Abstract

The crescent nanostructure with gain medium inside is theoretically studied to analyze the characteristic of plasmonic emitting with wide bandwidth. An accurate analytical model is built based on the transformation optics. In this model, the poles of the electrostatic potential function are in the second and the fourth quadrant of the complex plane if the imaginary part of the relative permittivity of the gain medium is larger than the loss compensation threshold, and then the extinction cross section is to be negative by integrating the electrostatic potential over the half complex plane via an inverse Fourier transform. The positive extinction cross section corresponds to absorption, and the negative corresponds to emission. The proposed analytical model agrees well with the numerical simulation results based on the finite element method, to give a physical insight into the loss compensation property of the plasmonic nanostuctures. Results show that the negative extinction cross section is realizable by introducing the gain medium into a plasmonic crescent nanowire, which is equivalent to an emitting device with wide bandwidth.

© 2015 Optical Society of America

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 εair=1, the relative permittivity of the metal slab and gain medium are εm and εg, 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 factork. l1 is the integration path along the real axis, which is from to+. l2 is the integration path along a semicircle with the radius approaching +. is the area surrounded by the lines l1 and l2.

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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 ϕΔ of the dipole is expanded as a Fourier series in y

ϕΔ(r)=12πε0Δrr2=12πdkϕΔ(k)eiky,
ϕΔ(k)=ϕΔ(x,y)eikydy=a(k)e|k||x|,
a(k)=sgn(x)Δx+isgn(k)Δy2ε0.

Then the induced field ϕ(k) located at slab structure can be expressed as

ϕ(k)={b(k)e|k|x,c(k)e|k|x+d(k)e|k|x,e(k)e|k|x,x<aa<x<a+dx>a+d.

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

a(k)e|k|a+b(k)e|k|a=c(k)e|k|a+d(k)e|k|a,
c(k)e|k|(a+d)+d(k)e|k|(a+d)=e(k)e|k|(a+d),
a(k)e|k|a+b(k)e|k|a=εm[c(k)e|k|a+d(k)e|k|a],
εm[c(k)e|k|(a+d)+d(k)e|k|(a+d)]=εge(k)e|k|(a+d).

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:

b(k)=[εmεgεm+εgεm1εm+1e2|k|d]e2|k|ae2|k|de2αa(k),
c(k)=2εm+1e2|k|de2|k|de2αa(k),
d(k)=2(εmεg)(εm+1)(εm+εg)e2|k|ae2|k|de2αa(k),
e(k)=4εm(εm+1)(εm+εg)e2|k|de2|k|de2αa(k).

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

ϕ(x,y)=12π{b(k)eiky+|k|xdk,[c(k)e|k|x+d(k)e|k|x]eikydk,e(k)eiky|k|xdk,x<aa<x<a+dx>a+d.

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:

ϕ(x<a)=14πε0+[Δxisgn(k)Δy][εmεgεm+εgεm1εm+1e2|k|d]e2|k|ae2|k|de2αeiky+|k|xdk,
where Δ=(Δx,Δy) is the dipole. Note that

The pole of this integration is

|kd|=α=12ln(εm1)(εmεg)(εm+1)(εm+εg),(forεm1<εg1<1)
where the parameters εm=εm1+iεm2, εg=εg1+iεg2, and α=α1+iα2 are complex numbers.

The integration course of Eq. (14) is integrating the factor k from to +, 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 l1+l2 is equal to the residue of the poles in the area Σ. The line integral l1 is just Eq. (14), whereas the line integral l2 is zero for the radius of the semicircle approaching +. Then, due to the residue theorem, the results of Eq. (14) are calculated as

forα2>0ϕ(x<a)=1ε0dεmεm21(iΔx+sgn(y)Δy)eαd(x2a)eiαd|y|,
forα2<0ϕ(x<a)=1ε0dεmεm21(iΔx+sgn(y)Δy)eαd(x2a)eiαd|y|.

Furthermore, applying the equation E=ϕ, the electric fields at the dipole can be obtained as follows

forα2>0,E(z=0)=iαε0dεmεm21e2αadΔ,
forα2<0,E(z=0)=iαε0dεmεm21e2αadΔ,
where z=x+iy is a complex number in the complex plane xy.

Next, we apply the following conformal transformation

z=g2z,
where g is a real constant. The resulting structure is a crescent shape, as shown in Fig. 1(b). The diameters of the inner and outer circles are, respectively, Di=g2/(a+d) and Do=g2/a.

