Abstract

The Drude model for metal is extended to include complex relaxation rates. As a test for what happens to the surface plasmon resonances with such metals, the lifetime is examined for propagating waves across a single planar metal-dielectric interface. By analytically solving the dispersion relation being fourth-order in the complex frequency, group-velocity dispersion and quality factors are explicitly found. Due to the symmetry breaking between the forward and backward waves, standing waves are not allowed in general.

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1. Introduction

It is well-known that there should be trade-offs between the desired confinement effect and the unavoidable dissipation effect when metallic nature is utilized for realizing plasmonic devices [1,2]. For metallic media, the classical Drude model with real relaxation rates needs to be amended in some situations. For instance, thin films containing subwavelength metallic nanoparticles are well modeled by taking the relaxation rate (CRR) γ to be complex-valued [3]. The concept of CRRs has relationships to systems in both classical mechanics [4] and quantum mechanics [58]. Recently, superconducting materials themselves are explored for the incorporation into metamaterials [9], thanks to their low losses. The concept of CRRs has been successfully employed in dealing with the vortex dynamics for type-II superconductors through the time-dependent Ginzburg-Landau (TDGL) equation [7], where the longitudinal conductivity and the (perpendicular) Hall conductivity are affected by parameters like γr and γi, respectively.

Here, several consequences of incorporating the CRRs in the Drude model will be examined in terms of wave propagations [1012]. Besides, the resulting symmetry breaking will be examined in terms of the electromagnetic energy flows. Let us consider the conventional Drude model for the dielectric constant of metal as follows, with ωp as the bulk plasma frequency.

ε(ω)=1ωp2ω(ω+iγ),
where ω is frequency and γ is the relaxation rate. Furthermore, we define εεr+iεi with εrRe(ε) and εiIm(ε). Likewise, γγr+iγi. For a moment, we assume ω to be real-valued. We separate Eq. (1) explicitly into its real and complex parts as follows.

εr=1ωp2(ωγi)ωDε,   εi=ωp2γrωDε,   Dε(ωγi)2+γr2.

2. Electric conductivity

In view of the circuit theory for nanophotonics [13,14], let us examine the notion of conventional complex electrical conductivity σ0 defined through ε(ω)1+i(σ0/ω). In consequence, we are led to the relationship σ0=(γiω)1ωp2 [3]. In order to account for the fact γi0 of the CRR, let us consider the following modified conductivity σσr+iσi (without any subscript).

ε1+ωp2γiω|γ|2+iσω.
As a result, σr=ωp2γrDε1 and σi=ωp2γi|γ|2+ωp2(ωγi)Dε1, where |γ|2γi2+γr2=γγ¯. Here, the over-line indicates a complex conjugate so that γ¯γriγi. Let us examine the following low-frequency limit of σ under the assumption that |ω|<<|γ|.
limω0σr=ωp2γr|γ|2+2ωp2γrγi|γ|4ω,   limω0σi=γr2γi2|γ|4ωp2ω.
Hence, σr(ω=0)0, but σi(ω=0)=0. In the meantime, we need to pay a special attention to the sign of γi. In this regard, it is found from Eq. (4) that σi>0 for γr>γi in the case ω>0. The reverse σi<0 is true for γr>γi when we extend the frequency in the negative range ω<0. The choice γi<0 ensures that εr(γiω)1ωp2<0 as ω0, and hence the metallic nature is guaranteed for γi<0.

In this aspect, we learn that only positive imaginary parts of the complex relaxation time or Im(γ1)>0 or γi<0 work properly in the case with the TDGL model [6,7]. We ascribe this restriction to the nonlinearity imbedded in the TDGL model, which helps to select only one of the two possible signs. However, both signs of γi work fine in the present model, since the problem under current investigation is essentially linear. For matter of convenience, we take however that γi<0 in the ensuing numerical computations.

Meanwhile, we make an interesting comparison to the axial complex conductance examined in [5]. They examined a quantum wire by solving the Schroedinger equation under several assumptions, for instance, prescribed electronic screening functions and Gaussian radial profiles of electric fields. Their model can be represented as a parallel circuit of one L1R1C1 subcircuit and another L0R0 subcircuit. In line with the above-mentioned circuit model [13], they defined another complex electrical conductivity σc (the subscript “c” implying Cuniberti in [5]).

In Fig. 1(a) , we display σr and σi for a set of data: ωp=1, γr=0.2, and γi=0.5. Returning to our model in Eq. (3), we find that symmetry holds true with respect to ω such that σr(ω,γi,γi)=σr(ω,γi,γi) and σi(ω,γi,γi)=σi(ω,γi,γi). Therefore, the corresponding behaviors for γi=0.5 can be easily inferred. It is surprising that Fig. 1(a) turns out to be almost identical to Fig. 4 in [5]. Upon closer examinations, the dc limits as ω0 are different. In other words, we have a linear dependence limω0σr=a0+a2ω with (a0,a1) defined in Eq. (4), whereas there is a quadratic dependence limω0Re(σc)=b0+b2ω2 with (b0,b1) as another set of constants defined in [5].

 

Fig. 1 (a) The real and imaginary parts of complex conductance for the Drude model according to Eq. (3). (b) A single planar interface between the upper dielectric and the lower metal-like medium. The in-plane wave number is denoted by k directed in the x-direction. Non-zero γr induces cross-interface energy flows, whereas non-zero γi entails an imbalance between the forward (“F”) and backward (“B”) waves along the interface.

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Fig. 4 The degree of asymmetry e obtained by varying one parameter while fixing the remaining two: e(k) based on the data of Fig. 2(a) with γr as an abscissa; e(γr) based on the data of Fig. 3(a) with k as an abscissa; and e(γi) vs. γi.

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3. Dissipation-induced stability

For the Drude model in Eq. (1) with a CRR, let us consider the propagations of electromagnetic waves. The problem of surface plasmon resonances (SPRs) occurring across a planar metal-dielectric (M-D) single interface is simplest; yet, it reveals fundamental aspects of SPRs, since diverse geometrical configurations involve a single M-D interface in one way or another [1,15]. In addition, the lifetime of SPRs is of a major concern in applications such as biological sensing [16], in particular, as regards two counter-propagating waves [17].

Let us consider the schematic Fig. 1(b) for air (with j=a) and metal-like medium (with j=m) separated by a single M-D interface at z=0. Here, the “metal-like” zone is more general in implying not only pure metals but also metallic composites. The relative dielectric constant is taken such that εa=1 without loss of generality and εmε for simplicity. With c0 as the light speed in air, we define for nondimensionalization the reference time and length to be 1/ωp and c0/ωp, respectively. Furthermore, let x and z be the longitudinal (in-plane) and depth (surface-normal) coordinates, respectively. Besides, t is time. Then, the phase factor exp[i(kxωt)] represents propagating electromagnetic waves, where k is the longitudinal wave number. For the purpose of investigating the lifetime of SPRs, it is appropriate henceforth to take a complex-frequency approach [18,19], where ωωr+iωi. Instead, k is real-valued.

Owing to stability for t>0, it is required that ωi<0 as seen from the expanded phase factor exp[i(kxωrt)]exp(ωit). As a consequence, a long-life surface plasmon (LLSP) corresponds to the solution with a smaller temporal attenuation |ωi| [1821]. On the other hand, we allow ωr to take either positive or negative sign. In order to understand the meaning of different signs of ωr, let us take k>0 for convenience. For ωr>0, waves are propagating rightward, because dx/dt=ωr/k>0 for constant phase of kxωrt. In comparison, for ωr<0, waves are propagating leftward, because dx/dt=ωr/k<0 for constant kxωrt. Besides, the real-valued k corresponds either to an incident angle for |ωr|>k or to the degree of an evanescent coupling for |ωr|<k [10,15,17].

Transverse-magnetic (TM) fields are described by a set of non-zero field components (Ex,Hy,Ez). For such TM waves, the dispersion relation is given by the well-known formula ε(k2ω2)1/2=(k2εω2)1/2 [1]. We then square the latter to obtain the following fourth-order (quartic) polynomial in ω.

ω4+iγω3(1+2k2)ω22iγk2ω+k2=0.
It is no coincidence that Eq. (5) is almost identical to the dispersion relation for a single wire with the only non-zero magnetic field directed parallel to the wire axis [19]. Because the factor γ in Eq. (5) is multiplied only on the terms odd in ω, Eq. (5) is reduced to γ=iD1N, with Nω4(1+2k2)ω2+k2 and Dω(ω22k2). As a reference, we consider the lossless case with γ=0, for which ε(ω)=1ω2. For γ=0, γ=iD1N is identical to N=0, thus giving rise to the following twin real frequencies.
ωL±(k)=±12(1+2k2)121+4k4,ωH±(k)=±12(1+2k2)+121+4k4.
Because 0<|ωL±|<1, both low-frequency acoustic branches ωL± represent evanescent waves. On the other hand, both high-frequency optic branches ωH± refer to guided waves because 1<|ωH±| [2,22].

