## Abstract

We theoretically investigate the electromagnetic enhancement on a metallic surface patterned with periodic subwavelength structures. Fully-vectorial calculations show a large-area electromagnetic enhancement (LAEE) on the surface, which strongly contrasts with the previously reported “hot spots” that occur in specific tiny regions and which relieves the rigorous requirement of the nano-scale location of sample molecules. The LAEE allows for designing more practicable substrates for many enhanced-spectra applications. By building up microscopic models, the LAEE is shown due to a resonant excitation of surface waves that include both the surface plasmon polariton (SPP) and a quasi-cylindrical wave (QCW). The surface waves propagate on the substrate over a long distance and thus greatly enlarge the area of electromagnetic enhancement compared to the nano-sized hot spots caused by localized modes. Gain medium is introduced to further strengthen the large-area surface-wave resonance, with which an enhancement factor (EF) of electric-field intensity up to a few thousands is achieved.

© 2013 Optical Society of America

## 1. Introduction

Electromagnetic enhancement by metallic nano-structures has aroused intense interests in recent years, which enables a giant exterior field at the metal surface that is available for the light-matter interaction. This property is due to a coupling of the electromagnetic field at the metal surface and the moving electrons in metal, and is important for a variety of applications, such as the surface enhanced Raman scattering (SERS) [1–7], the fluorescence enhancement [8–10] and the enhanced optical nonlinearity [11–13]. The enhanced spectra signal is mainly due to the enhanced electric field at the structure surface where molecules are located. Present researches on the electromagnetic enhancement are commonly focused on the “hot spot”, which appears in specific tiny regions such as the tiny gap between metal nanoparticles [14–19], metallic tips [20,21], and metallic nanogrooves [5,7,22,23]. The enhancement factor (EF) of electric-field intensity at the hot spot can reach a huge value of several orders of magnitude, which even fulfills the requirement of single molecule detection [24–27]. However, the molecule sample needs to be precisely located at the nano-sized hot spot to achieve the huge enhancement of field, leading to a rigorous requirement of nano-location techniques. Recently, substrates patterned with metallic nano-structures over a large area have been proposed to increase the area of electromagnetic enhancement [28–31]. But these substrates are still based on the “hot spot” localized at the nano-structures, for which the ratio of the field-enhancement area relative to the whole substrate area is considerably low (not exceeding a few percentage).

In this work, nearly 50% of the overall substrate area can act as the field-enhancement area. Via fully vectorial numerical calculations [32], a large-area electromagnetic enhancement (LAEE) is observed on a metallic surface patterned with periodic subwavelength indentations. The LAEE is shown to be a quite general property that exhibits similar characteristics for different types of indentations (such as apertures or bumps). To achieve an understanding of the phenomenon for guiding the design, we build up microscopic models that consider the excitation and multiple scattering of surface waves on the periodically corrugated metallic surfaces. Analysis of the model shows that the LAEE is due to a resonant excitation of surface waves that include the surface plasmon polariton (SPP) and a quasi-cylindrical wave (QCW) which form a hybrid wave (HW). Under the resonance condition derived from the model, the surface waves will sum up constructively and propagate on the metal surface over a long distance, which greatly enlarges the area of field enhancement compared with the hot spot that is due to localized surface modes. Inspired by the analysis, gain medium is introduced into indentations to compensate the scattering loss of surface waves, which further enhances the resonance and the field at the surface. Compared with the hot spot, the LAEE relieves the requirement of accurate location of sample molecules, and increases the total emission of spectra signals on the whole surface.

## 2. Numerical observations

We will show the LAEE on a metallic surface patterned with a periodic array of subwavelength indentations. Two-dimensional (2D) arrays are considered so that their properties are independent of the illumination polarizations. The data are obtained with a fully-vectorial rigorous coupled wave analysis (RCWA) [32], which is a well-developed method that solves the frequency-domain Maxwell’s equations of periodic systems upon Fourier basis.

Figure 1(a) shows a square array of blind holes drilled in a gold substrate, with an array period *a* = 1μm and a hole side length and depth of 0.2μm. The perforated surface is illuminated by a normally-incident *x*-polarized plane wave from air. Since the enhanced emission of molecules mainly depends on the electric field at the molecule position [9,12,13,33,34], we define an enhancement factor of electric-field intensity on the metal surface (*z* = 0) at *x* = *a*/4 and *y* = 0, where the electric-field intensity peaks under resonance condition [see Figs. 1(d1) and 1(d2)]. The enhancement factor (EF) is written as

**E**

_{inc}represents the electric-field intensity of the incident plane wave.