The original dipole is transformed into a uniform electric field E0=Δ/(2πε0g2), 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 dipoleP=ω2Im{ΔE(z=0)}, 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 Pin=ε0c0|E|2/2, the extinction cross section σext=P/Pin of the crescent structure can be deduced as [17,18 ]

σext=4π2k0(ρ1ρ)2Do2Re{αεm1εm2e2αρ1ρ},(α2>0)
σext=4π2k0(ρ1ρ)2Do2Re{αεm1εm2e2αρ1ρ},(α2<0)
where k0=2π/λ0 is the wave number in the vacuum, λ0is the wavelength in the vacuum, and ρ=Di/Do is the ratio of the inner and outer diameters.

Under the conformal transformation, the potential is preserved by the equationϕ(x,y)=ϕ(x,y). The electric field E(x,y)in the crescent can then be deduced from the potential Eu=ϕxxuϕyyu(with u=x,y). Apparently the total field enhancement can be given as follows:

|EE0|=2π(ρ1ρ)2|ln(εm1)(εmεg)(εm+1)(εm+εg)εmεm21[(εm1)(εmεg)(εm+1)(εm+εg)]ρ2(1ρ)|exp(ρ1ρα2|tan(θ2)|)cos2(θ2),(α2>0)
|EE0|=2π(ρ1ρ)2|ln(εm1)(εmεg)(εm+1)(εm+εg)εmεm21[(εm1)(εmεg)(εm+1)(εm+εg)]ρ2(1ρ)|exp(ρ1ρα2|tan(θ2)|)cos2(θ2),(α2<0)
where the angle θ is defined in Fig. 2(c).

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

θEmax=π|arcsin(ρ1ρα2)|.

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, εm, 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 ρ, the outer diameter Do, and the imaginary part of the relative permittivity of the gain medium εg2, 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 σext/Do, and the abscissa is the wavelength in the vacuum λ0, 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 σext/Do as a function of the wavelength λ0 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 component value of the electric field (green) for the different ratios ρ=0.4, 0.6, 0.8, respectively, at the same wavelength λ0=380nm.

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The real part of the relative permittivity of the gain medium, εg1, was chosen to be 2.09 (silica), and the imaginary part, εg2, 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, εg2, became larger, the extinction cross section became negative, which indicates emission.

The outer diameter, Do, 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, Do, 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, ρ, 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 component of the electric field for the different ratios at the same wavelength. From the sketch maps, we found that if the ratio, ρ, 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 ρ=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, ρ. Considering the coincidence of the theoretical and numerical results at the smaller outer diameter Do, we set the value Do=20nm for further analysis, as follows.

The results for the outer diameter Do=20nm were consequently obtained, as shown in Fig. 4 , where the ratio is ρ=0.8, and the outer diameter is Do=20nm. The real part of the relative permittivity of the gain medium εg1 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 εg1=2.09. For εg1=4.0, the imaginary part εg2 was chosen as 0, −0.1, −0.2, which correspond to the sub Figs. 4(a), 4(b) and 4(c), respectively. For εg1=8.9, the imaginary part εg2 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), εg1=12.9, and the imaginary part, εg2, 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, εg1, is changed. The surface plasma wavelengths were 362 nm, 393 nm, 485 nm, and 552 nm when the real part, εg1, 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 σext/Do as a function of the wavelength λ0 for the ratio ρ=0.8, the outer diameter Do=20nm, and different relative permittivities of the gain medium.

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We found that as the absolute value of the imaginary part, εg2, 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 εg1=2.9, about 400 nm for εg1=4.0, about 600 nm for εg1=8.9, and about 700 nm for εg1=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, Do=20nm, ρ=0.8, εg=2.090.2i, and the wavelength λo=700nm. 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 Do=20nm, ρ=0.8, εg=2.090.2i, and the wavelength λ0=700nm.

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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, α2, is changed. Therefore, the parameter, α2, is an important factor in determining the threshold value.

From Eq. (15), applying the inequalities |εm2|<<|εm1|,|εg2|<<|εg1|, the form of the imaginary part of the pole, α2, can be obtained as follows:

α2εm2εm121+εm2εg1εm1εg2εm12εg12.