Let us examine some properties of the complex-valued ω’s as solutions to Eq. (5) for a complex-valued γ, but with a real-valued k. First of all, we notice a mirror symmetry in Eq. (5) for a given k2, because (ω,γ) is a solution once (ω,γ) is a solution [22]. Furthermore, let us denote the complex conjugate of ω by ω¯ωriωi and hence |ω|2ωω¯ωr2+ωi2. When eliminating γi from Eq. (5), we obtain a γi-free dispersion equation |D|2γr=Im(D¯N). When further processed, the last relation reduces to

γrωi=I(k2,ωr2,ωi2)|D(k2,ω)|2.
Here,
Iak4+bk2+c,a2+4|ω|2>0,   b3|ω|2+4|ω|44ωr2(1+2|ω|2),   c|ω|4+|ω|6>0.
At this point, we are to prove that I>0 for all choices of (ω,k). To this goal, it is necessary to deal with two cases depending on the sign of b. In the first case that b>0, we have a,b,c>0, thus easily ensuring I>0. In the second case that b<0, we need to additionally make sure that 4ac>b2. This last requirement is rewritten to be
16ωr2(2|ω|2+1)2(|ω|2ωr2)+(5|ω|2+8|ω|4)|ω|2>0,
which is always satisfied by the inequality 3|ω|2+4|ω|4<4ωr2(1+2|ω|2) or b<0. The proof is then complete, namely γr/ωi=|D|2I<0 always. Consequently, γrωi<0, thereby signifying that the conventional material loss γr>0 renders propagating waves absolutely stable irrespectively of the imaginary part γi of the CRR. We could call the inequality γrωi<0 the “dissipation-induced stability” [18]. As its direct consequence, we find for the conventional complex conductivity σ0(γiω)1 that Re(σ0)>0 independently of γi.

As regards the four complex roots of ω as solutions to Eq. (5) for a given set of real data (k,γr,γi), it is remarked that a quadratic equation can be analytically solved by the Ferrari-Cardano method. Its geometric interpretation is lucidly illustrated in [23], and it calls for the standard Cardano method in solving an intermediate equation which is cubic in ω [20,21]. Once four complex roots are analytically found, we have checked the validity of the roots so evaluated by ensuring that Eq. (5) is satisfied as well. Besides, we remark that it is non-trivial to sort a set of four roots into proper branches as a parameter is varied. As regards γr, we consider both 0<γr<<1 for low-loss noble metals and 0<<γr=O(1) for high-loss gas plasmas [21]. Here, we call the attention that the value of γr is referred to the plasma frequency ωp.

In order to reveal non-negligible effects of γi, we will work with a relatively large value of γi=0.2. On atomic scales, non-zero values of γi may be caused by the structural asymmetry [15]. On larger scales, let us suppose that fine metal nanoparticles, whose sizes are much less than the wavelength of interest, are uniformly dispersed throughout a thin dielectric film [3,13]. The relaxation rate can then be considered as an average value for such a composite film with moderately higher filling ratios. In this case, the way the nanoparticles are dispersed may be responsible for the apparent asymmetry. It is the reason why the lower zone in Fig. 1(b) is denoted as “metal-like” rather than just as “purely metallic”.

4. Lifetimes and Wave Characteristics

Henceforth, let us present typical numerical results. We define a state by an eigenvalue pair on the (ωr,ωi)-plane. Furthermore, we define a “branch” by a group of such states possessing similar characters as shown in Figs. 2(a) and 3(a) . Each branch is composed of 1000 states in equal intervals of the parametric variations for the current numerical evaluations. According to the naming convention in Eq. (6), we call four branches “L+”, “L-“, “H+”, and “H-”, respectively. The first letter “L” or “H” is decided by the largeness of |ωr|, whereas the second symbol “+” or “-” is determined by the sign of ωr.

 

Fig. 2 (a) Migrations of four roots on the complex (ωr,ωi)-plane. (b) The quality factor plotted against ωr. The curves are generated by varying kover the range 0k2, where the arrows mean the direction of increasing k. The filled circles in pink color indicate the starting states at k=0, whereas the diamonds indicate the states on the half-way at k=1. In the inset of panel (a), we show the symmetric trajectories for the specified data γr=0.2 and γi=0 (namely, a real-valued γ).

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Fig. 3 (a) Migrations of four roots on the complex (ωr,ωi)-plane. The arrows imply the direction of increasing γr. The filled circles in pink color indicate the starting states at γr=0 (where ωi=0, thus being neutrally stable), whereas the diamonds indicate the states on the half-way at γr=1.5. In the inset of panel (a), we show the symmetric trajectories for the prescribed data γi=0 (a real-valued γ) and k=0.5. (b) Migrations of four roots on the complex (vgr,vgi)-plane. The thick horizontal arrow in shaded colors indicates the region of the subluminal group velocity, namely, |vgr|<k=0.5.

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Figure 2(a) displays such four branches in the case of γ=0.20.2i with a variation over 0k2. The two middle branches L- and L + starting out right from the origin (ωr,ωi)=(0,0) are acoustic by definition, whereas the two side branches H- and H + starting off the origin are optic. Branches L- and H- with ωr<0 exhibit heavier attenuations (i.e., larger |ωi|) than the respective branches L + and H + with ωr>0. Therefore, the rightward- and leftward-propagating waves have unequal |ωr| for a given k. Consequently, there cannot be standing waves formed as a result from two counter-propagating waves of equal amplitude colliding each other [1012].

The fact that branches with ωr<0 exhibit larger attenuation in Fig. 2(a) is corroborated by the two electrostatic roots marked by “static” with filled symbols, which are located just below the respective branches L- and L + . This electrostatic resonance is described by the condition that 1+ε=0, which is equivalent to the small-wavelength-limit k as easily seen from k(ω)=ω(1+ε)1ε [22]. Substituting the Drude model into 1+ε=0 results in 2ω(ω+iγ)1=0, which is in turn solved to give rise to two roots ω=12iγ±12(2γ2)1/2 [4]. For γ=0.20.2i, it is found that ω=0.807i0.114 and ω=0.607i0.0859 on the electrostatic resonance. The inset of Fig. 2(a) exhibits four branches in this case of γi=0, where they are symmetrically placed with respect to the ωi-axis. In comparison, the four branches in the main panel of Fig. 2(a) are asymmetrically placed with respect to ωr=0, although the degree of asymmetry is still not quite large.

According to the circuit paradigm for metal optics [14], even a planar interface stands for a LRC circuit, for which we can define a quality factor by Qω12|ωr/ωi| [9,11,18]. In Fig. 2(b), we plot Qω against ωr based on the roots in Fig. 2(a). It shows on both branches H + and H- that Qω increases with k (or as |ωr| increases). In comparison, on both branches L + and L-, the reverse trend is visible. For both electrostatic roots as mentioned just before, we obtain the same quality factor of Qω=3.54 as marked by the horizontal broken line in Fig. 2(b). The existence of a minimum in Qω at a certain |ωr| is in rough agreement with the predictions made for periodic arrays of metal cylinders [24].

As another parametric study, we plot in Fig. 3(a) the four branches on the complex ω-plane as γr is varied over 0γr3. But, we fix the two remaining parameters such that γi=0.2 and k=0.5, for an incidence angle of sin1(1/2)=30o. As in Fig. 2(a), a thousand of states are computed for each branch. Figure 3(a) shows that all the four branches start on the ωr-axis at ωi=0 for γr=0, as has been proved earlier by Eq. (7). As γr is increased (indicated by the arrows), branches H-, L + , and H + migrate downward first, but they bend upwards. Eventually they almost touch the ωr-axis in the large γr-limit. Such a minimum value of |ωi| obtained at a finite value of γr has been observed for gas-phase plasmas in the case of isolated nanoparticles as well [21].

In contrast, branch L- appears to be stretched downward on this display window. Indeed, we found numerically that branch L- asymptotically approaches the ωr-axis in the large γr-limit. For this purpose, let us examine Fig. 3 in more detail for the asymptotic behaviors of the four branches as γr. In this limit, Eq. (5) lends itself to the following four component solutions, being labeled with the corresponding branch L-, H-, H + , and L + as shown in Fig. 4.

(L)ω=iγ,     (H)ω=2k,     (L+)ω=0,     (H+)ω=+2k.
For the data γi=0.2 and k=0.5 employed in Fig. 3, branch L- approaches ω=0.2iγr as γr. In comparison, the three remaining branches approach the imaginary axis at ωi=0 as γr. In particular, ω0.707 on branch H-, whereas ω+0.707 on branch H + . There is hence a chance as γr of a standing wave established as the result of a collision between a propagating wave on branch H- and a counter-propagating wave on branch H + . Meanwhile, ω0 on branch L + , hence becoming completely time-independent.

Therefore, the relative magnitude of |ωr| is exchanged between branches L- and H- as γr increases. As in Fig. 2(a), Fig. 3(a) shows an asymmetry in |ωr|. All the more clearly, Fig. 3(a) depicts that the accompanying attenuation |ωi| is widely different among the four branches. As a consequence, branch H- appears to live longest among the four, thanks to its smallest |ωi|.

