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Abstract
A one-dimensional photonic-crystal (PC) cavity with nanoholes is proposed for extreme enhancement of terahertz (THz) electric fields using the electromagnetic (EM) boundary conditions. Both slot (for the perpendicular component of the electric displacement field) and anti-slot (for the parallel component of the electric field) effects contribute to the considerable field enhancement. The EM energy density can be enhanced by a factor of (εh/εl)2 in the high-refractive-index material, where εh and εl are the permittivities of the high- and low-refractive-index materials, respectively. Correspondingly, the mode volume can be reduced by a factor of 288, compared with a conventional THz PC cavity, and is three orders of magnitude smaller than the diffraction limitation. Further, the proposed THz cavity design also supports modes with high quality factors (Q) > 104, which induces strong Purcell enhancement by a factor exceeding 106. Our THz cavity design is feasible and attractive for experimental demonstrations, because the semiconductor layer in which the EM is maximized can naturally be filled with quantum-engineered active materials. Thus, the proposed design can possibly be used to develop room-temperature coherent THz radiation sources.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The term “terahertz (THz) radiation” typically refers to electromagnetic waves of frequencies ranging from 0.1 to 10 THz, where 1 THz = 1012 Hz. Recently, excellent progress has been achieved in the field of THz technology, with THz waves being applied in biological spectroscopy, label-free biosensing, imaging, and security screening [1–4]. Efficient generation of coherent THz radiation [5,6] is of particular importance for these applications. However, the radiative emission rate of a quantum emitter in the THz frequency band is limited by the small electric dipole moment [7]. The Purcell effect facilitates manipulation and enhancement of the emitter spontaneous emission rate through local modification of the electromagnetic (EM) density of states (DOS) for photons in a microcavity [8]. The Purcell factor (Fp) [9], i.e., the enhancement factor of the spontaneous emission rate in the microcavity, scales with the ratio of the quality factor (Q) and mode volume (V) [10]. Therefore, confining the EM field to deep subwavelength volumes far beyond the diffraction limit of light can greatly modify the DOS and enhance Fp.
THz plasmonic microcavities based on metal-dielectric structures have been proposed [8,11,12], which have been proven to support highly localized resonant modes concentrated in subwavelength V. Unfortunately, metal absorption loss renders plasmons inadequate for construction of compact practical devices in either the THz frequency band or the optical domain. For example, THz plasmonic microcavities usually exhibit Q of less than 102 [13], and the Fp values are typically limited to less than 50 [7,8]. Very recently, all-dielectric THz whispering-gallery-mode (WGM) resonators [14–16] with high Q of 103−104 were implemented. The WGM is known for high Q, as the field is evenly distributed at the cavity equator via total internal reflection; however, V is relatively large compared to the wavelength to avoid radiation loss. Besides plasmonic and WGM-based devices, all-dielectric nanostructures such as slot [17–19] and bowtie structures [20–23] have been utilized to enhance the local field for optical waves. This enhancement is realized by the EM boundary conditions on the normal component of the electric displacement field (D) [17–19] or the parallel component of the electric field (E) [22,23]. However, in the THz frequency range, the potential of all-dielectric structures to greatly enhance the DOS has remained unexplored.
In this paper, we propose an all-dielectric semiconductor photonic-crystal (PC) slab cavity with nanoscale air holes, and systematically explore the EM enhancement mechanism in the THz frequency band. By twice applying EM boundary conditions, D and E are both greatly enhanced by a factor of εh/εl, where εh and εl are the permittivities of the high- and low-refractive-index materials, respectively. Hence, the electric energy density (W) is greatly enhanced by a factor of (εh/εl)2. The THz radiation is squeezed into a V that is three orders of magnitude smaller than the diffraction-limited V. With this excellent mode confinement, the radiation loss of the mode is suppressed and the Q of the resonant mode exceeds 104, being limited by the material absorption loss. The DOS in the proposed microcavity is greatly modified and the emitter spontaneous emission rate is enhanced by an Fp of ~106. Compared to the slot or bowtie structures in which the EM field is concentrated in the low-refractive-index materials, the proposed structure maximizes the EM field in the high-refractive-index material, i.e., inside the cavity material. The proposed PC slab is monolithic and compatible with the current semiconductor fabrication technique. Thus, the proposed THz microcavity is particularly attractive for future realization of a coherent THz source.