The relative permittivity of the metal (εm1andεm2) is determined by the wavelength according to the silver dispersions [27]. Then, the other parameter that determines the imaginary part of the pole, α2, is the relative permittivity of the gain medium (εg1andεg2).

More specifically, for the wavelength λo=800nm and four kinds of gain media (εg1=2.09, 4.0, 8.9, 12.9), we obtained the numerical and theoretical extinction cross section σext/Do, Emax and θEmax as a function of the imaginary part of the relative permittivity of the gain medium εg2, as shown in Fig. 6 , where θEmax is determined theoretically by Eq. (25). All the inflexions in Fig. 6 corresponded to the zero value of the imaginary part of the pole, α2. When the imaginary part of the pole, α2, 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 Emax of the whole field demonstrated a maximum enhancement over about five orders of magnitude at the changing point of α2. 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 λ0, 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 σext/Do, Emax and θEmax as a function of the imaginary part of the relative permittivity of the gain medium, εg2.

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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 λ0.

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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, α2, 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.

Acknowledgments

Parts of this work were supported by the National Natural Science Foundation of China (No. 61378091, 11204226); the National Basic Research Program of China (grant No. 2015CB352005, 2012CB825802); the Shenzhen Research Project (ZDSYS20140430164957663, KQCX20140509172719305); Guangdong Natural Science Foundation (2014A030312008); the Training Plan of Guangdong Province Outstanding Young Teachers in Higher Education Institutions (Yq2013142); and the Science and Technology Innovation Project of Guangdong Province (2013KJCX0158).

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13. D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90(2), 027402 (2003). [CrossRef]   [PubMed]  

14. M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express 19(22), 22029–22106 (2011). [PubMed]  

15. V. Apalkov and M. I. Stockman, “Proposed graphene nanospaser,” Light: Sci. Appl. 3(7), e191 (2014). [CrossRef]  

16. A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10(7), 2574–2579 (2010). [CrossRef]   [PubMed]  

17. A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82(12), 125430 (2010). [CrossRef]  

18. A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B 82(20), 205109 (2010). [CrossRef]  

19. A. I. Fernández-Domínguez, S. A. Maier, and J. B. Pendry, “Transformation optics description of touching metal nanospheres,” Phys. Rev. B 85(16), 165148 (2012). [CrossRef]  

20. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef]   [PubMed]  

21. U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef]   [PubMed]  

22. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010). [CrossRef]   [PubMed]  

23. L. Xu and H. Chen, “Conformal transformation optics,” Nat. Photonics 9(1), 15–23 (2014). [CrossRef]  

24. J. B. Pendry, A. Aubry, D. R. Smith, and S. A. Maier, “Transformation optics and subwavelength control of light,” Science 337(6094), 549–552 (2012). [CrossRef]   [PubMed]  

25. Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005). [CrossRef]   [PubMed]  

26. A. V. Dorofeenko, A. A. Zyablovsky, A. P. Vinogradov, E. S. Andrianov, A. A. Pukhov, and A. A. Lisyansky, “Steady state superradiance of a 2D-spaser array,” Opt. Express 21(12), 14539–14547 (2013). [CrossRef]   [PubMed]  

27. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]  

