\overleftrightarrow{\epsilon }=[{\epsilon }_x,{\epsilon }_y,{\epsilon }_z],$ where ${\epsilon }_x={\epsilon }_z=p{\epsilon }_M+(\text{1}-p){\epsilon }_D$ and ${\epsilon }_y={\epsilon }_M{\epsilon }_D\text{/}[(\text{1}-p){\epsilon }_M+p{\epsilon }_D]$ represent the in-plane and out-plane permittivity component. ε_M and ε_D are the permittivities of metal and dielectric, the filling percentage of metal is fixed at p = 30%. The permittivity of metal is described by the Drude model and all the detailed parameters are taken from [28]. Besides, the IFC in zy plane is expressed as $k_z^2/{\epsilon }_y+k_y^2/{\epsilon }_z={\left(\omega /c\right)}^{\text{2}}$ and will be shown in Fig. 2.

 

Fig. 1 (a) 3D perspective view of tapered hollow hyperbolic metamaterial (THHM). (b) The normalized absorption cross section of original tapered hyperbolic metamaterial (THM, red curve), THHM (blue curve) and its effective medium counterpart (EMC, green curve). The magnetic resonant modes at partial marked peaks are illustrated in Fig. 2.

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Fig. 2 Magnetic field intensity distribution under resonances for THHM (W_h = 90 nm) at (a) 2.01 μm, (b) 2.60 μm, (c) 3.04 μm. Magnetic field intensity distribution for THM (W_h = 0) at (d) 2.31 μm and EMC of THHM (W_h = 90 nm) at (f) 2.76 μm. All the resonances are marked in Fig. 1(b) by different signs. All subfigures are H_x components of the resonant modes. (e) Isofrequency contours (IFC) for the multilayer structures. The blue and green curves are plotted under the wavelengths of 2.6 μm and 2.76 μm, which correspond to the resonant wavelengths in panels (b) and (f). The dashed and solid lines represent IFCs predicted by the effective medium theory (EMT) and transfer matrix method (TMM) in despite of colors.

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First of all, absorption property and field enhancement of the structure are examined in order to measure the influence of perforation. Here, THHM is assumed to be non-periodic. The whole structure is suspended in vacuum and illuminated by a TM wave with normal incidence along + y direction and polarized along z direction, so the fields inside THHM take the form of E_x = 0, H_y = 0 and H_z = 0. Numerical results in this paper are calculated by Lumerical FDTD Solutions [30]. In FDTD simulation, all six boundaries are set to perfectly-matched layers to reduce reflections from simulation boundaries, non-uniform mesh is adopted with a minimum size of 3nm in all three spatial dimensions. Multiple overridden mesh regions are also employed with 1.5 nm size in y direction at each metal layer to ensure the calculation precision. The time step is 0.004 fs, which further ensures the simulation stability. By extracting data from the FDTD simulation, the normalized absorption cross section in Fig. 1(b) can be calculated by${\sigma }_{abs}/W_b^2$, where σ_abs represents the absorption cross section of THHM, W 2 b is the projected area of the structure on the plane perpendicular to k, i.e. the xz plane. Figure 1(b) shows the absorption spectrum of original THM (red curve, W_h = 0), THHM (blue curve, W_h = 90 nm) and its effective medium counterpart (EMC, green curve, W_h = 90 nm). In order to verify the influence of perforation, EMC is treated as a hollow quadrangular frustum pyramid composed of homogeneous effective medium. Three conclusions can be made from Fig. 1(b), as listed below.

Firstly, rather than broadband absorption, the original THM clearly exhibits an typical characteristic of multiband absorption with multiple peaks. Then, the absorption spectrum of THHM further proves this multiband property will not be impaired by the operation of perforation. The presence of multiband spectra for both THM and THHM can be attributed to the non-periodic structure in this paper. For periodic THM absorber, broadband absorption is commonly observed [19–21]. This broadband absorption originates from the intercoupling between each resonant mode among neighboring THM structures, which leads to the merging of resonant peaks and thus the broadband spectrum [21]. In [21], when increasing period of THM array, the broadband absorption will gradually convert into multiband absorption. For an isolated THM or THHM in this paper, this intercoupling will thus be canceled. Therefore, the standalone THHM here remains multiband, which is also verified by the spectrum of its EMC in Fig. 1(b).

