1. Introduction

Localized surface plasmons (LSPs) have been attracting interest in the scientific community for the past decade because of the ability to collect and concentrate light into deeply subwavelength volumes with high efficiency [1–6 ]. Focusing light into nanoscale greatly heightens light-matter interactions, for examples surface enhanced Raman scattering (SERS) [7], enhanced fluorescent emission [8], and high harmonic generation experiments [9]. A problem for many applications of plasmonics and metamaterials is posed by losses inherent in the interaction of light with metals. To overcome this limitation, integrating plasmonic nanostructures with gain materials has been proposed as an effective strategy for local field enhancement, which compensates for the loss of conventional LSP structures [10,11 ]. Recently, we proposed a method to significantly enhance the local field of a gap plasmonic system by placing a metallic nanoparticle in close proximity to a substrate covered with an ~100 nm film that contains a gain material. Compared with a conventional dielectric substrate, the thin gain film can significantly enhance the local electric fields in the gap between the particle and the substrate; this enhancement, in turn, results in a substantially improved sensitivity of conventional SERS applications [12]. In addition, similar loss compensation method by gain medium in metallic nanostructures has also been applied in the design of SPASER (surface plasmon amplification by stimulated emission of radiation) [13,14 ], which leads to the coherent generation of surface plasmons with nano-localized, intense, coherent optical fields [15].

There have been theories published describing the loss compensation in metallic nanostructures, using the numerical simulations on the basis of classic electrodynamics by considering the gain medium as a dielectric with a negative imaginary part of the permittivity [10–12 ]. However, to achieve a clear physical insight into its working mechanism, one has to build some analytical models to solve the central issues of the performance of similar devices. A powerful approach based on transformation optics has been proposed to analytically derive the optical response of some complex nanostructures [16–19 ]. Transformation optics is well-known method in metal-material community [20–24 ]. As a specific subset of it, conformal transformation optics can effectively be used to analyze the problem in two dimensions [21,23 ]. A milestone review of transformation optics, which we recommend to interested readers, has been given in [23]. However, there is almost no one successful analysis of plasmonic devices with gain medium based on the transformation optics, heretofore.

In the present paper, we will focus on the extension of the transformation optics into the analysis of crescent nanowire with gain medium as the core. In our model, by conformal transformation, a specific subset of transformation optics, the poles of the electrostatic potential function without gain medium are in the first and the third quadrant of the complex plane, but by introducing the gain medium, the poles of the electrostatic potential function can be in the second and the fourth quadrant of the complex plane, which will result in negative extinction cross section. Thereinto, the positive sign corresponds to absorption and the negative sign corresponds to emission for the extinction cross section.

Particularly, we will apply the improved analytical model by conformal transformation in the characteristic analysis of a broadband plasmonic emitting device, which consists of a gain nanowire core and a crescent silver shell. The plasmonic crescent nanowire device has been proposed for the efficient harvesting of light and concentration of the energy to the nanoscale over the whole visible spectrum [16]. In the present paper, we will show that the gain-assistant crescent plasmonic device could serve as a broad emission device when the imaginary part of the relative permittivity of the gain medium is larger than the loss compensation threshold. This kind of nanoscale broadband plasmonic source might lead to transformative technological applications, including all-optical on-chip information processing, miniaturized smart display technology and super-resolution biomedical imaging. Although clearly this emitting phenomenon is not a SPASER, it might be useful for the design of SPASERs.