References

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  1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, L. Thio, and P. A. Wolff, “Extraordinary optical transmission through small apertures,” Nature 391(6668), 667–669 (1998).
    [Crossref]
  2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
    [Crossref] [PubMed]
  3. K. Kim, S. J. Yoon, and D. Kim, “Nanowire-based enhancement of localized surface plasmon resonance for highly sensitive detection: a theoretical study,” Opt. Express 14(25), 12419–12431 (2006).
    [Crossref] [PubMed]
  4. T. J. Davis, K. C. Vernon, and D. E. Gómez, “Effect of retardation on localized surface plasmon resonances in a metallic nanorod,” Opt. Express 17(26), 23655–23663 (2009).
    [Crossref] [PubMed]
  5. R. E. Arias and A. A. Maradudin, “Scattering of a surface plasmon polariton by a localized dielectric surface defect,” Opt. Express 21(8), 9734–9756 (2013).
    [Crossref] [PubMed]
  6. H. Yang, W. Lee, T. Hwang, and D. Kim, “Probabilistic evaluation of surface-enhanced localized surface plasmon resonance biosensing,” Opt. Express 22(23), 28412–28426 (2014).
    [Crossref] [PubMed]
  7. S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997).
    [Crossref] [PubMed]
  8. S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97(1), 017402 (2006).
    [Crossref] [PubMed]
  9. S. Kim, J. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
    [Crossref] [PubMed]
  10. Z. Y. Li and Y. Xia, “Metal nanoparticles with gain toward single-molecule detection by surface-enhanced Raman scattering,” Nano Lett. 10(1), 243–249 (2010).
    [Crossref] [PubMed]
  11. S.-Y. Liu, J. Li, F. Zhou, L. Gan, and Z. Y. Li, “Efficient surface plasmon amplification from gain-assisted gold nanorods,” Opt. Lett. 36(7), 1296–1298 (2011).
    [PubMed]
  12. J. Xian, L. Chen, H. Niu, J. Qu, and J. Song, “Significant field enhancements in an individual silver nanoparticle near a substrate covered with a thin gain film,” Nanoscale 6(22), 13994–14001 (2014).
    [Crossref] [PubMed]
  13. D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90(2), 027402 (2003).
    [Crossref] [PubMed]
  14. M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express 19(22), 22029–22106 (2011).
    [PubMed]
  15. V. Apalkov and M. I. Stockman, “Proposed graphene nanospaser,” Light: Sci. Appl. 3(7), e191 (2014).
    [Crossref]
  16. A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10(7), 2574–2579 (2010).
    [Crossref] [PubMed]
  17. A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82(12), 125430 (2010).
    [Crossref]
  18. A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B 82(20), 205109 (2010).
    [Crossref]
  19. A. I. Fernández-Domínguez, S. A. Maier, and J. B. Pendry, “Transformation optics description of touching metal nanospheres,” Phys. Rev. B 85(16), 165148 (2012).
    [Crossref]
  20. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
    [Crossref] [PubMed]
  21. U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
    [Crossref] [PubMed]
  22. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
    [Crossref] [PubMed]
  23. L. Xu and H. Chen, “Conformal transformation optics,” Nat. Photonics 9(1), 15–23 (2014).
    [Crossref]
  24. J. B. Pendry, A. Aubry, D. R. Smith, and S. A. Maier, “Transformation optics and subwavelength control of light,” Science 337(6094), 549–552 (2012).
    [Crossref] [PubMed]
  25. Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005).
    [Crossref] [PubMed]
  26. A. V. Dorofeenko, A. A. Zyablovsky, A. P. Vinogradov, E. S. Andrianov, A. A. Pukhov, and A. A. Lisyansky, “Steady state superradiance of a 2D-spaser array,” Opt. Express 21(12), 14539–14547 (2013).
    [Crossref] [PubMed]
  27. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
    [Crossref]

2014 (4)

H. Yang, W. Lee, T. Hwang, and D. Kim, “Probabilistic evaluation of surface-enhanced localized surface plasmon resonance biosensing,” Opt. Express 22(23), 28412–28426 (2014).
[Crossref] [PubMed]

J. Xian, L. Chen, H. Niu, J. Qu, and J. Song, “Significant field enhancements in an individual silver nanoparticle near a substrate covered with a thin gain film,” Nanoscale 6(22), 13994–14001 (2014).
[Crossref] [PubMed]

V. Apalkov and M. I. Stockman, “Proposed graphene nanospaser,” Light: Sci. Appl. 3(7), e191 (2014).
[Crossref]

L. Xu and H. Chen, “Conformal transformation optics,” Nat. Photonics 9(1), 15–23 (2014).
[Crossref]

2013 (2)

2012 (2)

J. B. Pendry, A. Aubry, D. R. Smith, and S. A. Maier, “Transformation optics and subwavelength control of light,” Science 337(6094), 549–552 (2012).
[Crossref] [PubMed]

A. I. Fernández-Domínguez, S. A. Maier, and J. B. Pendry, “Transformation optics description of touching metal nanospheres,” Phys. Rev. B 85(16), 165148 (2012).
[Crossref]

2011 (2)

2010 (5)

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref] [PubMed]

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10(7), 2574–2579 (2010).
[Crossref] [PubMed]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82(12), 125430 (2010).
[Crossref]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B 82(20), 205109 (2010).
[Crossref]

Z. Y. Li and Y. Xia, “Metal nanoparticles with gain toward single-molecule detection by surface-enhanced Raman scattering,” Nano Lett. 10(1), 243–249 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (1)

S. Kim, J. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[Crossref] [PubMed]

2006 (4)

