Strong optical interaction of two adjacent rectangular ... - OSA Publishing

Strong optical interaction of two adjacent rectangular nanoholes in a gold film M. Janipour,1 T. Pakizeh,2,* and F. Hodjat-Kashani1 2

1 Department of Electrical Engineering, Iran University of Science and Technology, Tehran 1684613114, Iran Faculty of Electrical and Computer Engineering, K. N. Toosi University of Technology, Tehran 1631714191, Iran * [email protected]

Abstract: The strong near-field optical interaction between two adjacent nanoholes milled in a gold film is investigated. A single nanohole is modeled as a magnetic dipole described by the simple relation between the magnetic- and electric-polarization in electromagnetic theory. To elucidate the role of the electric and magnetic fields in near-field characteristics of a nanohole illuminated by an optical plane-wave, the normalized electric and magnetic power amplitudes are accordingly introduced. This is extended to model the strong optical interaction of the two adjacent nanoholes in the near-field regime, leading to the magnetic coupled-dipole approximation (MCDA). It is shown that the optical transmission spectrum of the nanostructure may exhibit hybridized resonant peaks, depending on the configuration or the polarization. Compared to the known effects in the optical properties of a pair of metal nanoparticles for which the electricfield of the incident light is crucial, here it is illustrated that the magneticfield of the incident light plays the dominant role in defining the optical properties of the complement structure. Thus, the strength of the interaction of the two adjacent nanoholes and the resulting hybridized plasmon resonances are strongly depends on the magnetic-field orientation in respect to the pair axis as well as on the separating distance of the nanoholes. The theoretical findings are supported by the electromagnetic computations. ©2013 Optical Society of America OCIS codes: (310.5448) Polarization, other optical properties; (110.1220) Apertures; (310.6628) Subwavelength structures, nanostructures; (250.5403) Plasmonics.

References and Links 1.

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#195540 - $15.00 USD Received 12 Aug 2013; revised 11 Oct 2013; accepted 29 Oct 2013; published 16 Dec 2013 (C) 2013 OSA 30 December 2013 | Vol. 21, No. 26 | DOI:10.1364/OE.21.031769 | OPTICS EXPRESS 31769

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1. Introduction Since exploration of enhanced and extraordinary optical transmission through subwavelength nanoholes, much attention has been attracted to phenomena and applications related to subwavelength apertures milled in a metal film [1]. The most important optical feature of metallic nanostructures is the circumvention of the diffraction limit in conventional optics [2]. Using this unique trait and the electrical properties of metallic nanostructures, plasmonics could take a major step toward achieving nanoscale photonic and electronic devices [3]. Recent advances in nanofabrication techniques and also electromagnetic theory for plasmonic effects has led to more new applications such as integrated nanophotonic circuits [4], spectroscopy [5], metamaterials [6–8], nanoantennas [9,10], and sensing [11,12]. The extraordinary optical transmission through nanoholes is due to the arrangement of surface plasmon polaritons (SPPs) and the coupling effects with localized resonances [1, 13–16].