The reflection spectrum and the EF as a function of wavelength are shown by the green triangles and red circles in Fig. 1(b), respectively. The wavelength-dependent gold refractive index takes tabulated values from [35]. The minimum reflection occurs at *λ* = 1.0153μm that is slightly larger than the period *a* = 1μm, where a considerable portion of energy is absorbed. This wavelength is exactly the peak wavelength of the EF with a peak value of about 100.

Figures 1(d1) and 1(d2) show the distribution of the electric-field intensity on the metal surface (*z* = 0) and on the cross section *y* = 0, respectively, which is obtained with the RCWA at the resonance wavelength of 1.0153μm. We only show the *E _{z}* component since the

*E*and

_{x}*E*components are negligibly weak. The null field of

_{y}*E*component at

_{z}*x*= 0 is due to its anti-symmetric property about

*x*= 0. Figure 1(d1) shows that the field away from the holes is almost

*y*-invariant. The results are consistent with those obtained with finite-different time-domain (FDTD) calculations [36]. A large-area (nearly 50% of the overall surface area) enhancement of electric field is observed on the metal surface, which much eases the location of the sample molecules. To evaluate the overall field-enhancement effect of the LAEE, we also define an average enhancement factor on the surface, ${\text{EF}}_{\text{average}}=\frac{1}{{a}^{2}}{\displaystyle {\int}_{-a/2}^{a/2}{\displaystyle {\int}_{-a/2}^{a/2}\text{EF}(x,y,0)\text{d}x\text{d}y}}$, where EF(

*x*,

*y*,0) represents a position-dependent enhancement factor on the metal surface. It is found that for the hole array, the EF

_{average}is about 50 at the resonance wavelength, which is at the same order of magnitude compared with the peak value of EF (~100). However, this is not the case for the “hot spot” localized in nano-sized regions, for which the EF

_{average}over the whole substrate surface is far smaller than the peak value of EF at the hot spot.

We have checked that the LAEE is not limited to the aperture structure. Figure 2(a) shows a 2D array of gold cubes (side length 0.2μm and period *a* = 1μm) on a gold substrate. The array is illuminated by a normally-incident *x*-polarized plane wave. As shown by the red circles in Fig. 2(b), for this structure the EF can reach a higher peak value of 220 at *λ* = 1.0029μm that is slightly larger than the period like the hole array. The corresponding EF_{average} on the surface is 144. The distribution of |*E _{z}*|

^{2}on the cross-section

*y*= 0 obtained with the RCWA at the peak wavelength of EF (

*λ*= 1.0029μm) is shown in Fig. 2(d1). It is seen that this structure also provides the LAEE. Interestingly, the field distribution of the cube array near the surface is fairly similar to that of the hole array (except for some minor difference in the vicinity of the structure), and it is also almost

*y*-invariant. The strong field close to the upper corners of the cubes could be useful for exciting spectra of molecules away from the substrate surface.

Above results imply that the LAEE is likely to be a quite general phenomenon that is independent of the structure geometry in the array. For instance, other computation results show that the LAEE can be also obtained for 2D arrays of circular blind holes or gold cylinders, with a field distribution fairly similar to that shown in Figs. 1 and 2. A theoretical demonstration is necessary to achieve an understanding of the LAEE for guiding the design, as discussed hereafter.

## 3. Microscopic surface-wave models

We will build up comprehensive microscopic models to reproduce the electromagnetic field of the observed LAEE and to analyze its general mechanism. The model will be derived based on a formalism of hybrid waves (HWs) [37–40].