K. Kim, S. J. Yoon, and D. Kim, “Nanowire-based enhancement of localized surface plasmon resonance for highly sensitive detection: a theoretical study,” Opt. Express 14(25), 12419–12431 (2006).
[Crossref] [PubMed]

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97(1), 017402 (2006).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

2005 (1)

Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005).
[Crossref] [PubMed]

2003 (1)

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90(2), 027402 (2003).
[Crossref] [PubMed]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref] [PubMed]

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, L. Thio, and P. A. Wolff, “Extraordinary optical transmission through small apertures,” Nature 391(6668), 667–669 (1998).
[Crossref]

1997 (1)

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997).
[Crossref] [PubMed]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Andrianov, E. S.

Apalkov, V.

V. Apalkov and M. I. Stockman, “Proposed graphene nanospaser,” Light: Sci. Appl. 3(7), e191 (2014).
[Crossref]

Arias, R. E.

Aubry, A.

J. B. Pendry, A. Aubry, D. R. Smith, and S. A. Maier, “Transformation optics and subwavelength control of light,” Science 337(6094), 549–552 (2012).
[Crossref] [PubMed]

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10(7), 2574–2579 (2010).
[Crossref] [PubMed]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82(12), 125430 (2010).
[Crossref]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B 82(20), 205109 (2010).
[Crossref]

Bergman, D. J.

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90(2), 027402 (2003).
[Crossref] [PubMed]

Chan, C. T.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref] [PubMed]

Chen, H.

L. Xu and H. Chen, “Conformal transformation optics,” Nat. Photonics 9(1), 15–23 (2014).
[Crossref]

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref] [PubMed]

Chen, L.

J. Xian, L. Chen, H. Niu, J. Qu, and J. Song, “Significant field enhancements in an individual silver nanoparticle near a substrate covered with a thin gain film,” Nanoscale 6(22), 13994–14001 (2014).
[Crossref] [PubMed]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Davis, T. J.

Dorofeenko, A. V.

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, L. Thio, and P. A. Wolff, “Extraordinary optical transmission through small apertures,” Nature 391(6668), 667–669 (1998).
[Crossref]

Emory, S. R.

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997).
[Crossref] [PubMed]

Fernández-Domínguez, A. I.

A. I. Fernández-Domínguez, S. A. Maier, and J. B. Pendry, “Transformation optics description of touching metal nanospheres,” Phys. Rev. B 85(16), 165148 (2012).
[Crossref]

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10(7), 2574–2579 (2010).
[Crossref] [PubMed]

Gan, L.

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, L. Thio, and P. A. Wolff, “Extraordinary optical transmission through small apertures,” Nature 391(6668), 667–669 (1998).
[Crossref]

Gómez, D. E.

Håkanson, U.

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97(1), 017402 (2006).
[Crossref] [PubMed]

Hwang, T.

Jin, J.

S. Kim, J. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[Crossref] [PubMed]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Kim, D.

Kim, J.

Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005).
[Crossref] [PubMed]

Kim, K.

Kim, S.

S. Kim, J. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[Crossref] [PubMed]

Kim, S. W.

S. Kim, J. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[Crossref] [PubMed]

Kim, Y.

S. Kim, J. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[Crossref] [PubMed]

Kim, Y. J.

S. Kim, J. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[Crossref] [PubMed]

Kühn, S.

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97(1), 017402 (2006).
[Crossref] [PubMed]

Lee, L. P.

Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005).
[Crossref] [PubMed]

Lee, W.

Lei, D. Y.

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10(7), 2574–2579 (2010).
[Crossref] [PubMed]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82(12), 125430 (2010).
[Crossref]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B 82(20), 205109 (2010).
[Crossref]

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, L. Thio, and P. A. Wolff, “Extraordinary optical transmission through small apertures,” Nature 391(6668), 667–669 (1998).
[Crossref]

Li, J.

Li, Z. Y.

S.-Y. Liu, J. Li, F. Zhou, L. Gan, and Z. Y. Li, “Efficient surface plasmon amplification from gain-assisted gold nanorods,” Opt. Lett. 36(7), 1296–1298 (2011).
[PubMed]

Z. Y. Li and Y. Xia, “Metal nanoparticles with gain toward single-molecule detection by surface-enhanced Raman scattering,” Nano Lett. 10(1), 243–249 (2010).
[Crossref] [PubMed]

Lisyansky, A. A.