2. Principle

This paper aims at extending the transformation optics to analyze plasmonic nano-structures with the loss compensation considered. As a numerical case, we will design and characterize a crescent plasmonic device by using our improved analytical model based on the transformation optics. The device can contribute to a broadband plasmonic emission. As a theoretical work, we will make emphasis on the feasibility analysis whether the transformation optics is suitable for the analysis of plasmonic nanostructures with gain materials. Therefore, detailed fabrication process does not involve in the present paper. However Fig. 1 presents a viable manufacturing process for our numerical structure in the following analysis. Figure 1(a) shows the proposed design of the emitting device based on gain-medium nanowire coated with the crescent metallic layer, where the red arrows in show the emission direction. Figure 1(b) presents a feasible fabrication process on the novel design. The first step is casting a monolayer of gain medium nanowires on a photoresist-coated substrate; the second step is coating a metal layer on the surfaces of the gain medium nanowires by electron beam evaporation; and the third step is lift-off of the metal-coated gain medium nanowires from the substrate. In fact, the similar fabrication process has been reported in Ref [25], although their work is based on crescent nano-spheres, as well restricted to conventional plasmonic structure (i.e. without gain medium). In this study, the metal is chosen as silver, and the gain medium core can be chosen to be silica doped with optical gain inclusions (e.g. organic dyes, rare earth ions), silicon nitride, gallium nitride, or other semiconductor structures.

 

Fig. 1 The 3D sketch (a) and the fabrication procedure (b).

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As shown in Fig. 1, an array of the plasmonic devices can be formed in the fabrication process, which might have some different phenomenon from a single nanostructure. The coupling between two arbitrary SPASER nanostructures has been confirmed to exist, which can contribute to an enhanced directivity of the far-field radiation [26]. However, as shown in the fabrication step III (see Fig. 1 (b)), the final metal-coated gain nanowires will be lifted off from the substrate. By using dilution and manipulation (e.g., optical tweezers), we can succeed in obtaining single plasmonic emitting device. Therefore, in the following discussion, we only consider a single gain nanowire with crescent metal coating and characterize its performance.

To simplify the processes of the analysis, the crescent nanowire is simplified into a 2-dimension structure, with the third dimension length being infinite, although the 3-dimension method has also been proposed [19].

We cannot directly get the electric field distribution map of the crescent. However, the coupling of a dipole with SPPs (Surface Plasmon Polaritons) supported by an infinite slab of metal is easy to analyze. Then, for the quasistatic approximation, we apply the conformal transformation, thereby transforming the infinite thin slab of metal into the crescent structure, with the electrostatic potential preserved. The mathematics of the conformal transformation closely links the physics of the different geometries, and then converts the infinite structure into a finite structure, at the same time preserving the spectrum. A continuous spectrum is usually associated with infinite physical size. The infinite thin slab can create a continuous spectrum of SPPs, and so is the property of the finite structure of crescent.

3. Analytical model

We use the conformal transformation to study the property of the crescent nanowire structure. Although conformal transformation has already been used to analyze the nanoparticles, it has not yet been used to investigate nanoparticles containing gain medium.

We first consider a simple structure, which is a thin metal slab, with air on the left side and the gain medium on the right side, as shown in Fig. 2(a) , where the thickness of the slab is $d$, the relative permittivity of air is ${\epsilon }_{air}=1$, the relative permittivity of the metal slab and gain medium are ${\epsilon }_m$ and ${\epsilon }_g$, respectively. This air-slab-gain structure is excited by a dipole source at the zero point of the $xy$ plane, and the distance between the dipole and the metal slab is $a$.

 

Fig. 2 (a) A thin layer of metal with air on the left domain and gain medium on the right domain. (b) The transformed crescent structure with air outside and gain medium inside. The dipole source is transformed into a uniform electric field. (c) The complex plane of the integration factor$k$. $l_1$ is the integration path along the real axis, which is from $-\infty$ to$+\infty$. $l_2$ is the integration path along a semicircle with the radius approaching $+\infty$. $\sum$ is the area surrounded by the lines $l_1$ and $l_2$.

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As shown in Fig. 2(a), air-slab-gain structure is excited by a dipole source at the zero point of $xy$ plane, and the distance between the dipole and the metal slab is $a$. Applying the conformal transformation, the dipole source can be transformed into a plane wave source, as shown in Fig. 2(b). Firstly the incident field ${\varphi }_{\Delta }$ of the dipole is expanded as a Fourier series in $y$

Then the induced field $\varphi (k)$ located at slab structure can be expressed as

These unknown coefficients $b(k)$, $c(k)$, $d(k)$, and $e(k)$ are then obtained by imposing continuity of the parallel component of the electric field and the normal component of the displacement field at the dielectric slab interfaces. So the resulting matching equations are

Note that Eqs. (1)-(6) are completely the same as those mentioned in Ref [17]. However Eqs. (7) and (8) have been improved by involving the gain medium.

