1. Introduction

Spontaneous emission (SE) plays a fundamental role in determining the performance of nanodevices such as lasers [1–3], high-resolution bioimaging [4], and single-photon sources [5–7]. Owing to the increase in the density of photon states in the cavity [8], a strong enhancement in the SE rate (compared to the intrinsic value γ₀) has been observed in various experimental settings such as photonic-crystal cavities [9, 10], dielectric microcavities [11], and plasmonic nanostructures [12–16]. In particular, surface plasmon polariton (SPP) structures are attractive owing to their ability to concentrate light within extremely small volumes. To date, it has been shown that the SE rate can be increased by orders of magnitude in a variety of SPP structures, e.g., bowtie antenna structures [12], film-coupled nanoparticle structures [13], and gap plasmon structures [14–16]. Although the SE rate can be modified by varying the size of nanopatch [16] or changing the refractive index of liquid gain medium [17], it cannot be changed once the structure has been fabricated. Such inflexibility limits its applications in tunable devices such as single-photon switches and on-chip single-photon sources [5–7].

In this paper, we report a novel subwavelength plasmonic waveguide cladded with a liquid crystal (LC) and low-index metamaterial (LIM) that will enable enhanced as well as tunable SE of an embedded emitter. The basic idea of the design is as follows. The LC is used to modulate SE, while the LIM is used to further enhance the SE rate. These mutual actions give rise to modulation as well as enhanced SE. LCs possess dielectric anisotropy by virtue of their anisotropic molecular shape and alignment, whose permittivity tensor can be changed by a variety of mechanisms such as temperature, electricity, or optics [18, 19]. It has been demonstrated that an SPP with a tunable propagation constant can be supported at the interface between the LC and a noble metal [20]. Using this property, Hsiao et al. have modulated the localized surface plasmon resonance by a light-driven method for an all-optical switch [21], and Chen et al. have manipulated the emission wavelength of quantum dots with an applied electric potential [22]. However, the enhanced modulation of SE based on an SPP tuned by an LC has not yet been studied. Additionally, we employ an LIM−an artificial material with ability to concentrate light in small volume−to greatly enhance the SE rate [23–25]. As described in Ref. [23–25], dispersing molecules or nanoparticles (rods, wires, and spheres) in some appropriate host material is introduced to create materials with novel refractive index values ranging from positive, through zero, and into the negative domain.

In such a hybrid system, by varying the refractive index of the LC cladding, thereby changing the density of states of the surface plasmons, the enhanced SE rate can be modulated. Through the optimization of material parameters, i.e., using LC with refractive index anisotropy Δn =0.8 and LIM with refractive index n_d=0.4+0.05i, the SE rate can be continuously modulated from 131γ₀ to 327γ₀. In this process, the modulation range can maintained at a constant value of 2.50 for λ= 500−1500 nm. Additionally, the modulation range increases as the refractive index anisotropy increases, and for a fixed range of modulation, an LIM with a lower refractive index results in a larger SE rate. This design provides active, dynamic, continuous, and reversible modulation of SE as well as enhanced SE rate across a broad emission spectrum from visible to near infrared wavelength, which may have potential applications in nanoscale tunable single-photon devices and low-threshold plasmon lasers.

2. Module setup

The proposed system is illustrated in Fig. 1. A semi-infinite LIM with the refractive index n_d occupies the lowest space, while a uniaxial LC is on top of the metallic film. A dipole emitter with λ= 632.8 nm is embedded in the LC layer at a distance d above the metal surface. In this LC-Au-LIM waveguide, a reduction in the thickness of the Au film can accelerate the SE, but this is accompanied by metallic loss increases. To achieve a balance, the thickness of the Au film is set at 50 nm, which is also the cutoff thickness of SPPs at λ= 632.8 nm, where only one SPP can be excited in the LC-Au-LIM waveguide. For the case of obvious modulation [26], the optical axis (OA) of LC is set in theY-Z plane and has azimuthal angle θ with respect to the Y axis. Under such circumstances, the permittivity tensor $\widehat{\epsilon }$ of LC can be expressed as follows:

 

Fig. 1 Schematic of the LC-Au-LIM subwavelength plasmonic waveguide. Here, the thickness of Au film is 50 nm and the dipole emitter is embedded in the LC layer.