As shown in Fig. 3(a), for a metal surface patterned with a two-dimensional array of subwavelength indentations and illuminated by a normally-incident *x*-polarized plane wave, a set of elementary surface waves will be launched by the indentations in the array [38,39]. These surface waves, which are called hybrid waves, will be launched by every *y*-periodic indentation chain in the array. Thus the scattered field at the metal surface can be written as a superposition of HWs that originate from every chains of indentations,

*x*∈[0,

*a*],

*a*being the array period,

**Ψ**= [

*H*,

_{y}*E*,

_{x}*E*] denotes all the field components of HWs,

_{z}**Ψ**

_{HW}

^{+}(

*x*,

*z*) and

**Ψ**

_{HW}

^{−}(

*x*,

*z*) are the

*y*-independent fields of HWs that propagate in the positive and negative

*x*-directions, and the

*P*

_{0}and

*Q*

_{0}are the unknown coefficients of the right-going and left-going HWs [Fig. 3(a)]. Analytical expressions of the HW can be derived by solving the radiation of a

*y*-invariant line source on the metal surface [41], which asymptotically represents a subwavelength line scatterer [38] and which thus does not depend on the scatterer geometry. As shown in Figs. 3(b1)-(b3) for a gold-air interface at

*λ*= 1μm, the hybrid wave (b1) is composed of the surface plasmon polariton (b2) and a quasi-cylindrical wave (b3), the former obeying an exponential propagation rule of exp(

*ik*

_{SPP}

*x*), while the latter decaying as

*x*

^{−1/2}exp(

*ik*

_{QCW}

*x*). Within the range of about one wavelength that corresponds to the considered array period for the LAEE, the QCW is shown comparable to the SPP. While at longer propagation distances, the QCW decays much faster than the SPP. The propagation constant of the SPP is

*k*

_{SPP}=

*k*

_{0}[

*ε*

_{d}

*ε*

_{m}/(

*ε*

_{d}+

*ε*

_{m})]

^{1/2}(

*ε*

_{d}and

*ε*

_{m}being the permittivities of the dielectric and the metal, respectively,

*ε*

_{d}= 1 for air), which is slightly larger than the free-space propagation constant

*k*

_{0}

*ε*

_{d}

^{1/2}(

*k*

_{0}= 2

*π*/

*λ*), while the propagation constant

*k*

_{QCW}of the QCW is quite close to

*k*

_{0}

*ε*

_{d}

^{1/2}[41].

Equation (2) implies that the scattered field at the metal surface is *y*-independent. This is valid for *y*-invariant line scatterers (such as the slit shown in Fig. 4) or approximately valid for *y*-periodic line scatterers with a subwavelength period (such as the *y*-periodic chain of holes or cubes shown in Figs. 1 and 2). The latter can be understood as a one-dimensional grating with a subwavelength period, and according to classical theories of one-dimensional gratings [42], its scattered field away from the scatterer is approximately *y*-invariant due to the invisibility of the evanescent higher-order scattered waves. This is confirmed by the previous work [37] and also by the present results such as those in Fig. 1(d1).

To solve the unknown *P*_{0} and *Q*_{0} in Eq. (2), we focus on the main scalar field *H _{y}* (from

*H*the electric vector can be simply obtained by derivatives). For the right-going HW originating from the indentation chain at

_{y}*x*= 0 [see Fig. 3(a)], its coefficient

*P*

_{0}should result from three contributions: the scattering from the incident plane wave to a right-going HW [described with a coefficient

*β*, as shown in Fig. 3(c)]; the transmission from the right-going HWs that originate from all the left-side chains at

*x*<0 [characterized with a coefficient

*τ*, see Fig. 3(d)]; and the reflection from the left-going HWs that originate from all the right-side chains at

*x*>0 [characterized with a coefficient

*ρ*, see Fig. 3(d)]. Thus

*P*

_{0}can be expressed as,

*τ*−1 characterizes the HW transmission for the scattered field [38], and Σ

*H*

_{HW}= $\sum _{n=1}^{+\infty}{H}_{\text{HW}}^{+}}(na,0)={\displaystyle \sum _{n=1}^{+\infty}{H}_{\text{HW}}^{-}}(-na,0)$ denotes a lattice summation of the HW magnetic field at multiples of the array period

*a*. Note that

*H*

_{HW}

^{+}(

*x*,

*z*) =

*H*

_{HW}

^{−}(−

*x*,

*z*) due to symmetry reasons. All the scattering coefficients

*β*,

*τ*and

*ρ*, which characterize the elementary scattering of HWs at the unit scatterer of a

*y*-periodic chain of indentations, can be calculated with the fully-vectorial aperiodic Fourier modal method (a-FMM) [43], and details of the calculation can be found in [38]. The scattering coefficients are dependent on the scatterer geometries (type, size, refractive index, …), and thus provide freedoms of design for the field enhancement. Similarly,

*Q*

_{0}can be expressed as,By solving Eqs. (3) and (4), we can obtain

*P*

_{0}and

*Q*

_{0},

*x*-symmetric indentations that are widely used in practice. For the more general case of nonsymmetric indentations, similar equations can be written by distinguishing the definition of the scattering coefficients

*β*,

*τ*and

*ρ*for left-going and right-going HWs [38]. The multiple-scattering picture of HWs and the resultant physics possess no essential difference for arrays of either symmetric or nonsymmetric indentations, and hereafter we will focus on symmetric indentations for simplicity.