Solving these four equations provides the following results:

Now that the induced potential is known in the $k$ space, it can be deduced in the real space via an inverse Fourier transform,

At last, for $x<a$, by injecting the expressions of $a(k)$ and $b(k)$, we obtain the electrostatic potential function of the dipole coupling to the metal slab can be obtained as follows:

The pole of this integration is

The integration course of Eq. (14) is integrating the factor $k$ from $-\infty$ to $+\infty$, which is in the real domain. To obtain the result of the integration, the integrating field is expanded to the complex plane, as shown in Fig. 2(c). According to the residue theorem, the line integral $l_1+l_2$ is equal to the residue of the poles in the area $\Sigma$. The line integral $l_1$ is just Eq. (14), whereas the line integral $l_2$ is zero for the radius of the semicircle approaching $+\infty$. Then, due to the residue theorem, the results of Eq. (14) are calculated as

Furthermore, applying the equation $\overrightarrow{E}=-\nabla \varphi$, the electric fields at the dipole can be obtained as follows

Next, we apply the following conformal transformation

The original dipole is transformed into a uniform electric field ${E^{\prime }}_0=\overrightarrow{\Delta }/(2\pi {\epsilon }_0{g}^2)$, and it is assumed that the dimensions of the crescent is much smaller than the incoming wavelength. This small dimension makes possible the use of the quasistatic approximation to neglect the time derivatives in the Maxwell equations. The electric field induced at the dipole is of particular interest, since it is directly related to the net dipole moment of the crescent in the transformed geometry. In the slab frame, the dipole energy pumped into the SPPs can be calculated from the electric field due to the excited modes evaluated at the dipole$P=-\frac{\omega }{2}\mathrm{Im}\left\{{\overrightarrow{\Delta }}^{\ast }\cdot \overrightarrow{E}(z=0)\right\}$, which is equivalent to the power absorbed by the crescent from the uniform electric field in the transformed frame. Renormalized the power by the incoming flux $P_{in}={\epsilon }_0{c}_0{\left|{\overrightarrow{E}}^{\prime }\right|}^2/2$, the extinction cross section ${\sigma }_{ext}=P/P_{in}$ of the crescent structure can be deduced as [17,18 ]

Under the conformal transformation, the potential is preserved by the equation$\varphi (x,y)={\varphi }^{\prime }(x^{\prime },y^{\prime })$. The electric field ${\stackrel{\rightharpoonup }{E}}^{\prime }(x^{\prime },y^{\prime })$in the crescent can then be deduced from the potential ${E^{\prime }}_{u^{\prime }}=-\frac{\partial \varphi }{\partial x}\frac{\partial x}{\partial u^{\prime }}-\frac{\partial \varphi }{\partial y}\frac{\partial y}{\partial u^{\prime }}$(with $u^{\prime }=x^{\prime },y^{\prime }$). Apparently the total field enhancement can be given as follows:

The SPPs are excited along the crescent surface, and the angle $\theta$ can be used to describe the position of the electric field. From Eqs. (23) and (24) , we can get the maximum electric field position:

3. Numerical results

Based on the physical model above, we have investigated carefully the property of super-broad emission bandwidth. To verify the correctness of our analytical model, we have also used the Finite Element Method (a software with the name Comsol Multiphysics) method to analyze numerically the crescent nanowire structure. The metal is assumed to be silver with the relative permittivity, ${\epsilon }_m$, according to Johnson and Christy [27], and the gain medium core can be chosen to be silica doped with optical gain inclusions (e.g. organic dyes, rare earth ions). The extinction cross section is generally used to reflect the absorption property of the nanoparticles, which is straightforward to obtain by experimentation.