Figure 3(a) displays two additional curves denoting the electrostatic states as found for Fig. 2(a). In particular, Fig. 3(a) shows that branch L- and L + run asymptotically parallel to their corresponding electrostatic branches. There are always propagating waves with ωr0 as long as γi0, irrespectively of the largeness of γr. In comparison, it is found that there are no propagating waves for large γr in the case of γi=0 because ωr=0, which is seen within the inset of Fig. 3(a) on the central portion of the branch (indicated by the upward arrow within the inset) [20,21].

Although the meaning of group velocity gets less clear for complex-valued γ, we can still formally define it by

vgωk=2ω2+2iγω14ω3+3iγω22(1+2k2)ω2iγk22k.
Based on the data of Fig. 3(a), Fig. 3(b) draws the γr-trajectories on the complex(vgr,vgi)-plane with vgvgr+ivgi. For convenience, we also define the two phase-velocity lines at |ωr|=k=0.5, recalling that velocities are based on c0 and the light line is hence given by |ωr|=k. Therefore, we have subluminal group velocities in the range |ωr|<0.5. Besides, Fig. 3(b) displays a wildest swing on branch L- among the four. As a result, the density of states is rather sparse on the middle portion of branch L- [11]. In addition, both signs of vgr are possible for branch L-. However, vgr on the remaining three branches is one-signed. In particular, vgr<0 on the whole of branch H-, whereas vgr>0 on both branches L + and H + . By the way, the condition of zero-group velocity 2ω2+2iγω1=0 as found from the numerator of Eq. (11) is identical to that for electrostatic states. Likewise, we can interpret the previous Fig. 2(a) in terms of the group velocity (not presented). That is, vg0>0 on both branches L + and H + , whereas vg0<0 on both branches L- and H- for all k [10].

In order to summarize the results in Fig. 3(a), we define the degree of asymmetry e as follows.

e=14ω/=14|ω|.
Of course, 0e1. We plot three kinds of asymmetry in Fig. 4 based on the data of both Figs. 2(a) and 3(a). First, e(k) is obtained for the data in Fig. 2(a) over 0k2. Second, e(γr) is drawn for the data in Fig. 3(a) over 0γr3. Of course, e0 because γi=0.2 for both curves. Besides, we obtained additional results by varying |γi|(but not presented). From such data, we find that the pair L- and L + get more asymmetric with increasing |γi|, as are the case with the other pair H- and H + . Third, we display e(γi) in Fig. 4 from such data obtained by varying γi over the range 0|γi|3, while fixing the two remaining parameters such thatγr=0.2 and k=0.5. Incidentally, there is no difference in e(γi) with either γi>0 or γi<0. In addition, we note trivially that e(γi=0)=0.

There are three conceivable levels of asymmetry as regards the SPRs for the M-D configuration shown in Fig. 1(b). First, the zero-level asymmetry is concerned with the direction normal to the M-D interface. For γ=0, the time-averaged Poynting vectors are directed along the M-D interface in both media, thus disallowing cross-interface energy flow. In this case, the energy flows are directed along the M-D interface, although the flow in air is opposite to that in the metal-like medium. On the other hand, a non-zero positive material loss γ>0 causes non-zero cross-interface energy flow [22]. This first-level asymmetry is therefore associated with the real part of the relaxation rate γr, and the resulting effect takes the form of energy flow in the direction normal to the M-D interface. The second-level asymmetry is incurred if γi0, for which there will be no standing waves in the direction along the M-D interface, because the forward- and backward-propagating waves viewed along the M-D interface do not balance among them (ωr’s being asymmetric with respect to ωr=0).

Other symmetry-breaking phenomena occur with the nonlinear second-harmonic generation due to applied magnetic fields [15], which is closely related to the asymmetry with magnetic fields as occurs with the TDGL for superconductors [6,7]. The occurrence of complex eigenvalues can also be found in the Hatano-Nelson model of superconductors due to the asymmetric effect of hopping electrons [8].

5. Complex frequency versus complex wave number

We have already taken frequency to be real-valued in Sec. 2, whereas frequency is complex-valued in Sec. 4. In general, whether frequency is real- or complex-valued depends on how to properly to describe physical phenomena under investigation [2022].

For a real-valued k, the complex frequency ω describes both the direction of wave propagations by ωr and the wave attenuation by |ωi| as shown by the numerical simulations ensuing after pulsed excitations in [10,25]. Besides, the complex frequencies are sought as solutions to the generalized eigenvalue problems for light propagations through photonic crystals where metallic wires or filaments are periodically embedded [19,24,26,27]. In particular, positive values of ωi are suitable for describing optical gains due to external pumping [9,18,26]. In general, system responses to short-duration signals such as encountered in biological or chemical sensing [16,17] are better to be described in terms of complex frequencies rather than complex wave numbers (the latter being discussed shortly). Transient responses after the application of pulsed lasers can also be adequately described via complex frequencies [15,17], although the laser pulses themselves are better described by real-valued frequencies. In addition, complex frequency is associated with the aforementioned “time-dependent” Ginzburg-Landau (TDGL) equation [7], instead of its steady-state version.

On the other hand, conventional waveguide theory has been discussed in terms of the real-valued frequency, say, with ω>0 such as in the case of external illuminations by continuous-wave (cw) lasers [1113,22]. In this approach, the complex-valued k is sought as a solution to the dispersion relation, which is obtained as follows for Eq. (5).

k=±ωω2+iγω12ω2+2iγω1.
We find here that Eq. (13) is nothing but the well-known formula k=±ω(1+ε)1ε with ε=1(ω2+iγω)1 in Eq. (1) with the dimensionless ωp=1. We vary the real-valued ω in Eq. (13) for complex-valued γ, thus obtaining explicit values of kkr+iki. For a prescribed set of positive values (ω,γr,γi), Eq. (13) gives rise to both a positive pair of (kr,ki) and another pair (kr,ki). Let us examine the following phase factor [10].

exp[i(kxωt)]=exp(kix)exp[i(krxωt)].

Hence, the solution pairs (kr,ki) and (kr,ki) refer to the right- and left-propagating waves. Of course, both type of propagating waves undergo attenuations in their respective propagation directions. Therefore, we could establish standing waves by illuminating two continuous waves of equal amplitude from opposite directions.

We plot in Fig. 5 the positive pair (kr,ki) against ω for the complex relaxation rate γ as indicated within each panel. The range 0ω<kr and kr<ω refer to the evanescent and guided waves, respectively. In all the cases, kr and ki0 as ω according to the limiting relation kr21/2ω from Eq. (13). Moreover, kr exhibits both a local maximum and a local minimum with respect to ω in all the cases. In comparison, ki exhibits one single local maximum. In addition, the character of acoustic branch is displayed in all the cases, namely, ω0 as kr0.

 

Fig. 5 The real and imaginary parts of k=ω(1+ε)1ε according to Eq. (13), where both of (kr,ki) are positive. Three values of the complex relaxation rate are examined: (a) γi=0.2i0.2, (b) γi=0.03i0.2, and (c) γi=0.2i.

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From comparison of Fig. 5(a) with 5(b), the local maxima get sharper for both of (kr,ki) with a decrease in γr. In particular, we could infer from Fig. 5(b) that there is a mini-frequency-gap (where kr0 as indicated by the large hatched horizontal arrows) approximately near 0.8<ω<1 as γr0 with γi=0.2 fixed [19,24,26,27]. On the other hand, the increase in |γi| does not alter the overall trends of (kr,ki) with γr=0.2 fixed, as seen from Figs. 5(a) and 5(c).

In order for standing waves to be established in the case of complex relaxation rates, we require that continuous-wave beams of equal frequency and amplitude are carefully illuminated from exactly opposite directions. However, in the case of pulsed illuminations, it is quite natural that counter-propagating waves hardly lead to standing waves. This notion is valid for linear geometry. In comparison, the case with non-linear geometry is not quite simple as shown by the next case with circular geometry.

An interesting measurement setting is that a continuous-wave laser beam introduced from an optical fiber through a subwavelength-sized nanoparticle bead is incident tangentially along a certain circumference of a micro-sphere resonator [11]. It is because a whispering-gallery-mode (WGM) is established by the interaction of a rotating wave (directed into the external laser beam) and a counter-rotating wave (advancing into its opposite direction) aroused by back scattering. As a result from one-sided external excitation by a laser beam, the resulting waves are composed of two components: one running (propagating) and the other standing. Moreover, both waves are attenuated with time due to both resonator and coupling. In fact, this notion of propagating and standing waves are crucial in properly designing whispering-gallery-resonator (WGR) lasers [12,28].

6. Summary

To summarize, by examining the effects of the complex relaxation rates, we are a little closer to understanding the true nature of the lifetime and asymmetry of the surface plasmon resonances in the context of the most fundamental single-interface configuration. Extension to more complicated geometries can be carried out as well. Ways of controlling the relative magnitudes among ωp, γr, and γi should depend on the choice of materials and the formulation of composite materials, which is under our current investigation.