Neglecting the QCW contribution in the HW model [simply replacing the subscript “HW” by “SPP” in model Eqs. (2) and (5)], we can get a pure-SPP model. For instance, for the SPP model the Σ*H*_{HW} in (5) becomes Σ*H*_{SPP} = 1/[exp(−*ik*_{SPP}*a*)*−*1], and the scattering coefficients *β*, *ρ* and *τ* now describe the excitation, reflection and transmission of SPPs at a single chain of indentations [38].

As seen in Fig. 1(b), for the structure of hole array, the HW model (black solid curve) coincides well with the fully-vectorial data (red circles) in predicting the EF. The slight difference is due to the model assumption that the scatterer (holes) is small enough compared to the wavelength [38]. The SPP model (blue dash-dot curve) predicts a lower EF. The difference between the HW model and the SPP model shows an important contribution of the QCW to the field enhancement. This is consistent with the considerable contribution of the QCW reported in other surface-wave driven phenomena such as the extraordinary optical transmission through subwavelength metallic hole arrays [37,39,40].

To achieve an understanding of the LAEE, Eqs. (2) and (5) show that the enhancement of field should be due to a resonant excitation of HWs at the surface, or due to a large value of the HW coefficient *P*_{0}. We focus on the denominator of *P*_{0} that represents a geometrical series (or an interference) of the infinite number of HWs excited on the surface. Figure 1(c1) shows that as the wavelength *λ* is slightly larger than the period *a*, the modulus of 1/Σ*H*_{HW} + 1 (black solid curve) is slightly larger than 1, since now Σ*H*_{HW} = $\sum _{n=1}^{+\infty}{H}_{\text{HW}}^{+}}(na,0)$ takes a large value (so that 1/Σ*H*_{HW} + 1≈1) in view of the propagation constants of the SPP and of the QCW that are close to *k*_{0} = 2*π*/*λ*. Additionally, the modulus of *τ* + *ρ* (red dashed curve) is slightly smaller than 1 due to the weak energy loss of the surface-wave scattering at holes (note that all the modes in the subwavelength holes are evanescent and do not support a waveguide mode resonance, see [44] for some detailed demonstrations on the values of |*τ* + *ρ*|). Thus the denominator will be quite close to zero when the phases of two terms match each other,

*λ*=

*λ*

_{res}given by Eq. (6), the modulus of

*P*

_{0}takes a value,

*P*

_{0}| requires large values of |

*τ*+

*ρ*| and |

*β*| which however form a trade-off as analyzed in the following Section. For the SPP model, the phase-matching condition reduces towhere the first term represents the phase shift of the SPP accumulated in a single period

*a*, and the second term represents the phase change of the total transmitted and reflected SPP (with a coefficient

*τ*+

*ρ*) at the scatterer. Thus under the condition of Eq. (8), the SPPs launched by all the indentations in the array will sum up constructively, leading to a strong field of SPPs at the metal surface. Under the condition of Eq. (6), similar constructive interference occurs for HWs (which is composed of SPPs and QCWs) launched by all the indentations in the array. Detailed analysis becomes more sophisticated but should be similar to the SPP model, since the SPP and the QCW possess similar propagation constants and they are excited with a moderate difference of initial phases [see Figs. 3(b2) and 3(b3)].