In the numerical analysis, the gain coefficients used are dispersionless. However, any practical gain material has its specific gain spectrum. Typically, some rare earth -doped materials can provide a gain spectrum with tens of nanometers bandwidth, and some semiconductor materials can provide a wider bandwidth of the gain spectrum, even up to 100 nm. Moreover, these gain materials have various distributions of the gain spectrum. We have shown that the results using actual gain spectra agree very well with those obtained from the dispersionless gain model. Only when the wavelength is far away from the resonant peak, does a slight difference occur [12]. Furthermore, the difference tends to be reduced as the bandwidth increases. Therefore, in the following numerical calculations, we will ignore the effect of the dispersion of the gain coefficient to focus on the analysis of the extinction cross section due to the gain medium.

We have first analyzed closely the relationship between the extinction cross section and the wavelength for the different structure parameters. As shown in Fig. 3 , we changed the ratio of the inner and outer diameters $\rho$, the outer diameter $D_o$, and the imaginary part of the relative permittivity of the gain medium ${\epsilon }_{g2}$, to obtain the numerical and theoretical extinction cross sections as a function of the wavelength. The ordinate is the extinction cross section over the outer diameter ${\sigma }_{ext}/D_o$, and the abscissa is the wavelength in the vacuum ${\lambda }_0$, which is varied from 360 nm to 800 nm. The solid curves are the numerical results, and the dotted curves are the theoretical results according to Eqs. (21) and (22) .

 

Fig. 3 The numerical (solid curves) and theoretical (dotted curves) extinction cross sections ${\sigma }_{ext}/D_o$ as a function of the wavelength ${\lambda }_0$ for different shapes of the crescents and different image parts of the relative permittivity of the gain medium. (I) (II) (III) (on the right side of the Fig. 3) are sketch maps of the normalized value of the electric field (blue) and of the $x^{\prime }$ component value of the electric field (green) for the different ratios $\rho =$0.4, 0.6, 0.8, respectively, at the same wavelength ${\lambda }_0=380$nm.

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The real part of the relative permittivity of the gain medium, ${\epsilon }_{g1}$, was chosen to be 2.09 (silica), and the imaginary part, ${\epsilon }_{g2}$, was chosen as 0 (Figs. 3(a), 3(d) and 3(g)), −0.1 (Figs. 3(b), 3(e) and 3(h)), and −0.2 (Figs. 3(c), 3(f) and 3(i)). As the absolute value of the imaginary part of the relative permittivity of the gain medium, ${\epsilon }_{g2}$, became larger, the extinction cross section became negative, which indicates emission.

The outer diameter, $D_o$, was varied from 20 nm to 100 nm, depicted by the different colors of the curves. The numerical results agreed well with the theoretical results for the smaller outer diameter, $D_o$, as seen from the different colors of the solid and dotted curves in Fig. 3.

For a typical plasmonic nanostructure, the radiative loss is also very important issue on its final characteristics, which is particularly important in the gain medium to enhance the effective dipole moment of the nanostructure. Related problems have detailed been discussed in Ref [18]. However, for the present plasmonic emitting device, the gain threshold mainly depends on the absorption and scattering loss from the metallic nanostructures [13–15 ]. Therefore, the influence of the radiative loss will be ignored in the present paper.

The ratio, $\rho$, was chosen as 0.4 (Figs. 3(a)-3(c)), 0.6 (Figs. 3(d)-3(f)), and 0.8 (Figs. 3(g)-3(i)), while on the right side of the Fig. 3 there are sketch maps (I), (II) and (III) of the normalized values of the electric field and of the $x^{\prime }$ component of the electric field for the different ratios at the same wavelength. From the sketch maps, we found that if the ratio, $\rho$, is bigger, the plasmonic resonance is stronger with a larger bandwidth, and the maximum value shifts to the longer wavelength. Therefore, this structure of the ratio $\rho =0.8$ is what we need to design the super-broad bandwidth emitting device.