2. Design and performance
In the following, the properties of the proposed design are systematically studied through numerical simulation, with the eigenmodes of the THz cavity being determined using the three-dimensional (3-D) finite-element method with COMSOL Multiphysics 4.3. Figure 1(a) is a schematic of the two proposed types of THz PC cavities, in which nanoholes are placed in the centers of the conventional THz PC slab cavity. First, we describe the structure of a conventional THz PC cavity. This cavity consists of one suspended semiconductor (Si) waveguide patterned with a 1-D line of air holes. The refractive index and extinction coefficient of Si are nSi = 3.42 and kSi = 2.4 × 10−5, respectively [24,25]. The width and thickness of the Si waveguide are 50 and 22 μm, respectively. The hole-to-hole distance a = 33 μm is constant. The hole-to-hole distance between the two central holes occupies the distance remaining between the two Gaussian-shaped mirrors formed between the center and the end [Fig. 1(a)]; i.e., there is no additional cavity length between the central holes and the two Gaussian-shaped mirrors. Note that the definition of Gaussian-shaped mirror is different from the conventional Gaussian mirror that are characterized by a degree of reflection which slopes radially from the centre of the optic in a Gaussian distribution. Here, Gaussian attenuated mirrors from the center to the end are generated by quadratic radius-modulated air holes implemented in accordance with the deterministic cavity design process [26,27]; such Gaussian-shaped mirrors can minimize the radiation loss [27,28]. The jth hole’s radius (where j increases from 0 to jmax) is defined as r(j) = rcenter + j2(rend − rcenter)/jmax2, where rcenter = 12.127 μm and rend = 8.5749 μm are the center and end holes of the Gaussian mirrors in the tapered region. These parabolic tapered air holes form Bragg mirrors and Gaussian-type confinement can be achieved [28]. In this study, jmax was set to 20 and the hole radius in the periodic mirrors on both sides [Fig. 1(a)] was set to rend. The calculation region is 2400 × 300 × 200 μm3 and scattering boundary condition is used. While THz wave is extremely localized around the center nanoholes, fine mesh size is needed. The maximum element size of the nanohole is set as 1 μm (< 0.01λ, λ is resonant wavelength); the maximum element size of the microholes in the Gaussian-shaped mirrors is set as 4 μm. The corresponding electric field (|E|) distribution is plotted in Fig. 1(b). As expected, the THz wave is well confined in the cavity. The resonant frequency is 2.176 THz and the resonant mode is transverse-electric (TE)-polarized (|E| is in the y direction). The Gaussian envelope of the 1-D distribution of |E| is attributed to the parabolic tapered air holes.
Fig. 1 (a) Schematic of proposed THz PC cavities with nanoholes. In the type-1 structure, an elliptical air hole with a convex surface is inserted into the cavity center. In the type-2 structure, two silicon holes are shifted in opposite directions on the y-axis, yielding a remaining air hole with a concave surface. 2-D and 1-D distributions of |E| in (b) conventional, (c) type-1, and (d) type-2 THz PC cavities. Right panels: Enlarged views of center |E| field distributions of conventional, type-1, and type-2 cavities and air holes in type-1 and type-2 cavities.
In this study, two types of air hole were added to the cavity center of the conventional THz PC cavity design, yielding two novel THz PC cavities, as shown in Fig. 1(a). In the type-1 THz PC cavity, an elliptical air hole with longer radius ra and shorter radius rb was inserted into the center; the air hole surface was convex. In the type-2 THz PC cavity, however, two elliptical holes with longer radius ra and shorter radius rb were shifted in the opposite direction along the y-axis and an air hole with a concave surface was formed. From the y-direction perspective, a low-refractive-index air slot sandwiched by high-refractive-index materials was formed. This is the typical slot structure [17] that confines and guides EM waves in a nanometer-sized low-refractive-index material, because of the electric field discontinuity at the interface between the high-index-contrast materials. The so-called “slot effect” can greatly enhance EM wave density in slot structures, especially for slots with concave surfaces [21,29–31], and has benefits for nonlinear processes [32], sensing [33], and nanolasing [30]. From Figs. 1(b) and 1(c), it is apparent that extreme |E| enhancement occurs at the centers of the two novel, proposed THz PC cavities.