Liu, G. L.

Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005).
[Crossref] [PubMed]

Liu, S.-Y.

Lu, Y.

Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005).
[Crossref] [PubMed]

Maier, S. A.

J. B. Pendry, A. Aubry, D. R. Smith, and S. A. Maier, “Transformation optics and subwavelength control of light,” Science 337(6094), 549–552 (2012).
[Crossref] [PubMed]

A. I. Fernández-Domínguez, S. A. Maier, and J. B. Pendry, “Transformation optics description of touching metal nanospheres,” Phys. Rev. B 85(16), 165148 (2012).
[Crossref]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B 82(20), 205109 (2010).
[Crossref]

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10(7), 2574–2579 (2010).
[Crossref] [PubMed]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82(12), 125430 (2010).
[Crossref]

Maradudin, A. A.

Mejia, Y. X.

Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005).
[Crossref] [PubMed]

Nie, S.

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997).
[Crossref] [PubMed]

Niu, H.

J. Xian, L. Chen, H. Niu, J. Qu, and J. Song, “Significant field enhancements in an individual silver nanoparticle near a substrate covered with a thin gain film,” Nanoscale 6(22), 13994–14001 (2014).
[Crossref] [PubMed]

Park, I. Y.

S. Kim, J. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[Crossref] [PubMed]

Pendry, J. B.

J. B. Pendry, A. Aubry, D. R. Smith, and S. A. Maier, “Transformation optics and subwavelength control of light,” Science 337(6094), 549–552 (2012).
[Crossref] [PubMed]

A. I. Fernández-Domínguez, S. A. Maier, and J. B. Pendry, “Transformation optics description of touching metal nanospheres,” Phys. Rev. B 85(16), 165148 (2012).
[Crossref]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B 82(20), 205109 (2010).
[Crossref]

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10(7), 2574–2579 (2010).
[Crossref] [PubMed]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82(12), 125430 (2010).
[Crossref]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref] [PubMed]

Pukhov, A. A.

Qu, J.

J. Xian, L. Chen, H. Niu, J. Qu, and J. Song, “Significant field enhancements in an individual silver nanoparticle near a substrate covered with a thin gain film,” Nanoscale 6(22), 13994–14001 (2014).
[Crossref] [PubMed]

Rogobete, L.

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97(1), 017402 (2006).
[Crossref] [PubMed]

Sandoghdar, V.

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97(1), 017402 (2006).
[Crossref] [PubMed]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

Sheng, P.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref] [PubMed]

Smith, D. R.

J. B. Pendry, A. Aubry, D. R. Smith, and S. A. Maier, “Transformation optics and subwavelength control of light,” Science 337(6094), 549–552 (2012).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

Song, J.

J. Xian, L. Chen, H. Niu, J. Qu, and J. Song, “Significant field enhancements in an individual silver nanoparticle near a substrate covered with a thin gain film,” Nanoscale 6(22), 13994–14001 (2014).
[Crossref] [PubMed]

Sonnefraud, Y.

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10(7), 2574–2579 (2010).
[Crossref] [PubMed]

Stockman, M. I.

V. Apalkov and M. I. Stockman, “Proposed graphene nanospaser,” Light: Sci. Appl. 3(7), e191 (2014).
[Crossref]

M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express 19(22), 22029–22106 (2011).
[PubMed]

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90(2), 027402 (2003).
[Crossref] [PubMed]

Thio, L.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, L. Thio, and P. A. Wolff, “Extraordinary optical transmission through small apertures,” Nature 391(6668), 667–669 (1998).
[Crossref]

Vernon, K. C.

Vinogradov, A. P.

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, L. Thio, and P. A. Wolff, “Extraordinary optical transmission through small apertures,” Nature 391(6668), 667–669 (1998).
[Crossref]

Xia, Y.

Z. Y. Li and Y. Xia, “Metal nanoparticles with gain toward single-molecule detection by surface-enhanced Raman scattering,” Nano Lett. 10(1), 243–249 (2010).
[Crossref] [PubMed]

Xian, J.

J. Xian, L. Chen, H. Niu, J. Qu, and J. Song, “Significant field enhancements in an individual silver nanoparticle near a substrate covered with a thin gain film,” Nanoscale 6(22), 13994–14001 (2014).
[Crossref] [PubMed]

Xu, L.