Acknowledgments

This work has been supported by the National Research Foundation (NRF) of Korea grant funded by the Korean government (MEST) (No. 2011-0001087). The first author is grateful to Prof. Dong Pyo Chi of Seoul National University (SNU) for the invaluable advice on quantum mechanics administered at Research Institute of Mathematics of SNU. In addition, we thank the reviewer(s) not only for pointing out to us the possibility of standing waves in the case of complex wave numbers (as discussed in Sec. 5), but also for other helpful suggestions.

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3. J. S. Yang, J.-H. Sung, and B.-H. O, “Novel elastic scattering model for the understanding of the Anomalous transmittance for Au nanoparticle layer,” Opt. Express 18(13), 13418–13424 (2010). [CrossRef]   [PubMed]  

4. T. Savin and P. S. Doyle, “Role of a finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(4), 041106 (2005). [CrossRef]   [PubMed]  

5. G. Cuniberti, M. Sassetti, and B. Kramer, “Ac conductance of a quantum wire with electron-electron interactions,” Phys. Rev. B 57(3), 1515–1526 (1998). [CrossRef]  

6. N. Nakai, N. Hayashi, and M. Machida, “Simulation studies for the vortex-depinning dynamics around a columnar defect in superconductors,” Physica C 468(15–20), 1270–1273 (2008). [CrossRef]  

7. N. Nakai, N. Hayashi, and M. Machida, “Direct numerical confirmation of pinning-induced sign change in the superconducting Hall effect in type-II superconductors,” Phys. Rev. B 83(2), 024507 (2011). [CrossRef]  

8. F. Hébert, M. Schram, R. T. Scalettar, W. B. Chen, and Z. Bai, “Hatano-Nelson model with a periodic potential,” Eur. Phys. J. B 79(4), 465–471 (2011). [CrossRef]  

9. V. A. Fedotov, A. Tsiatmas, J. H. Shi, R. Buckingham, P. de Groot, Y. Chen, S. Wang, and N. I. Zheludev, “Temperature control of Fano resonances and transmission in superconducting metamaterials,” Opt. Express 18(9), 9015–9019 (2010). [CrossRef]   [PubMed]  

10. H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94(12), 123901 (2005). [CrossRef]   [PubMed]  

11. A. Mazzei, S. Götzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99(17), 173603 (2007). [CrossRef]   [PubMed]  

12. S. Zou, “Electromagnetic wave propagation in a multilayer silver particle,” Chem. Phys. Lett. 454(4–6), 289–293 (2008). [CrossRef]  

13. N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317(5845), 1698–1702 (2007). [CrossRef]   [PubMed]  

14. M. Staffaroni, J. Conway, S. Vedantam, J. Tang, and E. Yablonovitch, “Circuit analysis in metal-optics,” arXiv:1006.3126 [physics.optics].

15. V. K. Valev, A. V. Silhanek, W. Gillijns, Y. Jeyaram, H. Paddubrouskaya, A. Volodin, C. G. Biris, N. C. Panoiu, B. De Clercq, M. Ameloot, O. A. Aktsipetrov, V. V. Moshchalkov, and T. Verbiest, “Plasmons reveal the direction of magnetization in nickel nanostructures,” ACS Nano 5(1), 91–96 (2011). [CrossRef]  

16. A. Z. Abbasi, F. Amin, T. Niebling, S. Friede, M. Ochs, S. Carregal-Romero, J.-M. Montenegro, P. Rivera Gil, W. Heimbrodt, and W. J. Parak, “How colloidal nanoparticles could facilitate multiplexed measurements of different analytes with analyte-sensitive organic fluorophores,” ACS Nano 5(1), 21–25 (2011). [CrossRef]   [PubMed]  

17. A. Haddadpour and Y. Yi, “Metallic nanoparticle on micro ring resonator for bio optical detection and sensing,” Biomed. Opt. Express 1(2), 378–384 (2010). [CrossRef]  

18. L. Prkna, J. Čtyroký, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36(1–3), 259–269 (2004). [CrossRef]  

19. V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55(12), 7427–7444 (1997). [CrossRef]  

20. H.-I. Lee and J. Mok, “On the cubic zero-order solution of electromagnetic waves. I. Periodic slabs with lossy plasmas,” Phys. Plasmas 17(7), 072108 (2010). [CrossRef]  

21. H.-I. Lee and J. Mok, “On the cubic zero-order solution of electromagnetic waves. II. Isolated particles with lossy plasmas,” Phys. Plasmas 17(7), 072109 (2010). [CrossRef]  

22. H.-I. Lee, “Wave classification and resonant excitations in lossy metal-dielectric multilayers,” Photonics Nanostruct. Fundam. Appl. 8(3), 183–197 (2010). [CrossRef]  

23. R. W. D. Nickalls, “A new approach to solving the cubic: Cardan's solution revealed,” The Mathematical Gazette 77(480), 354–359 (1993). [CrossRef]  

24. V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B Condens. Matter 50(23), 16835–16844 (1994). [CrossRef]   [PubMed]  

25. A. A. Govyadinov and V. A. Markel, “From slow to superluminal propagation: dispersive properties of surface plasmon polaritions in linear chains of metallic nanospheroids,” Phys. Rev. B 78(3), 035403 (2008). [CrossRef]  

26. A. Kaso and S. John, “Nonlinear Bloch waves in metallic photonic band-gap filaments,” Phys. Rev. A 76(5), 053838 (2007). [CrossRef]  

27. S. John and R. Wang, “Metallic photonic-band-gap filament architectures for optimized incandescent lighting,” Phys. Rev. A 78(4), 043809 (2008). [CrossRef]  

28. V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes-part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006). [CrossRef]  

References

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  1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
    [CrossRef] [PubMed]
  2. J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70(1), 1–87 (2007).
    [CrossRef]
  3. J. S. Yang, J.-H. Sung, and B.-H. O, “Novel elastic scattering model for the understanding of the Anomalous transmittance for Au nanoparticle layer,” Opt. Express 18(13), 13418–13424 (2010).
    [CrossRef] [PubMed]
  4. T. Savin and P. S. Doyle, “Role of a finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(4), 041106 (2005).
    [CrossRef] [PubMed]
  5. G. Cuniberti, M. Sassetti, and B. Kramer, “Ac conductance of a quantum wire with electron-electron interactions,” Phys. Rev. B 57(3), 1515–1526 (1998).
    [CrossRef]
  6. N. Nakai, N. Hayashi, and M. Machida, “Simulation studies for the vortex-depinning dynamics around a columnar defect in superconductors,” Physica C 468(15–20), 1270–1273 (2008).
    [CrossRef]
  7. N. Nakai, N. Hayashi, and M. Machida, “Direct numerical confirmation of pinning-induced sign change in the superconducting Hall effect in type-II superconductors,” Phys. Rev. B 83(2), 024507 (2011).
    [CrossRef]
  8. F. Hébert, M. Schram, R. T. Scalettar, W. B. Chen, and Z. Bai, “Hatano-Nelson model with a periodic potential,” Eur. Phys. J. B 79(4), 465–471 (2011).
    [CrossRef]
  9. V. A. Fedotov, A. Tsiatmas, J. H. Shi, R. Buckingham, P. de Groot, Y. Chen, S. Wang, and N. I. Zheludev, “Temperature control of Fano resonances and transmission in superconducting metamaterials,” Opt. Express 18(9), 9015–9019 (2010).
    [CrossRef] [PubMed]
  10. H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94(12), 123901 (2005).
    [CrossRef] [PubMed]
  11. A. Mazzei, S. Götzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99(17), 173603 (2007).
    [CrossRef] [PubMed]
  12. S. Zou, “Electromagnetic wave propagation in a multilayer silver particle,” Chem. Phys. Lett. 454(4–6), 289–293 (2008).
    [CrossRef]
  13. N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317(5845), 1698–1702 (2007).
    [CrossRef] [PubMed]
  14. M. Staffaroni, J. Conway, S. Vedantam, J. Tang, and E. Yablonovitch, “Circuit analysis in metal-optics,” arXiv:1006.3126 [physics.optics].
  15. V. K. Valev, A. V. Silhanek, W. Gillijns, Y. Jeyaram, H. Paddubrouskaya, A. Volodin, C. G. Biris, N. C. Panoiu, B. De Clercq, M. Ameloot, O. A. Aktsipetrov, V. V. Moshchalkov, and T. Verbiest, “Plasmons reveal the direction of magnetization in nickel nanostructures,” ACS Nano 5(1), 91–96 (2011).
    [CrossRef]
  16. A. Z. Abbasi, F. Amin, T. Niebling, S. Friede, M. Ochs, S. Carregal-Romero, J.-M. Montenegro, P. Rivera Gil, W. Heimbrodt, and W. J. Parak, “How colloidal nanoparticles could facilitate multiplexed measurements of different analytes with analyte-sensitive organic fluorophores,” ACS Nano 5(1), 21–25 (2011).
    [CrossRef] [PubMed]
  17. A. Haddadpour and Y. Yi, “Metallic nanoparticle on micro ring resonator for bio optical detection and sensing,” Biomed. Opt. Express 1(2), 378–384 (2010).
    [CrossRef]
  18. L. Prkna, J. Čtyroký, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36(1–3), 259–269 (2004).
    [CrossRef]
  19. V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55(12), 7427–7444 (1997).
    [CrossRef]
  20. H.-I. Lee and J. Mok, “On the cubic zero-order solution of electromagnetic waves. I. Periodic slabs with lossy plasmas,” Phys. Plasmas 17(7), 072108 (2010).
    [CrossRef]
  21. H.-I. Lee and J. Mok, “On the cubic zero-order solution of electromagnetic waves. II. Isolated particles with lossy plasmas,” Phys. Plasmas 17(7), 072109 (2010).
    [CrossRef]
  22. H.-I. Lee, “Wave classification and resonant excitations in lossy metal-dielectric multilayers,” Photonics Nanostruct. Fundam. Appl. 8(3), 183–197 (2010).
    [CrossRef]
  23. R. W. D. Nickalls, “A new approach to solving the cubic: Cardan's solution revealed,” The Mathematical Gazette 77(480), 354–359 (1993).
    [CrossRef]
  24. V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B Condens. Matter 50(23), 16835–16844 (1994).
    [CrossRef] [PubMed]
  25. A. A. Govyadinov and V. A. Markel, “From slow to superluminal propagation: dispersive properties of surface plasmon polaritions in linear chains of metallic nanospheroids,” Phys. Rev. B 78(3), 035403 (2008).
    [CrossRef]
  26. A. Kaso and S. John, “Nonlinear Bloch waves in metallic photonic band-gap filaments,” Phys. Rev. A 76(5), 053838 (2007).
    [CrossRef]
  27. S. John and R. Wang, “Metallic photonic-band-gap filament architectures for optimized incandescent lighting,” Phys. Rev. A 78(4), 043809 (2008).
    [CrossRef]
  28. V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes-part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006).
    [CrossRef]