Throughout the remainder of this study, rb was set to 100 nm; i.e., the slot width was as large as 200 nm. In contrast, ra ranged from 100 to 4000 nm. Note that it is possible to prepare such a large slot (2rb) in experiment, because a sub-10-nm slot can be fabricated via atomic layer lithography [34]. The dependencies of the most relevant characteristics, including Q, V, and the resonant wavelength (λ), on ra in both types of THz PC cavity were calculated; the results are presented in Fig. 2. Here, we defined ra as positive and negative in the type-1 and −2 structures, respectively, with the two structures converging at ra = 0. The calculated Q is shown in Fig. 2(a), with the total Q (Qtot) being determined by the material absorption-related Q (Qabs) and the radiation-related Q (Qrad), as 1/Qtot = 1/Qabs + 1/Qrad. Note that Qrad was calculated by neglecting the imaginary component of the Si permittivity. Alternatively, Qrad can also be calculated by perturbation theory [35]. The stars in Fig. 2 indicate the conventional cavities without nanoholes (i.e., ra = 0). Similar tendencies for the two THZ PC cavities with increasing |ra| are apparent in Fig. 2 (noting that the ra values for the type-2 structure are given a negative sign in the plot, but the distances themselves are non-negative). In Fig. 2(a), Qrad decreases as |ra| increases in both cavities, which may be attributed to the air-slot-induced coupling between the cavity resonance mode and free-space radiation modes. As the field distribution of the cavity mode rarely changes, except in the region close to the nanohole, Qabs remains constant and Qtot decreases as |ra| increases for both types of THz PC cavity. Qtot is first dominated by the absorption loss as |ra| increases and then dominated by the radiation loss when |ra| exceeds approximately 3000 nm.
Fig. 2 (a) Quality factor (Q), (b) normalized mode volume (V/V0), (c) resonant wavelength (λ), and (d) Purcell factor (Fp) versus longer radius ra for both proposed THz PC cavities, where ra > 0 for the type-1 structure and ra < 0 for the type-2 structure.
V was calculated from the ratio of the total electrical energy to the maximum electric energy density [20]:
In Fig. 2(b), V is normalized by V0 ( = (λ/2)3), which represents the diffraction-limited mode volume in free space. In Fig. 2(b), the normalized mode volume (V/V0) decreases dramatically as |ra| increases for both THz PC cavities, which indicates that the field enhancement becomes stronger for larger |ra|. In the conventional THz PC, V is approximately one order of magnitude smaller than V0. In the type-2 structure, however, V can be reduced to approximately two orders of magnitude less than V0. In the type-1 structure, V can be further squeezed to more than three orders of magnitude smaller than V0. The mode-volume enhancement factor (Ev), which is defined as the ratio of the V in the proposed THz PC cavity against that in the conventional THz PC cavity, can reach 288 for the type-1 structure.There exists a trade-off between the Q factor and V. Therefore, another key parameter is shown in Fig. 2(d); namely, Fp, which represents the spontaneous-emission-rate enhancement factor of the emitters. The Purcell factor is calculated from [20]
where n is the material refractive index. Fp first increases and then decreases as |ra| increases; thus, the optimal values of ra are approximately 1000 and 2000 nm for the type-1 and type-2 structures, respectively. In the type-1 structure, strong enhancement of Fp to exceed 106 is achieved; this value is several orders larger than the values obtained in metal cavities [7,8,12] at THz frequencies. The radiation emission rate is known to be particularly low in the THz frequency range [7]. Thus, the proposed structures, especially the type-1 structure, provide an excellent platform for spontaneous emission enhancement. In addition, enhanced Purcell effect will also improve the quantum coherence properties of the emission [36,37]. In the following section, the underlying physical mechanisms of the field enhancement achieved in our proposed cavity structures are revealed.3. Extreme field enhancement mechanisms: slot and anti-slot effects
In this section, we discuss the physical mechanisms behind the extreme field enhancement, by considering the detailed field distribution around the nanohole. As apparent from the field distribution in Fig. 3(b, inset), the main E is on the y-axis, with the air hole serving as a traditional slot structure from the y-direction perspective. The EM field is highly confined in the low-refractive-index region, because of the so-called “slot effect” [17,38]. Theoretically, this field enhancement arises from the EM boundary condition that the normal component of D should be continuous at the interface. As a result, E must undergo a large discontinuity with much higher amplitude on the low-index side. Therefore, at point A, the normal components of D and E must satisfy
where subscripts h and l denote the high- and low-refractive-index materials, respectively. In this study, Eqs. (3) and (4) were confirmed by the numerical results for the normal components of D and E. D was continuous at the silicon−air interface, as for point A in Fig. 3(b, inset), while E was greatly enhanced in the air slot. The amplitudes of the normal components of E, i.e., |E|, at the silicon−air interface were EA,h = 7.26 × 105 a.u. (arbitrary unit) and EA,l = 7.72 × 106 a.u. The enhancement factor was EA,l/EA,h ~10.6, which agrees well with the theoretical value of εh/εl ~11.7 predicted by Eq. (4). As a result, W = DE/2 in the air slot was also enhanced [Fig. 3(c, insets)].Fig. 3 (a) W(x, 0)-field and (b) W(0, y)-field distributions in type-1 structure for different ra. (c) Enlarged view of W(0, y)-field distribution (yellow region) in (b). (b, insets): 2-D |E| and |D| distributions. (c, insets): 2-D W distributions with ra = 100 and 500 nm, respectively.