As shown by the intersection between the black solid curve and the red dashed curve in Fig. 1(c2), Eq. (6) of the HW model accurately predicts the peak wavelength (*λ* = 1.0153μm) of EF shown in Fig. 1(b). For the SPP model [see the blue dash-dot curves in Figs. 1(c1) and 1(c2), the phase-matching wavelength obtained with Eq. (8) shifts a little leftwards from that of the HW model, predicting the slight blue shift of the EF peak wavelength compared with the HW model, as shown in Fig. 1(b) (blue dash-dot curve). Compared with the HW model, the lower peak value of EF predicted by the SPP model is due to the larger difference between the moduli of 1/Σ*H*_{SPP} + 1 and of *τ* + *ρ* at the phase-matching wavelength [see Eq. (7)]. As the wavelength deviates from the phase-matching condition, the EF decreases rapidly due to the destructive interference of surface waves.

The analysis clarifies that to achieve the LAEE, the indentations in the array should be almost energy conservative for the in-plane scattering of surface waves (|*τ* + *ρ*|≈1, for instance, indentations that absorb or scatter away the energy of surface waves cannot fulfill this condition [44], see Fig. 4 for such an example), and the LAEE will happen at a wavelength slightly larger than the period that is accurately predicted by the phase-matching condition of Eq. (6).

It is notable that the phase-matching condition imposes a rigorous requirement on the wavelength to achieve the LAEE (a few nanometers), which could be lifted with the use of tunable lasers. On the other hand, the high sensitivity of the LAEE relative to the wavelength (or the array period) could provide a beneficial way for the sake of sensing.

Under the phase-matching condition (*λ* = 1.0153μm) of Eq. (6), we calculate the *x*-*z* distribution of the electric-field intensity with the SPP model [Fig. 1(d3)] and with the HW model [Fig. 1(d4)]. The LAEE obtained with the RCWA calculation [Fig. 1(d2)] is well reproduced by the HW model, but is underestimated by the SPP model that neglects the QCW. The slight difference between the HW model and the RCWA calculation for the near field close to the hole is due to the model assumption that the scatterer (*y*-periodic chain of holes) is *y*-invariant and small enough compared to the wavelength [38]. The model predictions are intrinsically *y*-invariant [see Eq. (2)], which is consistent with the RCWA result shown in Fig. 1(d1). The electric-field intensity peaks at *x* = *a*/4 due to the interference nodes of HWs under the resonance condition, at which the enhancement factor of Eq. (1) is defined.

Our analysis with the model evidences that the LAEE is due to a constructive interference of two surface waves, the SPP and the QCW. They propagate on the metal surface over a long distance [see Fig. 3(b)], resulting the LAEE instead of the nano-sized “hot spot” that is caused by localized modes [5,7,16,17,22,23]. Additionally, the large area of field enhancement is located at the planar metal surface, which is more convenient for assembling sample molecules than the hot spots at curved or inner metal surfaces [5,7,16,17,19–23].

For the two-dimensional array of gold cubes [Fig. 2(a)], the HW model again well predicts the fully-vectorial RCWA results, as shown in Fig. 2(b) for the EF and in Figs. 2(d1) and 2(d2) for the intensity distribution under the phase-matching condition (*λ* = 1.0029μm). The minor error of the model in predicting the near field close to the cube is due to the model assumption (similar to the hole array). The previous analysis of the LAEE with the model for the hole array still holds for the cube array, except for considering different values of the scattering coefficients *β*, *τ* and *ρ* that depend on the scatterer geometry.

Note that according to Eq. (2) of the model, the indentation-dependent coefficients *β*, *τ* and *ρ* have no impact on the relative distribution (the profile) of the scattered field at the metal surface [characterized with the indentation-independent **Ψ**_{HW}^{+}(*x*,*z*) and **Ψ**_{HW}^{−}(*x*,*z*)], but only affect the coefficient *P*_{0} of the field distribution [see Eq. (5)]. This explains the observed similarity between the relative distribution of the scattered field for the hole array and that for the cube array (see the RCWA results in Figs. 1 and 2). A big virtue of this point is that the area of the enhanced field at the metal surface (or the about 50% field-enhancement area of the LAEE), which is defined by the relative field distributions, is independent of the choice of the indentations in the array.

In previous experimental works on the SERS of subwavelength metallic hole arrays, the enhanced Raman spectra are found to be accompanied with the enhanced light transmission through the array of holes [45], and the enhanced Raman signal is observed as the array period fulfills some resonance condition [36]. With the model developed here, we now understand that these experimental observations are related to the general LAEE phenomenon, and the condition for observing the enhanced SERS signals is given by the phase-matching condition of Eq. (6) (for instance, the phase-matching condition is also the condition for the enhanced light transmission through array of holes [39]).