L. Xu and H. Chen, “Conformal transformation optics,” Nat. Photonics 9(1), 15–23 (2014).
[Crossref]

Yang, H.

Yoon, S. J.

Zhou, F.

Zyablovsky, A. A.

Light: Sci. Appl. (1)

V. Apalkov and M. I. Stockman, “Proposed graphene nanospaser,” Light: Sci. Appl. 3(7), e191 (2014).
[Crossref]

Nano Lett. (3)

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10(7), 2574–2579 (2010).
[Crossref] [PubMed]

Z. Y. Li and Y. Xia, “Metal nanoparticles with gain toward single-molecule detection by surface-enhanced Raman scattering,” Nano Lett. 10(1), 243–249 (2010).
[Crossref] [PubMed]

Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005).
[Crossref] [PubMed]

Nanoscale (1)

J. Xian, L. Chen, H. Niu, J. Qu, and J. Song, “Significant field enhancements in an individual silver nanoparticle near a substrate covered with a thin gain film,” Nanoscale 6(22), 13994–14001 (2014).
[Crossref] [PubMed]

Nat. Mater. (1)

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref] [PubMed]

Nat. Photonics (1)

L. Xu and H. Chen, “Conformal transformation optics,” Nat. Photonics 9(1), 15–23 (2014).
[Crossref]

Nature (2)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, L. Thio, and P. A. Wolff, “Extraordinary optical transmission through small apertures,” Nature 391(6668), 667–669 (1998).
[Crossref]

S. Kim, J. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[Crossref] [PubMed]

Opt. Express (6)

Opt. Lett. (1)

Phys. Rev. B (4)

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82(12), 125430 (2010).
[Crossref]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B 82(20), 205109 (2010).
[Crossref]

A. I. Fernández-Domínguez, S. A. Maier, and J. B. Pendry, “Transformation optics description of touching metal nanospheres,” Phys. Rev. B 85(16), 165148 (2012).
[Crossref]

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Phys. Rev. Lett. (3)

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90(2), 027402 (2003).
[Crossref] [PubMed]

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97(1), 017402 (2006).
[Crossref] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref] [PubMed]

Science (4)

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

J. B. Pendry, A. Aubry, D. R. Smith, and S. A. Maier, “Transformation optics and subwavelength control of light,” Science 337(6094), 549–552 (2012).
[Crossref] [PubMed]

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Figures (7)

Fig. 1
Fig. 1 The 3D sketch (a) and the fabrication procedure (b).
Fig. 2
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 to + . l 2 is the integration path along a semicircle with the radius approaching + . is the area surrounded by the lines l 1 and l 2 .
Fig. 3
Fig. 3 The numerical (solid curves) and theoretical (dotted curves) extinction cross sections σ e x t / D o as a function of the wavelength λ 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 component value of the electric field (green) for the different ratios ρ = 0.4, 0.6, 0.8, respectively, at the same wavelength λ 0 = 380 nm.
Fig. 4
Fig. 4 The numerical (solid curves) and theoretical (dotted curves) extinction cross sections σ e x t / D o as a function of the wavelength λ 0 for the ratio ρ = 0.8 , the outer diameter D o = 20 nm, and different relative permittivities of the gain medium.
Fig. 5
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, ρ = 0.8 , ε g = 2.09 0.2 i , and the wavelength λ 0 = 700 nm.
Fig. 6
Fig. 6 The numerical (solid curves) and theoretical (dotted curves) extinction cross section σ e x t / D o , E max and θ E max as a function of the imaginary part of the relative permittivity of the gain medium, ε g 2 .
Fig. 7
Fig. 7 The numerical (solid curves) and theoretical (dotted curves) threshold value of the loss compensation as a function of the wavelength λ 0 .