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The LC-Au-LIM waveguide is simulated by the COMSOL Multiphysics software. The module dimensions are 4 × 4 ×1 μm3, and a perfectly matched layer with a thickness of 200 nm is introduced to minimize boundary reflections. The thickness of the LC, Au, and LIM layers are 750 50, and 200 nm, respectively. The emitter embedded in the LC is simulated as an oscillating classical point dipole; it can simulate the SE of single photons and fluorescent molecules. Using a similar model, we have studied the nanoparticle surface plasmon resonance images of electrocatalytic activity [27] and the efficient emission of single photons based on gap plasmons [15]. Specifically, we consider the modulation of the SE using an LC with Δn = 0 to Δn = 1.0, while the real part of the refractive index of the LIM changes from 0.4 to 0.8. The dielectric constant of the metal is taken from experimental data [28]. All materials have been reported recently [29, 30], which strongly suggests that it will be feasible to experimentally fabricate these materials.

In principle, three decay channels make up the total decay channel with a decay rate γ_total [31]: direct decays into free space with γ_r, decays into a nonradiative channel due to the metallic loss with γ_nr, and decays into the SPP channel with γ_spp. γ_total can be obtained from γ_total/γ₀=P/P₀ [32], where P is the radiative power of the dipole, and P₀ is its value in vacuum. P is calculated by performing the surface integral over the Poynting vector S on a 4-nm-radius sphere that envelops the dipole emitter. γ_nr is caused by the ohmic losses due to electron – hole pair generation induced by the dipole emitter [33]. γ_nr can be calculated by using the volume integral for the Joule heating (J·E) in the metal layer, where J is the current density, and E is the electric field intensity. γ_spp is equal to the total decay minus the other two decay rates in the channel: γ_spp=γ_total-γ_nr-γ_r (the computational details are shown in Appendix A.). We note here that the dipole emitter is so close to the metal that the power decays more rapidly into the SPP and nonradiative channels than into the free-space channel. Therefore, in the following figures, only γ_spp and γ_nr are displayed.

3. Results and discussion

We first explore the modulation of the SE when the dipole emitter is placed at a distance d = 5 nm above the Au film with parallel polarization. As shown in Fig. 2(a), when θ is zero, the total SE rate γ_total is at its minimum value of 131γ₀ ; when varying the OA, γ_total increases with θ continually and reaches a maximum value of 327 at 90°. In this process, the modulation range, defined as the ratio of maximum SE rate to the minimum value, is 2.50. It also shows that γ_spp changes from 80γ₀ to 140γ₀. This part of the energy can couple into the dielectric nanofiber with a high efficiency, which can be routed into on-chip devices and used for tunable-threshold nanospacer. Inevitably, γ_nr caused by ohmic losses changes from 50γ₀ to 187γ₀ when θ increases. Within the small-angle region, γ_spp is larger than γ_nr, while the nonradiative part occupies a greater proportion of γ_total in the large-angle region. Although ohmic losses is harmful, γ_nr can improve the modulation range of the γ_total, which can be used for high contrast switching of spontaneous emission. Moreover, since the OA can be controlled and changed in an arbitrary manner between 0° and 90° reversibly by means of optics, temperature or electricity, the modulation of SE is continuous and reversible, i.e., the SE rate can be increased when the optical angle changes from 0° to 90°, or decreases then the optical angle decreases from 90° to 0°.