Secondly, the perforation operation will not change the absorprtion mechanism but weaken localization of resonant modes. The absorprtion mechanism predicts incident wave couples into THM at a cutoff position and evolves as waveguide modes [21]. Each waveguide mode is composed of multiple coaxial MD modes and gets absorbed while propagating. As λ increases, the cutoff position moves towards the bottom. This phenomenon can be interpreted by ${\lambda }_r=2w\sqrt{{\epsilon }_y}$ [19,31], where w is the width of THM and λ_r is the resonant wavelength. In order to verify this evolving law is preserved after perforation, Figs. 2(a)–(c) illustrate the normalized |H |² distributions at first three resonances in THHM (marked by blue star, dot and triangle in both Figs. 2 and 1(b)), which is defined as the ratio of |H |² at the same position with or without the structure [4]. The insets in Fig. 2 also show H_x (the dominant component of H filed) for all the resonances. From insets of Figs. 2(a)–(c), it is clearly shown that three adjacent resonances include one, two and three MD modes, where the resonant condition is generally expressed as [21]:

Thirdly, the spectra in Fig. 1(b) show two obvious redshifts, one of which is from THM to THHM, the other is from THHM to its own EMC. This can be concluded from Figs. 2(b), 2(d) and 2(f), which display the distributions of |H |² and H_x field under three resonances. First of all, the three insets prove the three resonances are identical resonances with two opposite-oscillating MD modes. By marking these resonances in Fig. 1(b) (shown in red, blue and green circular dots), it is clearly shown that the same resonances are excited at longer wavelengths for THHM and its EMC, which thus verifies the two redshifts. The first redshift can be attributed to the lower light confinement induced by perforation. For THM, the reduced light confinement usually reduces k_z and k_y of the waveguide modes, which will move the modes towards the bottom. Consequently, the width w will increase and thus the resonant wavelength according to${\lambda }_r=2w\sqrt{{\epsilon }_y}$. The second redshift originates from the deviation of IFCs between the realistic structure and the EMC. In real multilayers, the IFC follows effective medium theory only at small wavenumbers. Once the wavenumber increases, a more accurate form of IFC based on transfer matrix method should be considered as below [32]:

After analyzing the multiband absorption property and the resonant condition of THHM, we are ready to characterize the magnetic near-field enhancement and its effect on enhancing MD emissions. In Fig. 3, we first calculate the magnetic field intensity enhancement (i.e. normalized |H |²) at three positions: −400 nm (point A), 0 (point B), and 400 nm (point C). It is noted that both points A and B show a considerable enhancement of magnetic field intensity. Importantly, rather than a single peak, the enhancement is in the form of multiband, which would be suitable for different kinds of MD emitters in practical devices. Moreover, since point B is closer to the bottom than A, the enhancement of B is thus in longer wavelength range. Besides, almost no enhancement is obtained at point C, which is due to the larger mode volume formed at the bottom and thus the reduced field confinement.

 

Fig. 3 Normalized magnetic field intensity at three positions on the axis of the structure: point A (y = −400 nm), point B (y = 0 nm, the original point) and point C (y = 400 nm). The THHM structure starts from −500 nm to 500 nm along y axis. Five resonant peaks indicated by blue marks are in identical wavelengths with those by the same marks in Fig. 1(b).

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3. Multiband enhancement for magnetic dipole emissions

In order to explore the enhancing effect of THHM on MD emissions, an x-polarized MD emitter is placed into the central hole. After measuring the total radiated power from the emitter without the structure (P₀) and the radiated power into far field in presence of the structure (P_r), the radiative decay rate enhancement (RE) can be calculated as RE = P_r / P₀ [3,13]. Figure 4(a) records RE at three identical positions indicated in the inset of Fig. 3 in their respective colors. It can be found that the RE curves are quite similar to the normalized |H |², thus the multiband property are almost intactly preserved under dipolar excitation. This also indicates the importance of adopting spectrum profile of the normalized |H |² to estimate RE in THHM. For example, both normalized |H |² in Fig. 3 and RE in Fig. 4(a) for point C stay in a low level. As mentioned earlier, the weak light confinement at the bottom leads to the low normalized |H |², which further reduces the magnetic local density of states (LDOS) at point C. Therefore, the spontaneous emission rate for the MD, i.e. γ_m, will also decrease according to the the Fermi rule, which can be written as [2]:

 

Fig. 4 (a) Radiative decay rate enhancement of the MD emitters located at points A, B and C shown in Fig. 3. (b) Radiative decay rate enhancement as function of y position of MD emitter.

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To fully reveal the multiband enhancement for MD radiations in THHM, the RE spectra as a function of y position of the emitter are illustrated in Fig. 4(b). Due to the stronger light confinement, the top of THHM supports the largest magnetic field enhancement. Consequently, an MD emitter near the top provides a higher RE. As the position moves towards the bottom (towards + y), the enhancement value decays gradually and the multiband peaks redshifts as well. In order to provide a better illustration for the typical multiband enhancement, spectrum positions of each band are also highlighted with white dashed lines, which are 2.01, 2.60, 3.04, 3.43 and 3.76 μm, respectively. These bands are almost identical to those in Figs. 1(b) and 3, revealing the consistency of resonances in the spectra of absorpotion, field enhancement and emission enhancement.