Equations (26)

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ϕ Δ ( r ) = 1 2 π ε 0 Δ r r 2 = 1 2 π d k ϕ Δ ( k ) e i k y ,
ϕ Δ ( k ) = ϕ Δ ( x , y ) e i k y d y = a ( k ) e | k | | x | ,
a ( k ) = sgn ( x ) Δ x + i sgn ( k ) Δ y 2 ε 0 .
ϕ ( k ) = { b ( k ) e | k | x , c ( k ) e | k | x + d ( k ) e | k | x , e ( k ) e | k | x , x < a a < x < a + d x > a + d .
a ( k ) e | k | a + b ( k ) e | k | a = c ( k ) e | k | a + d ( k ) e | k | a ,
c ( k ) e | k | ( a + d ) + d ( k ) e | k | ( a + d ) = e ( k ) e | k | ( a + d ) ,
a ( k ) e | k | a + b ( k ) e | k | a = ε m [ c ( k ) e | k | a + d ( k ) e | k | a ] ,
ε m [ c ( k ) e | k | ( a + d ) + d ( k ) e | k | ( a + d ) ] = ε g e ( k ) e | k | ( a + d ) .
b ( k ) = [ ε m ε g ε m + ε g ε m 1 ε m + 1 e 2 | k | d ] e 2 | k | a e 2 | k | d e 2 α a ( k ) ,
c ( k ) = 2 ε m + 1 e 2 | k | d e 2 | k | d e 2 α a ( k ) ,
d ( k ) = 2 ( ε m ε g ) ( ε m + 1 ) ( ε m + ε g ) e 2 | k | a e 2 | k | d e 2 α a ( k ) ,
e ( k ) = 4 ε m ( ε m + 1 ) ( ε m + ε g ) e 2 | k | d e 2 | k | d e 2 α a ( k ) .
ϕ ( x , y ) = 1 2 π { b ( k ) e i k y + | k | x d k , [ c ( k ) e | k | x + d ( k ) e | k | x ] e i k y d k , e ( k ) e i k y | k | x d k , x < a a < x < a + d x > a + d .
ϕ ( x < a ) = 1 4 π ε 0 + [ Δ x i sgn ( k ) Δ y ] [ ε m ε g ε m + ε g ε m 1 ε m + 1 e 2 | k | d ] e 2 | k | a e 2 | k | d e 2 α e i k y + | k | x d k ,
| k d | = α = 1 2 ln ( ε m 1 ) ( ε m ε g ) ( ε m + 1 ) ( ε m + ε g ) , ( for ε m 1 < ε g 1 < 1 )
f o r α 2 > 0 ϕ ( x < a ) = 1 ε 0 d ε m ε m 2 1 ( i Δ x + sgn ( y ) Δ y ) e α d ( x 2 a ) e i α d | y | ,
f o r α 2 < 0 ϕ ( x < a ) = 1 ε 0 d ε m ε m 2 1 ( i Δ x + sgn ( y ) Δ y ) e α d ( x 2 a ) e i α d | y | .
f o r α 2 > 0 , E ( z = 0 ) = i α ε 0 d ε m ε m 2 1 e 2 α a d Δ ,
f o r α 2 < 0 , E ( z = 0 ) = i α ε 0 d ε m ε m 2 1 e 2 α a d Δ ,
z = g 2 z ,
σ e x t = 4 π 2 k 0 ( ρ 1 ρ ) 2 D o 2 Re { α ε m 1 ε m 2 e 2 α ρ 1 ρ } , ( α 2 > 0 )
σ e x t = 4 π 2 k 0 ( ρ 1 ρ ) 2 D o 2 Re { α ε m 1 ε m 2 e 2 α ρ 1 ρ } , ( α 2 < 0 )
| E E 0 | = 2 π ( ρ 1 ρ ) 2 | ln ( ε m 1 ) ( ε m ε g ) ( ε m + 1 ) ( ε m + ε g ) ε m ε m 2 1 [ ( ε m 1 ) ( ε m ε g ) ( ε m + 1 ) ( ε m + ε g ) ] ρ 2 ( 1 ρ ) | exp ( ρ 1 ρ α 2 | tan ( θ 2 ) | ) cos 2 ( θ 2 ) , ( α 2 > 0 )
| E E 0 | = 2 π ( ρ 1 ρ ) 2 | ln ( ε m 1 ) ( ε m ε g ) ( ε m + 1 ) ( ε m + ε g ) ε m ε m 2 1 [ ( ε m 1 ) ( ε m ε g ) ( ε m + 1 ) ( ε m + ε g ) ] ρ 2 ( 1 ρ ) | exp ( ρ 1 ρ α 2 | tan ( θ 2 ) | ) cos 2 ( θ 2 ) , ( α 2 < 0 )
θ E max = π | arc sin ( ρ 1 ρ α 2 ) | .
α 2 ε m 2 ε m 1 2 1 + ε m 2 ε g 1 ε m 1 ε g 2 ε m 1 2 ε g 1 2 .

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