 

Fig. 2 Mechanism of modulating spontaneous emission. (a) Modulation of normalized spontaneous emission rates for a dipole emitter oriented parallel along the Y-axis with d=5 nm above the Au film. (b) The normalized propagation constant, propagation length and (c) penetration depths of SPP as a function of θ. (d) Electric field intensity distributions at three specific θ. When θ increases, the permittivity tensor of LC is changed. This leads to a decreasement of penetration depths Δ, which means electric field is more concentrated. As a result, the total decay rate γ_total can be changed from 131γ₀ to 327γ₀.

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To explain the mechanism of the modulation of the SE, we numerically analyze the SPPs existing in the LC-Au-LIM waveguide. According to the derived dispersion relationship for SPPs [20], the LC-Au-LIM waveguide supports only one SPP at λ= 632.8 nm. As shown in Fig. 2(b), the normalized propagation constant Re[k_y/k₀] of the SPP changes from 2.11 to 2.82 as θ increases. The propagation length L is nearly 0.5 um and most of the energy concentrated in the LC layer. Thus, when the dipole emitter is embedded in the LC layer, the SPP is excited more efficiently, leading to a large enhancement in the SE rate. The process of modulation is close to the confinement of SPPs, which is described by the penetration depth Δ, quantified by the distance where |E| decreases to 1/e of its maximum value. As shown in the Fig. 2(c), when θ increases, both of the penetration depth into LIM and LC decreases; in particular, the penetration depth into the LC decreases from 690 nm to 90 nm. For smaller values of Δ, the light in the SPP waveguide is more confined, i.e., more energy is compressed within a small volume. Thus, the density of SPP states increases with θ, resulting in the modulation of SE.

An alternative and perhaps more intuitive way to understand the mechanism may be gained by considering the electric field intensity. The distribution of |E|² for three specific angles, θ= 0°, 45°, and 90° is shown in Fig. 2(d). Here, |E|² is normalized by its maximum value at the interface between the LC and the metal. It is shown that most of the energy is distributed in the LC rather than in the LIM layer, and |E|² increases as θ increases. By varying the OA, i.e., changing θ, |E|² active modulation of the SE.

The effect of the LC on the modulation of the SE is manifested in the following plots of the SE for different values of Δn for the same LIM with n_d= 0.4 + 0.05i. As shown in Fig. 3(a), as Δn increases, γ_max (maximum value of total decay rate) increases while γ_min (minimum value of total decay rate) remains at almost the same value. Thus, the difference between γ_max and γ_min becomes large and a prominent modulation range is obtained, which can be up to 3 when the Δn =1.0. In addition, as shown in Fig. 3(b), the modulation range linearly increases with Δn and can be fitted with the relationship γ_max/γ_min=1.91Δn+0.98 very well.

 

Fig. 3 Effect of LC’s refractive index anisotropy and LIM’s refractive index on spontaneous emission modulation. (a) Normalized total decay rates γ_max/γ₀ (θ = 90^◦), γ_min/γ₀ (θ = 0°) and (b) modulation range γ_max/γ_min as a function of Δn. Other parameters are the same as those in Fig. 2. The modulation range increases with Δn as a linear relationship, γ_max/γ_min=1.91Δn+0.98. (c) Normalized total decay rate γ_total/γ₀ and (d) penetration depths into LIM Δ_LIM for different refractive index n_d as a function of θ. The enhancement of total decay rate is larger for smaller n_d due to the more concentrated SPP, but the modulation range remains almost constant.