Some studies have been centered on the coupling interaction between linear chains of nanoholes through antisymmetric surface plasmon polaritons consequently, the chains are treated like the linear wire antennas [17], where the propagating SPPs on the interface connecting dielectric medium and metal film, have been considered as the coupling parameter between two nanoholes. The SPP coupling occurs when the separation distance between holes is comparable to the SPP wavelength, or in other words each nanohole locates in the farfield optical region of its neighboring hole [17]. In the present study, the coupling mechanisms between two closely spaced, i.e. in the nearfield region, subwavelength holes, arranged in sor p-configuration are investigated. Each nanohole is modeled using a magnetic-dipole, thus the strong optical interaction of a pair is theoretically studied using an extended coupled dipoles approximation method. In contrast to the optical interaction of the nanoparticles, it is found that the incident magnetic field plays the dominant role. Interestingly, similar to the strong interaction of a pair of nanoparticles, the hybridized resonant modes are observed in the near- and far-field calculations of two coupled nanoholes. 2. Theoretical Model Based on the small size of the structure relative to the incident field wavelength, the most of electromagnetic problems in nano-optics can be studied in the quasistatic limits. Often, the optical properties of the metal nanoparticles or nanostructures composed of them are simply modeled based on the electric dipoles with energies defined by the localized surface plasmon resonances (LSPRs) [18]. It is showed that a similar resonance is expected at the edges of a nanohole perforated in a noble metal [19, 20]. In this context, the dielectric functions of noble metals like gold (Au) which is dispersive in optical frequency region are also modeled using the Drude-Lorentz (DL) classical model with few poles, accounting for the interband transitions of the electrons [21, 22]. The main theoretical issue is to determine whether the optical properties of a nanohole, in a metal film, can be adequately modeled using an electricor a magnetic-dipole. Accordingly, the impacts of the electric and magnetic dipoles moments on the optical properties of a single nanohole should be elaborated. These properties can be extended for the coupling mechanism between two adjacent nanoholes. 2.1. Single Nanohole The incident light arranges the free electrons of the Au film according to the polarization of the incident light. The electrodynamic simulations demonstrate that a localized surface plasmon resonance (LSPR) with an electric dipole nature can be allocated to the single nanohole’s resonance [17]. The Babinet’s principle shows that a small hole in a thin metal film can be evaluated using induced dipoles, according to Bethe’s sketch of scattered fields by means of a single aperture milled in a thin screen [23, 24], which are one magnetic dipole parallel to the film and an electric dipole normal to it. Such a statement is applicable in microwave engineering, in which the thickness of the film is not considered and the metal supposed to be perfect electric conductor (PEC). A rectangular nanoaperture with dimensions a × b × d milled in an Au film with thickness d is considered. The electric polarizability of such a single nanoaperture or nanohole can be approximately defined [25–27]. The schematic of a single rectangular nanohole with cross section of a × b milled in a d thickness gold film, and the possible electric (P) and magnetic (M) dipole moments are shown in Fig. 1.


Fig. 1. The schematic of the electric (P) and magnetic (M) dipoles induced in a subwavelength rectangular hole (a × b), milled in an Au film with thickness d.

Based on a microwave concept which deals with the normalized amplitude of each dipole moment, it is possible to identify the weight of each dipole. Primarily, it is necessary to obtain the electric ( α e ) and magnetic ( α m ) polarizabilities of the considered subwavelength aperture. Usually, α e and α m of an arbitrary geometry are obtained by solving elliptic

integrals which depend on the hole/particle shape [24, 26]. However, by knowing α e of the hole [27], its α m can be preferably obtained by relating α m to α e . This relationship can be conveniently assessed based on Maxwell's equations [24, 28]: ∇ × E = jωμ0 H - jωμ0 Pm − M s ,

(1)

∇ × H = jωε 0 E - jω Pe − J s ,

(2)

where M s and J s are electric and magnetic current density sources, respectively. In addition, Pe and Pm are the electric and magnetic polarizations, correspondingly [28]: Pe = ε 0α e E,

(3)

Pm = −α m H,

(4)

Using Eqs. (1) and (2), and considering a source-free region ( M s = J s = 0 ), the following general relation between electric and magnetic polarizations is obtained: ∇ × Pm = − jω Pe . (5) As mentioned before, the electric and magnetic fields of a rectangular nanohole, milled in a metal film and exposed to the incident plane-wave light, can be identified using the quasistatic approximation method: E = −∇Φe ,

(6)

H = −∇Φm ,

(7)

 Φe   Be sin(k x x) cos(k y y )   =  × [ C cos h(kz ) + D sin h(kz ) ] , Φm   Bm cos(k x x) sin(k y y ) 