Above analysis explains the observed general existence of the LAEE phenomenon and confirms the general validity of the microscopic models.

## 4. Design strategies to improve the large-area electromagnetic enhancement

With the knowledge gained from the microscopic models, the design objective of enhancing the LAEE reduces to the enhancement of the surface-wave resonance (i.e. increasing the modulus of *P*_{0}). Following the analysis in the previous Section, this design requires that the indentations in the array should be almost energy conservative for the in-plane scattering of surface waves (|*τ* + *ρ*|≈1), and that the HW excitation coefficient *β* under external illuminations should not be too low.

To illustrate above strategies of design, we first provide an example without the LAEE due to the dissatisfaction of |*τ* + *ρ*|≈1. As shown in Fig. 4 for a periodic array of *y*-invariant infinite-depth subwavelength slits cut in a gold substrate [Fig. 4(a)], the enhancement factor only takes a peak value of about 6 [Fig. 4(b)], which is much lower than the previous observed high enhancement factor for the hole array or the cube array. Analysis with the model shows that under the phase-matching condition of Eq. (6) [the intersection between the curves in Fig. 4(c2)], |*τ* + *ρ*| only takes a value of about 0.83 that is distinctly smaller than 1, so that the denominator of the HW coefficient *P*_{0} [see Eq. (7)] is not close to zero under the phase-matching condition, yielding a low resonant value of |*P*_{0}| or EF. The dissatisfaction of |*τ* + *ρ*|≈1 is due to the fact that the subwavelength slit always supports a propagative fundamental TEM_{00} mode, which carries away the energy as an incident surface wave of HW is transmitted and reflected at the slit [44] [see the sketch of the scattering process in Fig. 3(d) but for a slit]. Despite of the weak resonance, the field distribution of the slit array [Fig. 4(d)] is still similar to those of two-dimensional indentation arrays.

Inspired by the results in Fig. 4, one may expect that the resonance of surface waves or the EF should be stronger as the energy loss of the in-plane scattering of surface waves is lower (i.e. |*τ* + *ρ*| is more close to 1). However, this is not true since according to Eq. (7) of the model, under resonance condition a large value of the HW coefficient *P*_{0} requires not only a large value of |*τ* + *ρ*|, but also a large value of |*β*|. Generally there exists a trade-off between a large value of |*τ* + *ρ*| and that of |*β*|, since the former requires a weak in-plane scattering of surface waves while the latter requires a strong scattering of the incident plane wave.

To illustrate this point, we look back on the results in Figs. 1 and 2 for the array of blind holes and for the array of gold cubes. It is seen that under the phase-matching condition of Eq. (6), |*τ* + *ρ*| = 0.9904 for the cube array that is lower than |*τ* + *ρ*| = 0.9969 for the hole array. However, due to the stronger scattering effect of the cubes than the holes, under the phase-matching condition |*β*| = 0.1015 for the cube array that is distinctly higher than |*β*| = 0.0315 for the hole array. The trade-off between the lower values of |*τ* + *ρ*| and the higher values of |*β*| finally results in a higher value of EF (or equivalently, |*P*_{0}|) for the cube array than for the hole array. Similar illustration can be performed by comparing the results of the gold-cube array (Fig. 2) and those of the dielectric-cube array (results in Fig. 6 with *n*_{cube} = 1.5−0*i*).

To reconcile the trade-off between |*τ* + *ρ*| and |*β*|, in the following we will consider the introduction of gain medium into scatterers to compensate the in-plane scattering loss of surface waves, with which we expect to increase |*τ* + *ρ*| without the cost of decreasing |*β*|.

We first consider a 2D array of holes filled with gain material [Fig. 5(a)]. The structure size and the illumination are the same as the previously considered array of air holes. The refractive index of the gain medium filled in holes is *n*_{hole} = 1.5−0.15*i* that is achievable in practice. For example, quantum dots (with dimensions up to 10 nm) [46] or gain-filled nanoboxes (32 × 40 nm^{2} in [24]) may provide this imaginary part of *n*_{hole}. However, filling of cubic holes of 0.2μm side length with these materials could be problematic, especially for nanoboxes. Another option is to use smaller dye molecules as the gain medium. According to the theoretical estimation [24], −Im(*n*_{hole}) = (*λ*/4*π*)*Nσ _{e}*, the gain coefficient −Im(