2011 (4)

N. Nakai, N. Hayashi, and M. Machida, “Direct numerical confirmation of pinning-induced sign change in the superconducting Hall effect in type-II superconductors,” Phys. Rev. B 83(2), 024507 (2011).
[CrossRef]

F. Hébert, M. Schram, R. T. Scalettar, W. B. Chen, and Z. Bai, “Hatano-Nelson model with a periodic potential,” Eur. Phys. J. B 79(4), 465–471 (2011).
[CrossRef]

V. K. Valev, A. V. Silhanek, W. Gillijns, Y. Jeyaram, H. Paddubrouskaya, A. Volodin, C. G. Biris, N. C. Panoiu, B. De Clercq, M. Ameloot, O. A. Aktsipetrov, V. V. Moshchalkov, and T. Verbiest, “Plasmons reveal the direction of magnetization in nickel nanostructures,” ACS Nano 5(1), 91–96 (2011).
[CrossRef]

A. Z. Abbasi, F. Amin, T. Niebling, S. Friede, M. Ochs, S. Carregal-Romero, J.-M. Montenegro, P. Rivera Gil, W. Heimbrodt, and W. J. Parak, “How colloidal nanoparticles could facilitate multiplexed measurements of different analytes with analyte-sensitive organic fluorophores,” ACS Nano 5(1), 21–25 (2011).
[CrossRef] [PubMed]

2010 (6)

H.-I. Lee and J. Mok, “On the cubic zero-order solution of electromagnetic waves. I. Periodic slabs with lossy plasmas,” Phys. Plasmas 17(7), 072108 (2010).
[CrossRef]

H.-I. Lee and J. Mok, “On the cubic zero-order solution of electromagnetic waves. II. Isolated particles with lossy plasmas,” Phys. Plasmas 17(7), 072109 (2010).
[CrossRef]

H.-I. Lee, “Wave classification and resonant excitations in lossy metal-dielectric multilayers,” Photonics Nanostruct. Fundam. Appl. 8(3), 183–197 (2010).
[CrossRef]

V. A. Fedotov, A. Tsiatmas, J. H. Shi, R. Buckingham, P. de Groot, Y. Chen, S. Wang, and N. I. Zheludev, “Temperature control of Fano resonances and transmission in superconducting metamaterials,” Opt. Express 18(9), 9015–9019 (2010).
[CrossRef] [PubMed]

J. S. Yang, J.-H. Sung, and B.-H. O, “Novel elastic scattering model for the understanding of the Anomalous transmittance for Au nanoparticle layer,” Opt. Express 18(13), 13418–13424 (2010).
[CrossRef] [PubMed]

A. Haddadpour and Y. Yi, “Metallic nanoparticle on micro ring resonator for bio optical detection and sensing,” Biomed. Opt. Express 1(2), 378–384 (2010).
[CrossRef]

2008 (4)

A. A. Govyadinov and V. A. Markel, “From slow to superluminal propagation: dispersive properties of surface plasmon polaritions in linear chains of metallic nanospheroids,” Phys. Rev. B 78(3), 035403 (2008).
[CrossRef]

S. John and R. Wang, “Metallic photonic-band-gap filament architectures for optimized incandescent lighting,” Phys. Rev. A 78(4), 043809 (2008).
[CrossRef]

N. Nakai, N. Hayashi, and M. Machida, “Simulation studies for the vortex-depinning dynamics around a columnar defect in superconductors,” Physica C 468(15–20), 1270–1273 (2008).
[CrossRef]

S. Zou, “Electromagnetic wave propagation in a multilayer silver particle,” Chem. Phys. Lett. 454(4–6), 289–293 (2008).
[CrossRef]

2007 (4)

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317(5845), 1698–1702 (2007).
[CrossRef] [PubMed]

A. Mazzei, S. Götzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99(17), 173603 (2007).
[CrossRef] [PubMed]

A. Kaso and S. John, “Nonlinear Bloch waves in metallic photonic band-gap filaments,” Phys. Rev. A 76(5), 053838 (2007).
[CrossRef]

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70(1), 1–87 (2007).
[CrossRef]

2006 (1)

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes-part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006).
[CrossRef]

2005 (2)

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94(12), 123901 (2005).
[CrossRef] [PubMed]

T. Savin and P. S. Doyle, “Role of a finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(4), 041106 (2005).
[CrossRef] [PubMed]

2004 (1)

L. Prkna, J. Čtyroký, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36(1–3), 259–269 (2004).
[CrossRef]

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

1998 (1)

G. Cuniberti, M. Sassetti, and B. Kramer, “Ac conductance of a quantum wire with electron-electron interactions,” Phys. Rev. B 57(3), 1515–1526 (1998).
[CrossRef]

1997 (1)

V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55(12), 7427–7444 (1997).
[CrossRef]

1994 (1)

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B Condens. Matter 50(23), 16835–16844 (1994).
[CrossRef] [PubMed]

1993 (1)

R. W. D. Nickalls, “A new approach to solving the cubic: Cardan's solution revealed,” The Mathematical Gazette 77(480), 354–359 (1993).
[CrossRef]

Abbasi, A. Z.

A. Z. Abbasi, F. Amin, T. Niebling, S. Friede, M. Ochs, S. Carregal-Romero, J.-M. Montenegro, P. Rivera Gil, W. Heimbrodt, and W. J. Parak, “How colloidal nanoparticles could facilitate multiplexed measurements of different analytes with analyte-sensitive organic fluorophores,” ACS Nano 5(1), 21–25 (2011).
[CrossRef] [PubMed]

Aktsipetrov, O. A.

V. K. Valev, A. V. Silhanek, W. Gillijns, Y. Jeyaram, H. Paddubrouskaya, A. Volodin, C. G. Biris, N. C. Panoiu, B. De Clercq, M. Ameloot, O. A. Aktsipetrov, V. V. Moshchalkov, and T. Verbiest, “Plasmons reveal the direction of magnetization in nickel nanostructures,” ACS Nano 5(1), 91–96 (2011).
[CrossRef]

Ameloot, M.

V. K. Valev, A. V. Silhanek, W. Gillijns, Y. Jeyaram, H. Paddubrouskaya, A. Volodin, C. G. Biris, N. C. Panoiu, B. De Clercq, M. Ameloot, O. A. Aktsipetrov, V. V. Moshchalkov, and T. Verbiest, “Plasmons reveal the direction of magnetization in nickel nanostructures,” ACS Nano 5(1), 91–96 (2011).
[CrossRef]

Amin, F.

A. Z. Abbasi, F. Amin, T. Niebling, S. Friede, M. Ochs, S. Carregal-Romero, J.-M. Montenegro, P. Rivera Gil, W. Heimbrodt, and W. J. Parak, “How colloidal nanoparticles could facilitate multiplexed measurements of different analytes with analyte-sensitive organic fluorophores,” ACS Nano 5(1), 21–25 (2011).
[CrossRef] [PubMed]

Bai, Z.

F. Hébert, M. Schram, R. T. Scalettar, W. B. Chen, and Z. Bai, “Hatano-Nelson model with a periodic potential,” Eur. Phys. J. B 79(4), 465–471 (2011).
[CrossRef]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Benson, O.