(8)

in which,

where k x = iπ / (a + 2δ ) , k y = π / (b + 2δ ) , k = k y2 − k x2 , and δ is the skin depth of Au [12]. It should be noted that kx is pure imaginary to account for the excited surface plasmons localized on the edges because of the incident x-polarized light. In Eq. (8) coefficients Be , Bm , C , and D are constant values. Having α e and inserting the fields evaluated from Eqs. (6) and (7) in Eqs. (3)–(5), the relationships between different components of α m and α e ,


using a 3 × 3 matrix equation, which depends on k x , k y , k , and α e , is obtained. Thus for an x-polarized incident light ( Ex , H y ), propagating along z-direction with wavenumber k0,

α ym = (C xe / C ym )α xe , where C ym and Cxe are constants obtained from left and right sides of Eq. (5), respectively. This is simplified to the following equation by ignoring the negligible variations of the potentials (Φe and Φm) along z-axis around the nanohole:  M − Nk02  e αx ,  M − Nkk0 

α ym ≈ −i 

(9)

where M = kk xωε 0 and N = ε 0 / μ0 (k y2 / k x ) . Obviously, the polarizability of x-oriented electric polarization ( α xe ) is related to the polarizability of y-oriented magnetic polarization ( α ym ). It can be shown that the latter is stronger than the former. Thus, it is expected that optical properties of a single plasmonic nanohole are modeled using the dominant magnetic dipole [Eq. (9)], oscillating at the LSPR energy of the nanohole and excited by the incident magnetic-field.

2.2. Two Adjacent Nanoholes Based on the quasistatic theory and described modeling of a single rectangular nanohole in sec. 2.1, the two adjacent nanoholes are theoretically modeled as two strongly interacting magnetic dipoles. Therefore, we propose Eq. (10) for the optical interaction of the magnetic dipoles of the subwavelength holes based on the magnetic coupled dipole approximation (MCDA) method: n   (q) (q) (10) M i( q ) (ω ) = −α qmi (ω )  H 0 (r , ω ) +  Rij (r , ω ) M j (ω )  , j =1, j ≠ i   where H 0 is the amplitude of the incident magnetic field, q = x, y, and z directions. In Eq. (10), M i( q ) and α qmi are the dipole magnetization and the magnetic polarizability, respectively. Moreover, Rij( q ) (r , ω ) M (j q ) (ω ) is the effect of jth on ith magnetic dipole because of the strong optical interaction. The proposed MCDA method is the extension of the known coupled dipole approximation (CDA) method which is often applied to model the electromagnetic interactions of nanoparticles [29, 30]. Although in an obvious contrast to the CDA, the magnetic field and magnetization are playing the crucial roles in the MCDA method. Thus, the optical coupling mechanism of two nanoholes which are placed closely, i.e. in the nearfield region, can be investigated using the described method. Furthermore, to demonstrate that the optical properties of the considered nanostructures can be well modeled using the dipoles, the normalized electric (Ae) and magnetic (Am) powers are evaluated as below [28]: Ae = Am =

−1 E ⋅ Jdv, P0 v

(11)

1 H ⋅ Mdv, P0 v

(12)

where J and M are the volume distributions of the electric and magnetic current densities induced in the metal region around the nanohole defined by volume v, respectively. In Eqs. (11) and (12), P0 is defined using the Lorentz reciprocity theorem as [24, 28]: P0 = 2  ( E0 × H 0 ) ⋅ nˆ ds,

(13)

S0


Renormalized Transmission

where the E0 and H 0 are the electric and magnetic fields of the incident plane-wave, and s0 is the part of the enclosed surface of the volume v, which is normal to Poynting vector of the incident light. Considering η0 as the free space characteristic impedance, Eq. (13) can be simplified to P0 = 2ab / η0 for a single nanohole with s0 = a × b [28]. Once E and H are theoretically or numerically obtained in the nearfield region, then using Eqs. (11) and (13) it will be trivial to calculate Ae and Am for a single nanohole or a pair of adjacent nanoholes. Hence, it can be considered appropriate to evaluate the electric and magnetic field powers in the near-field region as criteria to identify the electric or magnetic nature of the optical coupling of two nanoholes. 1