*n*

_{hole}) can be increased by increasing the concentration

*N*of dye molecules, where

*σ*is the emission cross section (a comparison of

_{e}*Nσ*for different gain molecules can be found in [47]). For instance, a concentration

_{e}*N*≈3.14 × 10

^{19}/cm

^{3}of dye molecule T5oCx (

*σ*= 6.0 × 10

_{e}^{−16}cm

^{2}[48]) can achieve a gain coefficient of 0.15 at

*λ*= 1μm. The array of gain-filled holes can be achieved with photoresist mixed with dye molecules that is spin coated on the metal surface with pre-etched air holes [49]. The redundant photoresist layer on the metal surface can be removed with the process of exposure and develop. Similar process can be applied to obtain the dielectric gain-cubes shown in Fig. 6(a).

For the array of gain-filled holes, Fig. 5 shows for different wavelengths the moduli (c1) and phases (c2) of the two terms 1/Σ*H*_{HW} + 1 (black solid curves) and *τ* + *ρ* (red dashed curves) that appear in the resonance denominator of *P*_{0}. For a direct comparison, the corresponding results by setting *n*_{hole} = 1.5−0*i* without gain are also shown in Figs. 5(e1) and 5(e2). The comparison shows that due to the compensation effect of the gain medium, |*τ* + *ρ*| increases from 0.9944 for *n*_{hole} = 1.5−0*i* to 1.0004 (even slightly larger than 1) for *n*_{hole} = 1.5−0.15*i* under the phase-matching condition (*λ* = 1.0173μm and 1.0175μm without and with gain, note that 1/Σ*H*_{HW} + 1 is independent of the scatterer), so that the difference between |*τ* + *ρ*| and |1/Σ*H*_{HW} + 1| becomes smaller which implies a larger *P*_{0} or EF [see Eq. (7)]. The use of gain hardly changes the argument of *τ* + *ρ*, thus hardly changing the wavelength of resonance. This prediction is confirmed by the fully-vectorial calculation with the RCWA, which gives a peak value of EF~2000 [red circles in Fig. 5(b)] and an EF_{avergae}~1000 for gain-filled holes, being one order of magnitude higher than those of the hole array without gain [Fig. 5(d)]. It is notable that while increasing |*τ* + *ρ*|, the use of gain has a weak impact on the value of the HW excitation coefficient *β* [see Figs. 5(c3) and 5(e3)], thus avoiding the trade-off between the increase of |*τ* + *ρ*| and of |*β*| which is encountered when choosing scatterers of different types or sizes as discussed before.

Consistently, the RCWA data of EF are well reproduced by the HW model [black solid curves in Figs. 5(b) and 5(d) for the cases with and without gain]. As shown in Figs. 5(f) and (g) for the cases with and without gain, the distribution of the electric-field intensity under the resonance condition still exhibits the LAEE, similar to that shown in Fig. 1(d2) for air holes.

A comparison between Fig. 1 (array of air holes) and Fig. 5 with *n*_{hole} = 1.5−0*i* shows the impact of filling material on the LAEE. The peak value of EF only increases slightly when filling holes with dielectric instead of air. However, a comparison between Fig. 2 (array of gold cubes) and Fig. 6 with *n*_{cube} = 1.5−0*i* (array of dielectric cubes) shows a much higher EF of the former than the latter, which is due to the much higher |*β*| of the former than the latter (although |*τ* + *ρ*| of the former is lower than the latter concerning the trade-off between |*τ* + *ρ*| and |*β*|).

We also consider a 2D array of gain cubes [Fig. 6(a)] to check the general validity of the gain medium in enhancing the surface-wave resonance. The structure size and the illumination are the same as the previously considered array of gold cubes. Similar to the analysis in Fig. 5, as the cube refractive index *n*_{cube} changes from 1.5−0*i* to 1.5−0.15*i*, |*τ* + *ρ*| increases from 0.9991 to 1.005 under the phase-matching condition (*λ* = 1.0129μm and 1.0128μm without and with gain), so that the difference between |*τ* + *ρ* | and |1/Σ*H*_{HW} + 1| becomes smaller which implies a larger *P*_{0} or EF. This prediction is confirmed by the fully-vectorial RCWA results [red circles in Fig. 6(b)], which gives a peak value of EF~2000 and an EF_{avergae}~1000 for gain cubes, being two orders of magnitude higher than those without gain [Fig. 6(d)]. The intensity distribution shown in Figs. 6(f) and 6(g) with and without gain under resonance condition still exhibits the LAEE, and the use of gain material again has a weak impact on the value of *β* [Figs. 6(c3) and 6(e3)].