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We now turn to the influence of the LIM on the enhancement in the SE rate. Figure. 3(c) shows the modulation of the SE in the LC-Au-LIM waveguide with three different values of n_d while Δn is fixed at a value of 0.8. When varying the OA, γ_total changes from 131γ₀ to 327γ₀ for an LIM with n_d= 0.4 + 0.05i, while it changes from 117γ₀ to 305γ₀ when n_d= 0.8 + 0.05i. The enhancement in the SE rate is small with n_d= 1.2, which changes from 100γ₀ to 270γ₀. It is found that when the value of n_d decreases, the modulation range remains nearly the same value of 2.5, but the enhancement in the SE rate is larger. To reveal the effect of the LIM, the penetration depth into the LIM Δ_LIM for three values of n_d is shown in Fig. 3(d). For fixed θ, Δ_LIM is smallest when n_d= 0.4 + 0.05i, whereas the value is the largest for n_d= 1.2, which means that the ability to confine light is increased as n_d decreases. Thus, for a fixed modulation range, a small value of n_d is preferred for an enhanced SE rate.

With the development of nanotechnology, nanodevices with a broad emission band are clearly more desirable. Indeed, our design can meet this requirement. We show here that the above findings are not only suitable for the specific wavelength of λ= 632.8 nm, but are also valid across a broad range of emission wavelengths from 500 nm to 1500 nm. The results are shown in Figs. 4(a) and 4(b); except for the wavelength, the other parameters for the LC-Au-LIM waveguide are the same as those in Fig. 2(a). In visible wavelength, both γ_max and γ_min are correspondingly reduced with wavelength increases. For example, γ_max could reach 2000γ₀ at λ= 500 nm, while it is 10γ₀ when λ= 900 nm; on the other hand, γ_min is 790γ₀ at λ= 500 nm while it is 4γ₀ at λ= 900 nm. On the contrary, both γ_max and γ_min increase with wavelength in near infrared spectrum. Although the enhancement in the SE rate fluctuates with wavelength, the modulation range remains almost constant at 2.50 across this broad emission band and therefore meets the broad-emission-band requirement for single-photon sources.

 

Fig. 4 (a) Normalized total decay rate γ_max/γ₀ (θ = 90°), γ_min/γ₀ (θ = 0^◦) and (b) modulation range γ_max/γ_min as a function of wavelength. The parameters are the same as those in Fig. 2. With red shift of wavelength, the γ_max/γ₀ and γ_min/γ₀ decrease within visible spectrum while increase within infrared spectrum, but γ_max/γ_min remains almost constant 2.5 over a broad spectrum region.

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In the discussion above, the dipole is placed at the distance d = 5 nm above the Au film for the sake of the large SE rate. The modulation of the SE when the dipole emitter is placed at different distances d is shown in Fig. 5. When the dipole is very close to the Au film, the interaction between the dipole and the metal film is strongest, leading to large decay rate, e.g., γ_total changes from 1900γ₀ to 5100γ₀ when d=2 nm [Fig. 5(a)] while it changes from 610γ₀ to 1750γ₀ when d=3 nm [Fig. 5(b)]. The large enhancement of SE rate is accompanied by inevitable loss, which occupies more than 90% of γ_total, preventing it to practical applications. When the dipole emitter is further away from the Au film, less energy decays into the SPP channel owing to the lower excitation efficiency. Meanwhile, owing to the decrease in the number of electron−hole pairs [28], γ_nr also becomes small. Thus, when varying the OA of the LC, γ_total changes from 30γ₀ to 86γ₀ at d = 10 nm [Fig. (5c)] and changes from 17γ₀ to 36γ₀ at d = 15 nm [Fig. (5d)].

 

Fig. 5 Modulation of normalized spontaneous emission rates for a dipole emitter placed at (a) d=2 nm, (b) d=3 nm, (c) d=10 nm and (d)=15 nm. The enhancement of spontaneous emission rate is large when dipole is close to the Au film while it is small when dipole is away from the Au film, but the modulation range nearly unchanged.

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We again note that although the enhancement in the SE rate is small as d increases, the modulation range remains almost the same. Moreover, γ_spp is larger than γ_nr when the dipole emitter is far away from Au film, which is beneficial for collecting and guiding photons. In addition to the specific situations explored above, we also explore other two situations (see the Appendix B). One is the situation where the dipole is embedded in the LIM layer. The other is the situation where the OA lies in the X−Y plane. In both situations, the modulation range of the SE is small owing to the low excitation efficiency of the SPP.