A. Mazzei, S. Götzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99(17), 173603 (2007).
[CrossRef] [PubMed]

Biris, C. G.

V. K. Valev, A. V. Silhanek, W. Gillijns, Y. Jeyaram, H. Paddubrouskaya, A. Volodin, C. G. Biris, N. C. Panoiu, B. De Clercq, M. Ameloot, O. A. Aktsipetrov, V. V. Moshchalkov, and T. Verbiest, “Plasmons reveal the direction of magnetization in nickel nanostructures,” ACS Nano 5(1), 91–96 (2011).
[CrossRef]

Bogaerts, W.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94(12), 123901 (2005).
[CrossRef] [PubMed]

Buckingham, R.

Carregal-Romero, S.

A. Z. Abbasi, F. Amin, T. Niebling, S. Friede, M. Ochs, S. Carregal-Romero, J.-M. Montenegro, P. Rivera Gil, W. Heimbrodt, and W. J. Parak, “How colloidal nanoparticles could facilitate multiplexed measurements of different analytes with analyte-sensitive organic fluorophores,” ACS Nano 5(1), 21–25 (2011).
[CrossRef] [PubMed]

Chen, W. B.

F. Hébert, M. Schram, R. T. Scalettar, W. B. Chen, and Z. Bai, “Hatano-Nelson model with a periodic potential,” Eur. Phys. J. B 79(4), 465–471 (2011).
[CrossRef]

Chen, Y.

Chulkov, E. V.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70(1), 1–87 (2007).
[CrossRef]

Ctyroký, J.

L. Prkna, J. Čtyroký, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36(1–3), 259–269 (2004).
[CrossRef]

Cuniberti, G.

G. Cuniberti, M. Sassetti, and B. Kramer, “Ac conductance of a quantum wire with electron-electron interactions,” Phys. Rev. B 57(3), 1515–1526 (1998).
[CrossRef]

De Clercq, B.

V. K. Valev, A. V. Silhanek, W. Gillijns, Y. Jeyaram, H. Paddubrouskaya, A. Volodin, C. G. Biris, N. C. Panoiu, B. De Clercq, M. Ameloot, O. A. Aktsipetrov, V. V. Moshchalkov, and T. Verbiest, “Plasmons reveal the direction of magnetization in nickel nanostructures,” ACS Nano 5(1), 91–96 (2011).
[CrossRef]

de Groot, P.

de S. Menezes, L.

A. Mazzei, S. Götzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99(17), 173603 (2007).
[CrossRef] [PubMed]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Doyle, P. S.

T. Savin and P. S. Doyle, “Role of a finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(4), 041106 (2005).
[CrossRef] [PubMed]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Echenique, P. M.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70(1), 1–87 (2007).
[CrossRef]

Engelen, R. J. P.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94(12), 123901 (2005).
[CrossRef] [PubMed]

Engheta, N.

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317(5845), 1698–1702 (2007).
[CrossRef] [PubMed]

Fedotov, V. A.

Friede, S.

A. Z. Abbasi, F. Amin, T. Niebling, S. Friede, M. Ochs, S. Carregal-Romero, J.-M. Montenegro, P. Rivera Gil, W. Heimbrodt, and W. J. Parak, “How colloidal nanoparticles could facilitate multiplexed measurements of different analytes with analyte-sensitive organic fluorophores,” ACS Nano 5(1), 21–25 (2011).
[CrossRef] [PubMed]

Gersen, H.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94(12), 123901 (2005).
[CrossRef] [PubMed]

Gillijns, W.

V. K. Valev, A. V. Silhanek, W. Gillijns, Y. Jeyaram, H. Paddubrouskaya, A. Volodin, C. G. Biris, N. C. Panoiu, B. De Clercq, M. Ameloot, O. A. Aktsipetrov, V. V. Moshchalkov, and T. Verbiest, “Plasmons reveal the direction of magnetization in nickel nanostructures,” ACS Nano 5(1), 91–96 (2011).
[CrossRef]

Götzinger, S.

A. Mazzei, S. Götzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99(17), 173603 (2007).
[CrossRef] [PubMed]

Govyadinov, A. A.

A. A. Govyadinov and V. A. Markel, “From slow to superluminal propagation: dispersive properties of surface plasmon polaritions in linear chains of metallic nanospheroids,” Phys. Rev. B 78(3), 035403 (2008).
[CrossRef]

Haddadpour, A.

Hayashi, N.

N. Nakai, N. Hayashi, and M. Machida, “Direct numerical confirmation of pinning-induced sign change in the superconducting Hall effect in type-II superconductors,” Phys. Rev. B 83(2), 024507 (2011).
[CrossRef]

N. Nakai, N. Hayashi, and M. Machida, “Simulation studies for the vortex-depinning dynamics around a columnar defect in superconductors,” Physica C 468(15–20), 1270–1273 (2008).
[CrossRef]

Hébert, F.

F. Hébert, M. Schram, R. T. Scalettar, W. B. Chen, and Z. Bai, “Hatano-Nelson model with a periodic potential,” Eur. Phys. J. B 79(4), 465–471 (2011).
[CrossRef]

Heimbrodt, W.

A. Z. Abbasi, F. Amin, T. Niebling, S. Friede, M. Ochs, S. Carregal-Romero, J.-M. Montenegro, P. Rivera Gil, W. Heimbrodt, and W. J. Parak, “How colloidal nanoparticles could facilitate multiplexed measurements of different analytes with analyte-sensitive organic fluorophores,” ACS Nano 5(1), 21–25 (2011).
[CrossRef] [PubMed]

Hubálek, M.

L. Prkna, J. Čtyroký, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36(1–3), 259–269 (2004).
[CrossRef]

Ilchenko, V. S.

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes-part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006).
[CrossRef]

Jeyaram, Y.

V. K. Valev, A. V. Silhanek, W. Gillijns, Y. Jeyaram, H. Paddubrouskaya, A. Volodin, C. G. Biris, N. C. Panoiu, B. De Clercq, M. Ameloot, O. A. Aktsipetrov, V. V. Moshchalkov, and T. Verbiest, “Plasmons reveal the direction of magnetization in nickel nanostructures,” ACS Nano 5(1), 91–96 (2011).
[CrossRef]

John, S.

S. John and R. Wang, “Metallic photonic-band-gap filament architectures for optimized incandescent lighting,” Phys. Rev. A 78(4), 043809 (2008).
[CrossRef]

A. Kaso and S. John, “Nonlinear Bloch waves in metallic photonic band-gap filaments,” Phys. Rev. A 76(5), 053838 (2007).
[CrossRef]

Karle, T. J.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94(12), 123901 (2005).
[CrossRef] [PubMed]

Kaso, A.

A. Kaso and S. John, “Nonlinear Bloch waves in metallic photonic band-gap filaments,” Phys. Rev. A 76(5), 053838 (2007).
[CrossRef]

Korterik, J. P.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94(12), 123901 (2005).
[CrossRef] [PubMed]

Kramer, B.

G. Cuniberti, M. Sassetti, and B. Kramer, “Ac conductance of a quantum wire with electron-electron interactions,” Phys. Rev. B 57(3), 1515–1526 (1998).
[CrossRef]

Krauss, T. F.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94(12), 123901 (2005).
[CrossRef] [PubMed]

Kuipers, L.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94(12), 123901 (2005).
[CrossRef] [PubMed]

Kuzmiak, V.

V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55(12), 7427–7444 (1997).
[CrossRef]

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B Condens. Matter 50(23), 16835–16844 (1994).
[CrossRef] [PubMed]

Lee, H.-I.

H.-I. Lee, “Wave classification and resonant excitations in lossy metal-dielectric multilayers,” Photonics Nanostruct. Fundam. Appl. 8(3), 183–197 (2010).
[CrossRef]

H.-I. Lee and J. Mok, “On the cubic zero-order solution of electromagnetic waves. II. Isolated particles with lossy plasmas,” Phys. Plasmas 17(7), 072109 (2010).
[CrossRef]

H.-I. Lee and J. Mok, “On the cubic zero-order solution of electromagnetic waves. I. Periodic slabs with lossy plasmas,” Phys. Plasmas 17(7), 072108 (2010).
[CrossRef]

Machida, M.

N. Nakai, N. Hayashi, and M. Machida, “Direct numerical confirmation of pinning-induced sign change in the superconducting Hall effect in type-II superconductors,” Phys. Rev. B 83(2), 024507 (2011).
[CrossRef]

N. Nakai, N. Hayashi, and M. Machida, “Simulation studies for the vortex-depinning dynamics around a columnar defect in superconductors,” Physica C 468(15–20), 1270–1273 (2008).
[CrossRef]

Maradudin, A. A.

V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55(12), 7427–7444 (1997).
[CrossRef]

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B Condens. Matter 50(23), 16835–16844 (1994).
[CrossRef] [PubMed]

Markel, V. A.

A. A. Govyadinov and V. A. Markel, “From slow to superluminal propagation: dispersive properties of surface plasmon polaritions in linear chains of metallic nanospheroids,” Phys. Rev. B 78(3), 035403 (2008).
[CrossRef]

Matsko, A. B.