Besides the introduction of gain medium into scatterers to compensate the scattering loss of surface waves, another choice could be using dielectric gain layers covered on the metal surface to compensate the propagation loss of surface waves. However, for this way of loss compensation there exists a trade-off between the increase of the compensation effect and the decrease of the field of surface waves at the dielectric-air interface as the thickness of the gain layer increases [50]. For instance, calculations show that under resonance condition a 5nm thickness gain layer (refractive index 1.5−0.15*i*) covered on the hole array shown in Fig. 5(a) leads to a EF of about 6000 at the dielectric-air interface [higher than the peak value of 2000 shown in Fig. 5(b)], while the EF decreases to about 400 as the gain layer thickness increases to 10nm. Similar results hold for dielectric gain layer covered on the cube array shown in Fig. 6(a).

## 5. Conclusion

Large-area electromagnetic enhancement (LAEE) is observed on a metallic surface patterned with a periodic two-dimensional array of subwavelength indentations, which strongly contrasts with the previously reported “hot spots” that occur in nano-sized tiny regions. Nearly 50% of the overall surface area provides a strong enhancement of electric field, which much eases the location of sample molecules and increases the total emission of spectra signals. The LAEE is found to generally exist and exhibit similar characteristics for different types of subwavelength indentations such as holes or bumps. To achieve an understanding of the LAEE for guiding the design, comprehensive microscopic models that consider the launching and scattering of surface waves by every indentation in the array are built up. Analysis of the model shows that the LAEE results from two surface waves, the surface plasmon polariton (SPP) and another quasi-cylindrical wave (QCW), which propagate on the metal surface over a long distance and which may sum up constructively under a phase-matching condition derived from the model. This leads to a greatly enlarged area of field enhancement compared to the nano-sized “hot spot” caused by localized modes. The model clarifies that to achieve the LAEE, the indentations in the array should be almost energy conservative for the in-plane scattering of surface waves, and the LAEE will happen at a wavelength slightly larger than the array period that is accurately given by the phase-matching condition.

Inspired by the model, in the design process to improve the LAEE the trade-off between reducing the in-plane scattering loss of surface waves and increasing the HW excitation efficiency should be considered when choosing scatterers of different types or sizes. To reconcile the trade-off, indentations with optical gain are considered to compensate the scattering loss of surface waves at indentations and thus to further enhance the resonance of surface waves. The enhancement factor (EF) of electric-field intensity is greatly increased (~2000), which is almost one order of magnitude higher than that without gain. Besides the two-dimensional array of indentations considered here, we have checked that the LAEE can be also achieved by one-dimensional periodic array of *y*-invariant indentations (such as ridges or grooves) patterned on the metal surface if the illumination is orientated to be TM-polarized (magnetic vector along the invariant *y*-direction) to support the propagation of surface waves, as hinted by the surface wave model that is intrinsically *y*-invariant. Aperiodic array of indentations could be another option to optimize the LAEE, which has been used for achieving broadband absorption enhancement in silicon solar cells [51]. Surface-wave models can be established as well for aperiodic array of indentations [38], and the LAEE is also expected to be achieved when the launched surface waves interfere constructively under certain conditions.

Our reported results are helpful for designing more practicable substrates for many electromagnetic-enhancement related applications, such as the surface enhanced Raman scattering (SERS), the fluorescence enhancement and the enhanced optical nonlinearity. The present analysis with microscopic models could be also beneficial for understanding a variety of phenomena related to the seminal topic of Wood’s anomaly [52], which causes rapid variations of the reflection or transmission spectra of subwavelength metallic gratings and which has been shown related to a resonant excitation of surface waves [53,54].

## Acknowledgments

This work is financially supported by the National Key Basic Research Program of China (973 Program) under Grant No. 2013CB328701, by the Natural Science Foundation of Tianjin under Grant No. 11JCZDJC15400 and by the Natural Science Foundation of China (NSFC) under Grant No. 61322508.

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