Circ. hole [31] Circ. hole, Drude Circ. hole, DL Sq. hole, DL Rect. hole, DL

0.8 0.6

(a)

0.4 0.2 0 500

600

700

800

Wavelength (nm)

900

1000

Fig. 2. (a) A comparison of the calculated normalized transmission of the single circular nanohole of radius r = 100 nm in a d = 100 nm Au film on a substrate (n = 1.5), using the Drude model (blue solid-line), compared with the results reported in Ref [31]. (circular-dots). Using the more accurate model (DL), the transmission spectra are calculated and shown for the circular nanohole (brown dashed-line), the same sized square shaped nanohole (red dashdotted line), and for a rectangular shaped nanohole with a = 100 nm and b = 200 nm (green thick solid-line). Normalized intensities of (b) the electric field, and (c) the magnetic field of the rectangular nanohole milled in an Au film with d = 100 nm at λ = 720 nm.

3. Results and discussions The investigation of the optical properties of a single rectangular nanohole is the basis for studying the plasmonic coupling effects of two neighbor rectangular nanoholes. The optical properties of an isolated circular shaped nanohole in an Au film have been presented [20, 27, 31]. However the present study deals with the rectangular shaped nanoholes, regenerating the simulations for the normalized transmission of a circular nanohole and verifying numerical simulation used for a single rectangular nanohole is not ungraceful.

3.1. Optical Properties of a Single Rectangular Nanohole The normalized optical transmission of a single circular nanohole with radius r = 100 nm in an Au film with d = 100 nm on a glass substrate ( ε glass = 2.25 ) is simulated and shown in Fig. 2(a). In this case, the Au is roughly modeled using the Drude model. The simulation is based on a three dimensional finite-difference time-domain (FDTD) method [32], with uniform mesh Δx = Δy = Δz = 2 nm. The incident optical Gaussian plane-wave is x-polarized ( Ex ,


H y ), and propagates along z-direction. The calculated results (blue solid-line) are compared