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes-part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006).
[CrossRef]

Mazzei, A.

A. Mazzei, S. Götzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99(17), 173603 (2007).
[CrossRef] [PubMed]

Mok, J.

H.-I. Lee and J. Mok, “On the cubic zero-order solution of electromagnetic waves. II. Isolated particles with lossy plasmas,” Phys. Plasmas 17(7), 072109 (2010).
[CrossRef]

H.-I. Lee and J. Mok, “On the cubic zero-order solution of electromagnetic waves. I. Periodic slabs with lossy plasmas,” Phys. Plasmas 17(7), 072108 (2010).
[CrossRef]

Montenegro, J.-M.

A. Z. Abbasi, F. Amin, T. Niebling, S. Friede, M. Ochs, S. Carregal-Romero, J.-M. Montenegro, P. Rivera Gil, W. Heimbrodt, and W. J. Parak, “How colloidal nanoparticles could facilitate multiplexed measurements of different analytes with analyte-sensitive organic fluorophores,” ACS Nano 5(1), 21–25 (2011).
[CrossRef] [PubMed]

Moshchalkov, V. V.

V. K. Valev, A. V. Silhanek, W. Gillijns, Y. Jeyaram, H. Paddubrouskaya, A. Volodin, C. G. Biris, N. C. Panoiu, B. De Clercq, M. Ameloot, O. A. Aktsipetrov, V. V. Moshchalkov, and T. Verbiest, “Plasmons reveal the direction of magnetization in nickel nanostructures,” ACS Nano 5(1), 91–96 (2011).
[CrossRef]

Nakai, N.

N. Nakai, N. Hayashi, and M. Machida, “Direct numerical confirmation of pinning-induced sign change in the superconducting Hall effect in type-II superconductors,” Phys. Rev. B 83(2), 024507 (2011).
[CrossRef]

N. Nakai, N. Hayashi, and M. Machida, “Simulation studies for the vortex-depinning dynamics around a columnar defect in superconductors,” Physica C 468(15–20), 1270–1273 (2008).
[CrossRef]

Nickalls, R. W. D.

R. W. D. Nickalls, “A new approach to solving the cubic: Cardan's solution revealed,” The Mathematical Gazette 77(480), 354–359 (1993).
[CrossRef]

Niebling, T.

A. Z. Abbasi, F. Amin, T. Niebling, S. Friede, M. Ochs, S. Carregal-Romero, J.-M. Montenegro, P. Rivera Gil, W. Heimbrodt, and W. J. Parak, “How colloidal nanoparticles could facilitate multiplexed measurements of different analytes with analyte-sensitive organic fluorophores,” ACS Nano 5(1), 21–25 (2011).
[CrossRef] [PubMed]

O, B.-H.

Ochs, M.

A. Z. Abbasi, F. Amin, T. Niebling, S. Friede, M. Ochs, S. Carregal-Romero, J.-M. Montenegro, P. Rivera Gil, W. Heimbrodt, and W. J. Parak, “How colloidal nanoparticles could facilitate multiplexed measurements of different analytes with analyte-sensitive organic fluorophores,” ACS Nano 5(1), 21–25 (2011).
[CrossRef] [PubMed]

Paddubrouskaya, H.

V. K. Valev, A. V. Silhanek, W. Gillijns, Y. Jeyaram, H. Paddubrouskaya, A. Volodin, C. G. Biris, N. C. Panoiu, B. De Clercq, M. Ameloot, O. A. Aktsipetrov, V. V. Moshchalkov, and T. Verbiest, “Plasmons reveal the direction of magnetization in nickel nanostructures,” ACS Nano 5(1), 91–96 (2011).
[CrossRef]

Panoiu, N. C.

V. K. Valev, A. V. Silhanek, W. Gillijns, Y. Jeyaram, H. Paddubrouskaya, A. Volodin, C. G. Biris, N. C. Panoiu, B. De Clercq, M. Ameloot, O. A. Aktsipetrov, V. V. Moshchalkov, and T. Verbiest, “Plasmons reveal the direction of magnetization in nickel nanostructures,” ACS Nano 5(1), 91–96 (2011).
[CrossRef]

Parak, W. J.

A. Z. Abbasi, F. Amin, T. Niebling, S. Friede, M. Ochs, S. Carregal-Romero, J.-M. Montenegro, P. Rivera Gil, W. Heimbrodt, and W. J. Parak, “How colloidal nanoparticles could facilitate multiplexed measurements of different analytes with analyte-sensitive organic fluorophores,” ACS Nano 5(1), 21–25 (2011).
[CrossRef] [PubMed]

Pincemin, F.

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B Condens. Matter 50(23), 16835–16844 (1994).
[CrossRef] [PubMed]

Pitarke, J. M.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70(1), 1–87 (2007).
[CrossRef]

Prkna, L.

L. Prkna, J. Čtyroký, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36(1–3), 259–269 (2004).
[CrossRef]

Rivera Gil, P.

A. Z. Abbasi, F. Amin, T. Niebling, S. Friede, M. Ochs, S. Carregal-Romero, J.-M. Montenegro, P. Rivera Gil, W. Heimbrodt, and W. J. Parak, “How colloidal nanoparticles could facilitate multiplexed measurements of different analytes with analyte-sensitive organic fluorophores,” ACS Nano 5(1), 21–25 (2011).
[CrossRef] [PubMed]

Sandoghdar, V.

A. Mazzei, S. Götzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99(17), 173603 (2007).
[CrossRef] [PubMed]

Sassetti, M.

G. Cuniberti, M. Sassetti, and B. Kramer, “Ac conductance of a quantum wire with electron-electron interactions,” Phys. Rev. B 57(3), 1515–1526 (1998).
[CrossRef]

Savin, T.

T. Savin and P. S. Doyle, “Role of a finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(4), 041106 (2005).
[CrossRef] [PubMed]

Scalettar, R. T.

F. Hébert, M. Schram, R. T. Scalettar, W. B. Chen, and Z. Bai, “Hatano-Nelson model with a periodic potential,” Eur. Phys. J. B 79(4), 465–471 (2011).
[CrossRef]

Schram, M.

F. Hébert, M. Schram, R. T. Scalettar, W. B. Chen, and Z. Bai, “Hatano-Nelson model with a periodic potential,” Eur. Phys. J. B 79(4), 465–471 (2011).
[CrossRef]

Shi, J. H.

Silhanek, A. V.

V. K. Valev, A. V. Silhanek, W. Gillijns, Y. Jeyaram, H. Paddubrouskaya, A. Volodin, C. G. Biris, N. C. Panoiu, B. De Clercq, M. Ameloot, O. A. Aktsipetrov, V. V. Moshchalkov, and T. Verbiest, “Plasmons reveal the direction of magnetization in nickel nanostructures,” ACS Nano 5(1), 91–96 (2011).
[CrossRef]

Silkin, V. M.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70(1), 1–87 (2007).
[CrossRef]

Sung, J.-H.

Tsiatmas, A.

Valev, V. K.

V. K. Valev, A. V. Silhanek, W. Gillijns, Y. Jeyaram, H. Paddubrouskaya, A. Volodin, C. G. Biris, N. C. Panoiu, B. De Clercq, M. Ameloot, O. A. Aktsipetrov, V. V. Moshchalkov, and T. Verbiest, “Plasmons reveal the direction of magnetization in nickel nanostructures,” ACS Nano 5(1), 91–96 (2011).
[CrossRef]

van Hulst, N. F.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94(12), 123901 (2005).
[CrossRef] [PubMed]

Verbiest, T.

V. K. Valev, A. V. Silhanek, W. Gillijns, Y. Jeyaram, H. Paddubrouskaya, A. Volodin, C. G. Biris, N. C. Panoiu, B. De Clercq, M. Ameloot, O. A. Aktsipetrov, V. V. Moshchalkov, and T. Verbiest, “Plasmons reveal the direction of magnetization in nickel nanostructures,” ACS Nano 5(1), 91–96 (2011).
[CrossRef]

Volodin, A.

V. K. Valev, A. V. Silhanek, W. Gillijns, Y. Jeyaram, H. Paddubrouskaya, A. Volodin, C. G. Biris, N. C. Panoiu, B. De Clercq, M. Ameloot, O. A. Aktsipetrov, V. V. Moshchalkov, and T. Verbiest, “Plasmons reveal the direction of magnetization in nickel nanostructures,” ACS Nano 5(1), 91–96 (2011).
[CrossRef]

Wang, R.

S. John and R. Wang, “Metallic photonic-band-gap filament architectures for optimized incandescent lighting,” Phys. Rev. A 78(4), 043809 (2008).
[CrossRef]

Wang, S.

Yang, J. S.

Yi, Y.

Zheludev, N. I.

Zou, S.

S. Zou, “Electromagnetic wave propagation in a multilayer silver particle,” Chem. Phys. Lett. 454(4–6), 289–293 (2008).
[CrossRef]

Zumofen, G.