Using modern chemistry technologies, an LC with a large refractive index anisotropy has been fabricated [34]; Δn could be more than unity, as reported recently [35]. There are many ways to change the refractive index of the LC; for example, it can be changed by light through a variety of nonlinear optical mechanisms or simply by temperature [19]. As for metamaterial fabrication, various nanotechnologies such as atomic layer deposition, molecular beam epitaxy, and e-beam lithography are employed to create finely patterned host materials [36]. These efforts have led to the creation of materials with novel refractive index values ranging from positive, through zero, and into the negative domain [29]. For example, LIM can be formed by nematic host containing Ag-silica coreshell nanospheres, through varying the diameter of the silica core (while keeping the shell thickness constant), it is possible to shift the refractive index into low region [23]. Another example is periodic nanostructures containing LC, such as photonic crystals and frequency selective surface- like structures nanospheres [37]. It can be seen from above that the materials are not the obstacle to carry out our design in experiments. Moreover, the designed LC-metal-LIM structure is similar to fishnet metamaterials containing an LC and Au, which has been fabricated in experiments recently [25]. The insertion of a single dipole emitter into the structure is challenging, but it can be done by coating with a dilute layer of fluorescent molecules because the enhancement in the SE is spatially independent [38]. Moreover, the separation between the dipole emitter and the Au film can be controlled precisely by self-assembled polyelectrolyte (PE) spacer layers [38]. This strongly suggests that it will be feasible to experimentally realize our design in the near future using the above techniques.

4. Summary

In conclusion, we have theoretically demonstrated the tunable enhanced SE of an embedded emitter in a designed LC-Au-LIM subwavelength plasmonic waveguide. Using SPPs tuned by the LC, the SE can be modulated effectively by varying the refractive index of the LC. We also point out that more effective modulation of the SE could be achieved with a larger Δn and smaller n_d. Further, the active modulation of SE is valid across a broad range of emission wavelengths. Because it is superior to general plasmonic waveguides, our design could provide active, dynamic, continuous, and reversible modulation of SE as well as enhanced SE rate, which can be used in many fields. In the field of plasmonic nanolasers, whose thresholds are determined by the laser line width as a function of the pump rate [1], our design can change the threshold by active modulation of the SE. Compared to microcavities whose bandwidths are generally narrow, the LC-Au-LIM waveguide with broadband operation can meet the needs of single-photons emitters, which further promotes the integration of single-photon sources on chips [39]. In addition, the active and reversible modulation can be used in optical amplifiers and single-photon switches. Therefore, we believe the enhanced modulation of the SE has a bright future in tunable nanodevices and plasmon-based nanolasers.

Appendix A Validity of computing SPP channel decay rate

As shown in Fig. 6(a), the dimensions of computing model are 4 × 4 × 1 μm³. It is used to simulate infinite boundary environment at the wavelength of λ = 632.8 nm. A perfect matched layer with a thickness of 200 nm is introduced to minimize boundary reflections. To verify the validity of the model, we calculate the electric dipole radiation power P at different polarizations and compare the result with the theoretical value. In vacuum, electric dipole radiates power in the following form:

(1)$$\mathbf{P}=\frac{{\omega }^4{\left|\mathrm{p}\right|}^2}{12\pi {\epsilon }_0{c}^3}$$

where ω represents the frequency of electric dipole and p is the electric dipole moment. The dipole polarization does not affect the result, in other words, the value should be the same in different polarizations. Using the model in COMSOL, the simulation results are shown in Table 1. The good agreement with theoretical value indicates the validity of the model.