At point B in Fig. 3(b, inset), the tangential component of E is continuous according to the EM boundary condition. That is, the tangential components of D and E must satisfy
Thus, the tangential components of D and W are enhanced in the high-refractive-index materials, different from the traditional slot effect. This effect is called the “anti-slot effect.” The calculation results verified that |E|(x, 0) was continuous and |D|(x, 0) was discontinuous at the silicon−air interface. Further, |D|B,h and |D|B,l at point B were 8 × 10−4 a.u. and 6.75 × 10−5 a.u., respectively. The enhancement factor of |D|B,h/|D|B,l was approximately 11.9, which also agrees well with the theoretical value of εh/εl ~11.7.
The field enhancement mechanism can be summarized as follows: W is enhanced in the air slot by a factor of εh/εl, because of the slot effect. Then, it is further enhanced in the high-index region by a factor of εh/εl, because of the anti-slot effect. Ultimately, W is enhanced by a factor of (εh/εl)2, as shown in Fig. 3. Importantly, W is maximized in the high-refractive-index material, different from the behavior for traditional slot structures [17,29], where W is maximized in the low-refractive-index slot region. This mechanism is very helpful for light and matter interaction enhancement by embedding emitters in high-refractive-index materials, especially at THz frequencies for which the radiation emission rate is limited. H. Choi et al [23] provided another scheme to maximize EM field in the high-refractive-index material by connecting the tips.
As shown in Fig. 3, W is always uniform in the air-hole region in the type-1 structure. For ra < 300 nm, the W in the air hole is smaller than that in the conventional PC cavity (dashed lines in Fig. 3). W is always maximized in the x direction; this is attributed to the anti-slot effect. Here, without loss of generality and for reduced calculation cost, a 2-D model was used to calculate the field distribution. For larger ra, local field enhancement around the two corners of the elliptical air hole can be determined intuitively from Fig. 3(c, right inset). W increases and becomes more localized as ra increases, but the W enhancement at point A is always kept to approximately (εh/εl)2. This local field enhancement in the type-1 structure has a slightly similar appearance to that obtained in tip or bowtie structures [21–23]. However, the essential difference is that, in the type-1 structure, the |E|-field is enhanced in the high-refractive-index material. In conventional tip or bowtie structures, however, the |E|-field is enhanced in the low-refractive-index material.
Similar to the type-1 structure, in the type-2 THz PC cavity, the |E|-field and W(x, 0)-field are enhanced by the slot effect in the y direction, while the |D|-field and W(0, y)-field are enhanced by the anti-slot effect in the x direction. However, the V in the type-1 structure is lower than that in type-2 structure at the same |ra|, especially for larger |ra|.
To reveal this difference between the structures, we plotted and compared the W field distributions in the type-2 structure for different |ra| (from 100 to 1000 nm), as shown in Fig. 4. In the type-2 structure, W is non-uniform in the air-hole region, because the air-hole surface is concave. For |ra| < 1000 nm, W is maximized in the silicon−air interface in the x direction (that is, Point B), as can be seen from Fig. 4(a). For |ra| > 1000 nm, W is maximized at the silicon−air interface in the y direction (that is, Point A), which can be seen from Figs. 4(b) and 4(c). This indicates that the anti-slot effect dominates the W enhancement when |ra| < 1000 nm, but the slot effect dominates the W enhancement when |ra| > 1000 nm. However, the slot effect enhances W throughout the entire slot region, as shown in Fig. 4(b, inset). Therefore, W is relatively spread compared with that in the type-1 structure, where W is maximized and localized at the two corners (point B) in the x direction (Fig. 3). Besides, W is not finally enhanced by a factor of (εh/εl)2 at Point B (silicon−air interface for y = 0) in the type-2 structure, because the |E|-field is not enhanced by a factor of εh/εl uniformly throughout the entire air hole. As a result, the V in the type-1 structure is smaller than that in the type-2 structure at the same ra. It is worth noting that there are four points at which the fields cannot converge in calculation, because of the sharp slot corners in the type-2 structure, as shown in Fig. 4(b, inset). Therefore, the maximum values of the W-field at those points were excluded from the V value calculation performed in this study. We selected the maximum of the W-field in the x direction (y = 0) or the y direction (x = 0) when calculating the V values. When |ra| is larger, the EM field is highly localized around the sharp corners both in type-1 and type-2 structures. To display the field distributions more clearly, logarithmic W-field distributions are adopted in Fig. 3(c, right inset) and Fig. 4(b, inset).