with the results adapted from ref [31]. (circular-dots). As clearly seen in Fig. 2(a), a very good agreement is achieved. By using the suitable DL-model instead of simple Drude-model for the Au film, the normalized transmission spectra for the circular nanohole (brown dashedcurve) and also for a square shaped nanohole with a = b = 200 nm (red dash-dot line) are also shown in Fig. 2(a). The differences between the calculated results of circular nanoholes using the Drude- and DL-model return to the characteristics of each model. Since the DL-model addresses the interband transitions in Au, it is considered more accurate and the spectra calculated based on that are clearly broader than the transmission based on the Drude-model. Moreover, it is obvious that the optical transmission spectrum of a square shaped nanohole is very similar to the spectrum of a circular nanohole with same dimensions. However, by making the nanohole elongated the normalized transmission spectrum can be efficiently tailored, depending on the polarization and the corresponding aspect-ratio. Thus, by reducing a to 100 nm [see Fig. 1], therefore b/a = 2, a rectangular nanohole which is elongated along yaxis is resulted, as shown in Fig. 1. The optical transmission of such rectangular-shaped nanohole with a = 100 nm, b = 200 nm, and d = 100 nm, floated in air, is presented in Fig. 2(a), denoted by a green thick solid-curve. Interestingly despite of removing the glass substrate, the transmission spectrum is remarkably red-shifted which is due to the variations of the LSPR properties from a square-shaped to a rectangular-shaped nanohole. This trend is unexpected in the complement Au nanoparticles illuminated by an x-polarized incident light, in which by decreasing dimension along x-axis, i.e. a, a blue shift can be expected. There, the nanoparticle is approximately modeled by an electric dipole, excited by the incident electricfield, i.e. E x . Thus, in obvious contrast with the nanoparticles, the aspect-ratio of the nanoholes should be defined based on the incident magnetic-field, irritating on the importance of the magnetic-field. In the considered coordinate system here, the incident magnetic field is along y-direction, the aspect-ratio of the nanohole increases by increasing the nanohole’s dimension along y-axis, i.e. b. Modeling the nanohole as a magnetic dipole oriented along ydirection ( M y ), excited by the incident magnetic-field, the observed remarkable red-shift of the spectrum can be properly justified. Figures 2(b) and 2(c) depict the calculated field profile of | Ex / E0 |2 and | H y / H 0 |2 in the nearfield region of the considered rectangular nanohole with the cross section of 100 × 200 nm2, milled in an Au film with thickness d = 100 nm, respectively. The profiles are plotted at wavelength λ = 720 nm, where the transmission exhibits a peak. It is obvious from Figs. 2(b) and 2(c), although the incident light is polarized along x-axis; the electric-field is mostly confined in the dielectric region (inside the nanohole) as shown in Fig. 2(c). This is in contrast to the complement nanoparticles, which dominantly enhance and confine the local electric-field along the incident electric-field polarization. This fact is due to a large difference between dielectric (air) and metal (Au) electric permittivities at the nanohole boundaries, which allows the normal component of the electric field of a nanoparticle, penetrate much greater in free space region than in the metal medium. The same concept is valid for a nanohole in a screen where the electric field is confined inside the nanohole region. However, the magnetic-field is enhanced and confined along y-axis. In the near-field region, it becomes clear that the nanohole dominantly acts on the incident magnetic-field, representing a dipolar nature of the localized magnetic-field around the nanohole. Thus the nanohole can be modeled by a magnetic dipole in the transverse screen for which its polarizability α m can be obtained using Eqs. (3), (4), and (9). Figure 3(a) depicts the normalized amplitudes for real (blue solid-curve) and imaginary (red dashed-curve) parts of α ym using this technique. Comparably, Fig. 3(b) shows the real and imaginary parts of the induced magnetic field ( H y ) at point R (0, y0 , z0 ) , denoted in Fig. 2(c), where y0 = 110 nm,


and z0 = 60 nm, verifying Eq. (9) and also Fig. 3(a). The relatively small differences between Figs. 3(a) and 3(b) are due to the effects of the incident light. Because this method is polarization dependent, finding α xm needs solving Eq. (5) for the y-polarized incident wave.

Normalized Amplitude

1

Simulation

Analytic Model

(a)

(b)

0.5

0

-0.5

Re{αy }, Eq. (9) m

Re{Hy } m y Im{H } m

Im{αy }, Eq. (9) m -1 500

600

700

800

900 1000 600 Wavelength (nm)

700

800

900

1000

Fig. 3. (a) The normalized real (blue solid-curve) and imaginary (red solid-curve) parts of

α y [Eq. (9)] for a nanohole with a = 100 nm, b = 200 nm, and d = 100 nm, illuminated by an m

x-polarized plane wave. (b) The simulated real and imaginary parts of the induced magneticfield ( H y ) around the nanohole at point R (0, y0 , z 0 ) , shown in Fig. 2(a).

In the next step, we calculate and evaluate the electric and magnetic power amplitudes of the considered single nanohole with volume v = a × b × d nm3 [see Fig. 1]. In this respect, Fig. 4(a) depicts the normalized magnetic power amplitude, Am , obtained from Eq. (12) using the FDTD simulation (solid-curve) and supported by the theoretical results obtained using the described quasistatic approximation method [Eq. (9), dashed-curve]. For the nanostructures considered here, it is observed that the calculated Am is much greater than the normalized electric power, Ae . In fact the calculated Ae compared to Am is in the order of 10−3 . Thus, it can be concluded that the incident light dominantly induces the magnetic dipole inside a single nanohole milled in an Au film.