 

Fig. 6 Schematic and validity of modelling in COMSOL. (a) dimensions of model, the dark blue area is the upper and lower integral surfaces for computing P_r. (b) integral sphere for computing total radiative power Ptotal. (c) integral volume for computing metal loss P_nr.(d) The method in the main text (denoted as Method I in black curve) and energy flux method (denoted as Method II in dotted curve) of computing SPP channel normalized decay rates γ_spp/γ₀

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Table 1. Radiative power of a dipole with different polarizatioin

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As stated in the main text, the emitter decays through three channels: decaying into SPP channel γ_spp, radiating into far field channel γ_r and decaying into metallic loss channel γ_nr. By considering all the possible decay channels of dipole emitter, SPP channel decay rates γ_spp can be obtained by eliminating all the other channels from the total decay rates (Method I). Thus SPP channel normalized decay rate is:

(2)$$\frac{{\gamma }_{\text{spp}}}{{\gamma }_0}=\frac{P_{total}-P_m-P_r}{P_0}$$

P_total is the dipole radiative power obtained by performing surface integrals over the Poynting vector S on the 4 nm radius sphere [shown in Fig. 6(c)]. For the radiative channel, the power P_r is calculated by performing surface integrals over the Poyning vector S_z on the upper and lower boundaries of the model [shown in Fig. 6(a)]. In this situation, because the dipole is so closed to the metal, the power decays into SPP channel and metallic loss channel more rapidly than its radiation into free space. So, in the main text, only γ_spp and γ_nr are displayed. γ_m here can be looked as the loss of dipole image in the gold nanofilm. It could be calculated by performing volume integral for J·E in metal. As shown in Fig. 6(b), the dimension of the volume is 10 × 10 ×50 nm³.

To demonstrate the validity of model, we also use energy flux method to calculated the γ_spp. The energy flux method has been mentioned in previous work [11] and we refer it as Method II In this way, we first consider the power delivered into both the SPPs and radiation modes of the nanofilm, computed by summing over the energy flows along the x and y direction. And the radiation modes can be computed by surface integral on upper and lower boundaries. Thus the γ_spp is obtained by:

(3)$$\frac{{\gamma }_{\text{spp}}}{{\gamma }_0}=\frac{P_x+P_y-P_r}{P_0}$$

The good agreement of two methods in Fig. 6(d) indicates our model is validity. Using the similar modeling in COMSOL, we have successfully studied the nanoparticle surface plasmon resonance images [27] and the efficient emission of single photons based on gap plasmons [15].

Appendix B Enhanced modulation of spontaneous emission in other cases

In the main text, we discuss the modulation of spontaneous emission where the optical axis (OA) lies in the Y-Z plane and the dipole emitter is embedded in the LC layer. Additionally, we explore two other situations when the OA lies in the X-Y plane [shown in Fig. 7(a) in the main text] and the dipole emitter is embedded in the LIM layer [shown in Fig. 7(b)].

 

Fig. 7 Modulation of spontaneous emission rates when OA lies in the X-Y plane for (a) and dipole embedded in LIM for (b). The parameters are the same as those in Fig. 2(a) of main text.

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As shown in Fig. 7(a), the spontaneous emission rate varies from 101γ₀ to 131γ₀, compared to the situation when the OA lie in the Y-Z plane, the modulation is small. It is because the variation of φ has smaller effect on the propagation constant of SPP than the variation of θ [26].

Figure 7(b) shows the situation when the dipole emitter is embedded in the LIM layer, the spontaneous emission is not sensitive to angle θ. As shown in the Fig. 7(d) in the main tex, the energy field intensity in LIM layer is smaller compared to its value in LC layer and is nearly unchanged with the θ. So, in this situation, the dipole emitter sees a little change of energy field intensity and the enhanced modulation of spontaneous emission is not obvious.

Funding

This work was supported by the National Key Basic Research Program under Grant No. 2013CB328700, the National Natural Science Foundation of China under Grant Nos. 11525414, 11374025, 91221304.

Acknowledgments

We thank Professor Xiaolong Hu for useful discussion.

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