Ideally, our emitting device should have a large negative extinction cross section over a broad, continuous spectrum of wavelength with a low threshold value. The results in Fig. 3 suggested to us to produce such crescent nanowire structures with a larger ratio, $\rho$. Considering the coincidence of the theoretical and numerical results at the smaller outer diameter $D_o$, we set the value $D_o=20$nm for further analysis, as follows.

The results for the outer diameter $D_o=20$nm were consequently obtained, as shown in Fig. 4 , where the ratio is $\rho =0.8$, and the outer diameter is $D_o=20$nm. The real part of the relative permittivity of the gain medium ${\epsilon }_{g1}$ was chosen as 4.0, 8.9 or 12.9 (these dielectric constant values cover the range of common gain material, e.g., silicon nitride, gallium nitride, and other semiconductor structures). The results for silica are shown in Fig. 3, where ${\epsilon }_{g1}=2.09$. For ${\epsilon }_{g1}=4.0$, the imaginary part ${\epsilon }_{g2}$ was chosen as 0, −0.1, −0.2, which correspond to the sub Figs. 4(a), 4(b) and 4(c), respectively. For ${\epsilon }_{g1}=8.9$, the imaginary part ${\epsilon }_{g2}$ was chosen as 0, −0.2, or −0.6, which correspond to the sub Figs. 4(d), 4(e) and 4(f), respectively. For the last three sub Figs. 4(g), 4(h) and 4(i), ${\epsilon }_{g1}=12.9$, and the imaginary part, ${\epsilon }_{g2}$, was also chosen as 0, −0.2 or −0.6, respectively. It is worth noting that the surface plasma wavelength is changed when the real part of the relative permittivity of the gain medium, ${\epsilon }_{g1}$, is changed. The surface plasma wavelengths were 362 nm, 393 nm, 485 nm, and 552 nm when the real part, ${\epsilon }_{g1}$, was 2.09, 4.0, 8.9, and 12.9, respectively, for the real part of the denominator of the Eq. (15) to be zero. The theory of conformal transformation was correct only in the case of plasmon resonance, and therefore, the wavelength ranges studied in the sub-figures of Fig. 4 were different.

 

Fig. 4 The numerical (solid curves) and theoretical (dotted curves) extinction cross sections ${\sigma }_{ext}/D_o$ as a function of the wavelength ${\lambda }_0$ for the ratio $\rho =0.8$, the outer diameter $D_o=20$nm, and different relative permittivities of the gain medium.

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We found that as the absolute value of the imaginary part, ${\epsilon }_{g2}$, increased the extinction cross section became negative for longer wavelengths, and as the imaginary part increased sufficiently the extinction cross section became negative for the entire wavelength range of visible light. However, the positive extinction cross section indicates absorption, while the negative extinction cross section indicates emission. The emission bandwidth, which is defined as the FWHM (Full Width at Half Maximum), is about 200 nm for ${\epsilon }_{g1}=2.9$, about 400 nm for ${\epsilon }_{g1}=4.0$, about 600 nm for ${\epsilon }_{g1}=8.9$, and about 700 nm for ${\epsilon }_{g1}=12.9$. The crescent structure has such a large bandwidth, because the large bandwidth property of the thin infinite slab is retained after the conformal transformation. Therefore, as long as the gain bandwidth of the gain material permits, our design for the real part of the relative permittivity of any gain material can have a sufficient plasmonic bandwidth to achieve super-continuum emission.

The polar plot is a very important data to obtain for the light-emitting device. In Fig. 5 , we show the numerical results of the polar plots of the intensities of the near-field electric field (Fig. 5(b)), the far-field electric field (Fig. 5(c)), and the power flow (Fig. 5(d)), where the outer diameter, $D_o=20$nm, $\rho =0.8$, ${\epsilon }_g=2.09-0.2i$, and the wavelength ${\lambda }_o=700$nm. Furthermore, Fig. 5(a) is a polar sketch map of the crescent structure. The far-field electric field polar plot performs as a dipole with a larger left loop; this is because the gain medium core undergoes a left shift in the structure. The near-field polar plot just indicates that the tips of the crescent structure has a strong concentrating effect. Thus, from Fig. 5, we can see that the emitting device of this structure has a very good directivity. However, with the same structure parameters, the directionality remains the same for the other wavelengths. Therefore the polar plots for the other wavelengths are not shown.