Next, we investigate the geometric dependence of RE on THHM. According to the blue curves in Fig. 4(a), emitters at the bottom of THHM are almost not enhanced. Therefore, it would be meaningful to explore if the bottom part of THHM could be prescinded. Figure 5(a) displays the RE spectra for the three MD emitters at the identical position of y = −400 nm but in three different THHMs with full length, three quarter length (which means the quarter part near the bottom is deleted) and half length. Although the enhancement peak at 2.01 μm stays almost unaffected, the spectra in longer wavelength range drop quickly as the length shrinks. This weakening effect is remarkably enlarged as the MD emitters move to the center (y = 0 nm), as shown in Fig. 5(b). The phenomenon above reveals the importance of bottom part in assisting MD emissions, especially for the longer wavelengths. Figures 5(c) and 5(d) exhibits the dependence of RE on W_h and hole shape. Figure 5(c) proves the RE could be significantly improved with the blue shifted peaks by gradually decreasing W_h. When W_h = 30 nm, the RE peak at 1.87 μm can reach up to 694. If further shrinking W_h to null as blue dashed line shown, RE will increase to an even larger value. Inset in panel c shows the relationship between λ_r and the hole area (W 2 b) for the four resonances (W_h = 30 nm ~120 nm), which demonstrates a nearly perfect linear dependence for the both. In Fig. 5(d), the influence of hole shape on the performance of THHM are examined. It demonstrates the spectrum of MD emissions in the circular hole with 30nm diameter (blue dotted line) are nearly identical with that in the square hole with a width of 30 nm (orange solid line), except the slightly increased peak values and decreased resonant wavelengths. Thus, Fig. 6(b) verifies the enhancing performance of THHM is almost uncorrelated to the hole shape but sensitive to the hole area. A tiny reduction of the hole area will cause the blue shift of the spectrum.

 

Fig. 5 Radiative decay rate enhancement for THHMs with different heights while the MD emitter is located at –400 nm (a) and 0 nm (b). W_h in (a) and (b) are kept at 90 nm. Dependence of the radiative decay rate enhancement on the hole size W_h (c) and hole shape (d). The dotted and solid curves in (d) correspond to the structures with a circular hole of 30 nm diameter and a square hole with 30 nm side length, respectively. Inset of (c) shows the linear relationship between wavelength of the main resonances and the hole area. The height of the whole structures in (c) and (d) are kept at 1 μm. All the abscissas and ordinates in Fig. 5 are wavelengths and radiative decay rate enhancement.

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Fig. 6 (a) Quantum efficiencies for THHMs with different hole dimension W_h. (b) Radiative decay rate enhancement when W_h = 90 nm. Shadow region in (b) shows the wavelength range with a quantum efficiency higher than 40%.

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Finally, quantum efficiencies (QE) for the proposed structure are investigated. Figure 6(a) demonstrates QE can reach up to 70% for THHM, which is much higher than traditional non-hollow THM. This phenomenon can be attributed to the introduction of air hole, which performs as a high efficiency channel for emissions. For an x-polarized dipole emitter, it is well known that the time averaged Poynting vector‾S_p of the radiation becomes largest along directions perpendicular to x and its P_r is simply written as $P_r={\displaystyle {\int }_S\left|\overline{S_p}\right|}ds$, where S is the surface enclosing the dipole. Thus, a hole along y will help to extract the power from the internal emitter. This can be verified by Fig. 6(a), where QE dramatically drops when shrinking the hole dimension. On the other hand, THM shows a much stronger field enhancement (shown in Fig. 2(d)) than THHM (Fig. 2(b)), which indicates the emissions could be greatly enhanced in THM. However, QE for THM in Fig. 6(a) stays in a low level, which implies this stronger field enhancement leads to a much higher total decay rate enhancement inside THM, rather than a higher RE. Thus, one can conclude that most emissions from the MD source are absorbed in THM, resulting in a large non-radiative decay rate enhancement. Moreover, since THM holds similar absorption spectrum and magnetic hot spots, it is reasonable to conclude THM could also provide such a multiband enhancement for MD emitters. Besides, Fig. 6(a) also indicates a larger hole provides a higher QE and this effect becomes saturated when W_h = 120 nm. Figure 6(b) shows a RE spectrum with W_h = 90 nm. Wavelength range where QE > 40% is illustrated by the shadow region (also shown by the red dashed line in Fig. 6(a)), indicating the main resonances on the RE spectrum are included.

4. Conclusions

A multiband enhancement for magnetic dipole emitters is realized in tapered hollow hyperbolic metamaterial. By opening an air hole through the multilayer structure, the internal magnetic fields become accessible for emitters. We firstly demonstrate this hollow multilayer structure supports a multiband light absorption, forming multiple coaxial magnetic hot spots. Inherited from the multi-wavelength resonant characteristic of hyperbolic metamaterials, the proposed structure could offer emission enhancement in five discrete bands. More importantly, each band provides multiple coaxial hot spots, which thus offers multiple spatial positions for enhancing the emissions. Further simulation shows both field confinement and emission enhancement could be further improved by decreasing the hole dimension, but quantum efficiency will then reduce accordingly. The results of this work reveal the unique ability of such structured metamaterials to enhance and manipulate light emission from magnetic dipole transitions. We anticipate these results will contribute to experimental investigations into structured metamaterials for magnetic dipole emissions.