Fig. 4 (a) W(x, 0)-field and (b) W(0, y)-field distributions in type-2 structure for different ra. (c) Enlarged view of W(0, y)-field distribution (yellow region) in (b). (b, inset) 2-D W distributions with ra = 500 nm.
4. Horizontally coupled air hole array
Finally, we propose another strategy to further enhance the THz waves by using cascaded type-1 structures, in which an array chain of horizontally coupled circular air holes is inserted into the centers of the THz PC cavities. Figures 5(a)-5(c) show the 1-D |E|-field, |D|-field, and W-field distributions in the x direction in THz PC cavities with two horizontal air holes separated by distance w [Fig. 5(a, insets)]. The |E|-field is continuous at the outer silicon−air interface, i.e., x = ± 205 nm, for w = 10 nm, as shown in Fig. 5(a). Further extremely large |E|-field enhancement occurs at the silicon ridge between the two air holes. Therefore, extremely large enhancement of the |D|-field and W-field also occurs at the silicon ridge, as shown in Figs. 5(b) and 5(c). The THz wave is highly confined in the nanoscale silicon ridge, as apparent from Fig. 5(c, inset). V decreases by one order of magnitude compared with the type-1 structure with one circular air hole [Fig. 5(e)]. Accompanied by squeezing of the EM field, the radiation loss is also suppressed and Q increases [Fig. 5(d)]. From simulation of cases involving greater numbers of horizontally aligned air holes and larger silicon ridges, we found that the enhancement increases with the hole number, but reduces with greater silicon ridge width. When five air holes are coupled with w = 10 nm [the field distribution is shown in Fig. 5(d, inset)], V is three orders smaller than the V0, which yields an extremely large Fp exceeding 106 [Fig. 5(f)].
Fig. 5 (a)−(c) 1-D |E|-field, |D|-field, and W-field distributions in x direction in THz PC cavities with two horizontally coupled air holes. Left inset in (a) shows the schematic of two coupled air holes separated by w; Right inset in (a) shows the 2-D |E|-field distribution. Inset in (b) show 2-D |D|-field distribution. Inset in (c) shows the 2-D W-field distribution. (d)−(f) Q-factor, V/V0, and Fp as functions of hole number. Inset in (d) shows 2-D W-field distribution with five horizontally coupled air holes. Scale bars (red lines) in the insets of (a)-(d) are 200 nm.
5. Summary
We have proposed a novel type of all-dielectric THz PC cavity in which air holes are inserted into the cavity center; this cavity can achieve extreme concentration of the THz-wave electric field. Utilizing the boundary conditions for EM waves at dielectric interfaces, the electric field can first be enhanced by a factor of εh/εl via the slot effect (where εh and εl are the permittivities of the high- and low-refractive-index materials, respectively). The electric displacement field can be further enhanced by a factor of εh/εl via the anti-slot effect. Then, the electric energy density is enhanced by a factor of (εh/εl)2, and the cavity mode volume can be reduced to three orders of magnitude less than the diffraction-limited mode volume. The mode volume can be reduced by a factor of 288 compared to that in a conventional THz PC cavity, while a high Q exceeding 104 can also be maintained. The highly localized field produced by our designed cavity structure can greatly enhance the light-matter interactions in the THz frequency band; this is highly desirable for applications ranging from sensing [39,40] to coherent THz sources [5,6,41].
Funding
National Key Basic Research Program of China (973 project) (Grant No. 2015CB352006); National Natural Science Foundation of China (Grant Nos. 61705039, 61505195, 91536219, and 61335011); China Postdoctoral Science Foundation (2017M610389); Fujian Provincial Program for Distinguished Young Scientists in University; Fujian Provincial Key Project of Natural Science Foundation for Young Scientists in University (JZ160423); Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT_15R10); Special Funds of the Central Government Guiding Local Science and Technology Development (2017L3009); CLZ is supported by the Anhui Initiative in Quantum Information Technologies (AHY130000).
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