A. Mazzei, S. Götzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99(17), 173603 (2007).
[CrossRef] [PubMed]

ACS Nano (2)

V. K. Valev, A. V. Silhanek, W. Gillijns, Y. Jeyaram, H. Paddubrouskaya, A. Volodin, C. G. Biris, N. C. Panoiu, B. De Clercq, M. Ameloot, O. A. Aktsipetrov, V. V. Moshchalkov, and T. Verbiest, “Plasmons reveal the direction of magnetization in nickel nanostructures,” ACS Nano 5(1), 91–96 (2011).
[CrossRef]

A. Z. Abbasi, F. Amin, T. Niebling, S. Friede, M. Ochs, S. Carregal-Romero, J.-M. Montenegro, P. Rivera Gil, W. Heimbrodt, and W. J. Parak, “How colloidal nanoparticles could facilitate multiplexed measurements of different analytes with analyte-sensitive organic fluorophores,” ACS Nano 5(1), 21–25 (2011).
[CrossRef] [PubMed]

Biomed. Opt. Express (1)

Chem. Phys. Lett. (1)

S. Zou, “Electromagnetic wave propagation in a multilayer silver particle,” Chem. Phys. Lett. 454(4–6), 289–293 (2008).
[CrossRef]

Eur. Phys. J. B (1)

F. Hébert, M. Schram, R. T. Scalettar, W. B. Chen, and Z. Bai, “Hatano-Nelson model with a periodic potential,” Eur. Phys. J. B 79(4), 465–471 (2011).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes-part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006).
[CrossRef]

Nature (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Quantum Electron. (1)

L. Prkna, J. Čtyroký, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36(1–3), 259–269 (2004).
[CrossRef]

Photonics Nanostruct. Fundam. Appl. (1)

H.-I. Lee, “Wave classification and resonant excitations in lossy metal-dielectric multilayers,” Photonics Nanostruct. Fundam. Appl. 8(3), 183–197 (2010).
[CrossRef]

Phys. Plasmas (2)

H.-I. Lee and J. Mok, “On the cubic zero-order solution of electromagnetic waves. I. Periodic slabs with lossy plasmas,” Phys. Plasmas 17(7), 072108 (2010).
[CrossRef]

H.-I. Lee and J. Mok, “On the cubic zero-order solution of electromagnetic waves. II. Isolated particles with lossy plasmas,” Phys. Plasmas 17(7), 072109 (2010).
[CrossRef]

Phys. Rev. A (2)

A. Kaso and S. John, “Nonlinear Bloch waves in metallic photonic band-gap filaments,” Phys. Rev. A 76(5), 053838 (2007).
[CrossRef]

S. John and R. Wang, “Metallic photonic-band-gap filament architectures for optimized incandescent lighting,” Phys. Rev. A 78(4), 043809 (2008).
[CrossRef]

Phys. Rev. B (4)

A. A. Govyadinov and V. A. Markel, “From slow to superluminal propagation: dispersive properties of surface plasmon polaritions in linear chains of metallic nanospheroids,” Phys. Rev. B 78(3), 035403 (2008).
[CrossRef]

V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55(12), 7427–7444 (1997).
[CrossRef]

G. Cuniberti, M. Sassetti, and B. Kramer, “Ac conductance of a quantum wire with electron-electron interactions,” Phys. Rev. B 57(3), 1515–1526 (1998).
[CrossRef]

N. Nakai, N. Hayashi, and M. Machida, “Direct numerical confirmation of pinning-induced sign change in the superconducting Hall effect in type-II superconductors,” Phys. Rev. B 83(2), 024507 (2011).
[CrossRef]

Phys. Rev. B Condens. Matter (1)

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B Condens. Matter 50(23), 16835–16844 (1994).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

T. Savin and P. S. Doyle, “Role of a finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(4), 041106 (2005).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94(12), 123901 (2005).
[CrossRef] [PubMed]

A. Mazzei, S. Götzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99(17), 173603 (2007).
[CrossRef] [PubMed]

Physica C (1)

N. Nakai, N. Hayashi, and M. Machida, “Simulation studies for the vortex-depinning dynamics around a columnar defect in superconductors,” Physica C 468(15–20), 1270–1273 (2008).
[CrossRef]

Rep. Prog. Phys. (1)

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70(1), 1–87 (2007).
[CrossRef]

Science (1)

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317(5845), 1698–1702 (2007).
[CrossRef] [PubMed]

The Mathematical Gazette (1)

R. W. D. Nickalls, “A new approach to solving the cubic: Cardan's solution revealed,” The Mathematical Gazette 77(480), 354–359 (1993).
[CrossRef]

Other (1)

M. Staffaroni, J. Conway, S. Vedantam, J. Tang, and E. Yablonovitch, “Circuit analysis in metal-optics,” arXiv:1006.3126 [physics.optics].

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Figures (5)

Fig. 1
Fig. 1

(a) The real and imaginary parts of complex conductance for the Drude model according to Eq. (3). (b) A single planar interface between the upper dielectric and the lower metal-like medium. The in-plane wave number is denoted by k directed in the x-direction. Non-zero γ r induces cross-interface energy flows, whereas non-zero γ i entails an imbalance between the forward (“F”) and backward (“B”) waves along the interface.

Fig. 4
Fig. 4

The degree of asymmetry e obtained by varying one parameter while fixing the remaining two: e ( k ) based on the data of Fig. 2(a) with γ r as an abscissa; e ( γ r ) based on the data of Fig. 3(a) with k as an abscissa; and e ( γ i ) vs. γ i .

Fig. 2
Fig. 2

(a) Migrations of four roots on the complex ( ω r , ω i ) -plane. (b) The quality factor plotted against ω r . The curves are generated by varying kover the range 0 k 2 , where the arrows mean the direction of increasing k. The filled circles in pink color indicate the starting states at k = 0 , whereas the diamonds indicate the states on the half-way at k = 1 . In the inset of panel (a), we show the symmetric trajectories for the specified data γ r = 0.2 and γ i = 0 (namely, a real-valued γ).

Fig. 3
Fig. 3

(a) Migrations of four roots on the complex ( ω r , ω i ) -plane. The arrows imply the direction of increasing γ r . The filled circles in pink color indicate the starting states at γ r = 0 (where ω i = 0 , thus being neutrally stable), whereas the diamonds indicate the states on the half-way at γ r = 1.5 . In the inset of panel (a), we show the symmetric trajectories for the prescribed data γ i = 0 (a real-valued γ) and k = 0.5 . (b) Migrations of four roots on the complex ( v g r , v g i ) -plane. The thick horizontal arrow in shaded colors indicates the region of the subluminal group velocity, namely, | v g r | < k = 0.5 .

Fig. 5
Fig. 5

The real and imaginary parts of k = ω ( 1 + ε ) 1 ε according to Eq. (13), where both of ( k r , k i ) are positive. Three values of the complex relaxation rate are examined: (a) γ i = 0.2 i 0.2 , (b) γ i = 0.03 i 0.2 , and (c) γ i = 0.2 i .

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

ε ( ω ) = 1 ω p 2 ω ( ω + i γ ) ,
ε r = 1 ω p 2 ( ω γ i ) ω D ε ,     ε i = ω p 2 γ r ω D ε ,     D ε ( ω γ i ) 2 + γ r 2 .
ε 1 + ω p 2 γ i ω | γ | 2 + i σ ω .
lim ω 0 σ r = ω p 2 γ r | γ | 2 + 2 ω p 2 γ r γ i | γ | 4 ω ,     lim ω 0 σ i = γ r 2 γ i 2 | γ | 4 ω p 2 ω .
ω 4 + i γ ω 3 ( 1 + 2 k 2 ) ω 2 2 i γ k 2 ω + k 2 = 0.
ω L ± ( k ) = ± 1 2 ( 1 + 2 k 2 ) 1 2 1 + 4 k 4 , ω H ± ( k ) = ± 1 2 ( 1 + 2 k 2 ) + 1 2 1 + 4 k 4 .
γ r ω i = I ( k 2 , ω r 2 , ω i 2 ) | D ( k 2 , ω ) | 2 .
I a k 4 + b k 2 + c , a 2 + 4 | ω | 2 > 0 ,     b 3 | ω | 2 + 4 | ω | 4 4 ω r 2 ( 1 + 2 | ω | 2 ) ,     c | ω | 4 + | ω | 6 > 0.
16 ω r 2 ( 2 | ω | 2 + 1 ) 2 ( | ω | 2 ω r 2 ) + ( 5 | ω | 2 + 8 | ω | 4 ) | ω | 2 > 0 ,
( L ) ω = i γ ,       ( H ) ω = 2 k ,       ( L + ) ω = 0 ,       ( H + ) ω = + 2 k .
v g ω k = 2 ω 2 + 2 i γ ω 1 4 ω 3 + 3 i γ ω 2 2 ( 1 + 2 k 2 ) ω 2 i γ k 2 2 k .
e = 1 4 ω / = 1 4 | ω | .
k = ± ω ω 2 + i γ ω 1 2 ω 2 + 2 i γ ω 1 .
exp [ i ( k x ω t ) ] = exp ( k i x ) exp [ i ( k r x ω t ) ] .

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