 

Fig. 5 The sketch of the crescent structure (a). The numerical results of the polar plots of the intensities of near-field electric field (b), the far-field electric field (c), and the power flow (d), where the outer diameter $D_o=20$nm, $\rho =0.8$, ${\epsilon }_g=2.09-0.2i$, and the wavelength ${\lambda }_0=700$nm.

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Then, we have analyzed the impact of gain materials with different relative permittivities on the loss compensation threshold. Here, the loss compensation threshold is defined as the value of the relative permittivity of the gain medium for a specific wavelength, which can cause the extinction cross section of the structure to switch from positive to negative. However, from Eqs. (21) and (22) , we can see that the expression of the extinction cross section is changed when the sign of the parameter, ${\alpha }_2$, is changed. Therefore, the parameter, ${\alpha }_2$, is an important factor in determining the threshold value.

From Eq. (15), applying the inequalities $\left|{\epsilon }_{m2}\right|<<\left|{\epsilon }_{m1}\right|,\left|{\epsilon }_{g2}\right|<<\left|{\epsilon }_{g1}\right|$, the form of the imaginary part of the pole, ${\alpha }_2$, can be obtained as follows:

The relative permittivity of the metal (${\epsilon }_{m1}$and${\epsilon }_{m2}$) is determined by the wavelength according to the silver dispersions [27]. Then, the other parameter that determines the imaginary part of the pole, ${\alpha }_2$, is the relative permittivity of the gain medium (${\epsilon }_{g1}$and${\epsilon }_{g2}$).

More specifically, for the wavelength ${\lambda }_o=800$nm and four kinds of gain media (${\epsilon }_{g1}=$2.09, 4.0, 8.9, 12.9), we obtained the numerical and theoretical extinction cross section ${\sigma }_{ext}/D_o$, $E_{\max }$ and ${\theta }_{E\max }$ as a function of the imaginary part of the relative permittivity of the gain medium ${\epsilon }_{g2}$, as shown in Fig. 6 , where ${\theta }_{E\max }$ is determined theoretically by Eq. (25). All the inflexions in Fig. 6 corresponded to the zero value of the imaginary part of the pole, ${\alpha }_2$. When the imaginary part of the pole, ${\alpha }_2$, changed from positive to negative, the equation of the extinction cross section changed theoretically from Eq. (21) to Eq. (22), and there was the same kind of inflexion in the Eq. (25). The $E_{\max }$ of the whole field demonstrated a maximum enhancement over about five orders of magnitude at the changing point of ${\alpha }_2$. This indicates that, at the threshold value, the local electric field enhancement reaches a maximum value, because the threshold value is the critical value between the absorption and emission [10]. In addition, we obtained the numerical and theoretical threshold values of the gain compensation as a function of the wavelength ${\lambda }_0$, as shown in Fig. 7 . The threshold value was very small for the entire wavelength range of visible light. Thus, it is easy to realize the emission phenomenon for the whole wavelength range of the visible light.

 

Fig. 6 The numerical (solid curves) and theoretical (dotted curves) extinction cross section ${\sigma }_{ext}/D_o$, $E_{\max }$ and ${\theta }_{E\max }$ as a function of the imaginary part of the relative permittivity of the gain medium, ${\epsilon }_{g2}$.

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Fig. 7 The numerical (solid curves) and theoretical (dotted curves) threshold value of the loss compensation as a function of the wavelength ${\lambda }_0$.

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4. Conclusion

We studied the crescent nanostructures with gain medium inside by the theoretical and numerical methods. We found that the imaginary part of the pole, ${\alpha }_2$, determines the sign of the extinction cross-section, where the positive sign corresponds to absorption, and the negative sign corresponds to emission. With the small threshold value for the whole wavelength range of visible light, we can obtain a nano plasmonic emitting device with a super-broad bandwidth.