3.2. Strong Optical Interaction of Two Rectangular Nanoholes Similar to the considered single nanohole, for two adjacent nanoholes, the incident light dominantly induces two magnetic dipoles which are electromagnetically interacting. However, in the case of two nanoholes, the structure is called serial (s-config.) and/or parallel (p-config.) configuration if both of holes lie along the major and/or minor axis, respectively. Figure 4(b) shows Am for the two rectangular nanoholes in s-config. which are positioned in the nearfield region of each other and are exposed by an x-polarized incident light. In all cases of interaction of two nanoholes with s- or p-config. arrangements it is considered that Δx = Δy = Δz = 1 nm. The separation distance between the holes is Δ = 5, 10, and 15 nm. It should be stressed that in the all cases considered in Fig. 4, Ae is not shown because it is negligible relative to Am . In the case of two nanoholes, the structure is called serial (s-config.) and/or parallel (pconfig.) configuration if the both of holes lie along the major (y) and/or minor (x) axis, respectively, as depicted in Fig. 5 for the nanoparticles and the complement nanostructure


composed of the nanoholes. Figure 4(b) shows Am for the two rectangular s-config. nanoholes which are positioned in the nearfield region of each other and are exposed by an xpolarized incident light, as schematically shown in Fig. 5(b2). Remarkably, the Am spectra demonstrate two resonant peaks for Δ = 5 (blue dash-dotted line), 10 (red solid-line), and 15 nm (green dashed-line). By decreasing distance Δ , the long-wavelength resonant peak redshifts while the short-wavelength resonant peak experiences a small blue-shift. Using the same procedure employed to theoretically obtain Am for a single nanohole, Am can be obtained for the nanostructure composed of two nanoholes considering s0 = 2a × b , and v = a × (2b + Δ ) × d for s-config., and v = (2a + Δ ) × b × d for p-config., in Eqs. (12) and (13). For instance, theoretical Am (circular markers) is shown in Fig. 4(b) for the case with Δ = 10 nm. However the both calculation and theory results similarly exhibit two hybridized peaks for the two s-config. nanoholes, the spectra are deviated at high-energy side of the spectrum due to the approximations made in the theory. 3

x 10-3

single nanohole Calculation Theory

(a)

-3 5 x 10

Two s-config. nanoholes

(b)

Δ = 15 nm

4

2 Am

Theory (Δ = 10 nm) 3

1

Δ = 10 nm Δ = 5 nm

2 0 500

600

700 800 Wavelength (nm)

900

1000 500

600

700 800 Wavelength (nm)

900

1000

Fig. 4. The calculated spectra of the normalized magnetic power amplitude, Am, for (a) the single nanohole, and (b) the two coupled nanoholes in the s-config. arrangement, with a = 100 nm, b = 200 nm, and distance Δ = 5, 10, and 15 nm. The calculated results in (a) and (b) are compared with the theoretical results.

The two resonant peaks observed in Fig. 4(b) can be assigned to the strong nearfield optical interactions of these magnetic dipoles. The idea of modeling of the single nanohole by a magnetic dipole can be extended to study the optical interaction of two adjacent nanoholes, arranged in the s- or p-configuration. Figure 5 shows the schematic descriptions of the coupling mechanisms in the two complement plasmonic nanostructures. In the left-column, Figs. 5(a1) and 5(a2) depict the possible coupling mechanisms of two cubic nanoparticles which are arranged in s- or pconfig., respectively. In this case, the two nanoparticles modeled as two electric dipoles (blue arrows), and clearly the stronger interaction occurs in the p-config. [Fig. 5(a2)]. In contrast, the coupling mechanisms of two nanoholes, modeled as two magnetic dipoles (red arrows) are shown in Figs. 5(b1) and 5(b2) for p- and s-config., respectively. However it is apparent that the coupling of nanoparticles is dependent on the polarization direction of the incident electric field, the coupling of nanoholes strongly depends on the direction of the incident magnetic field. It is expected that stronger interaction of nanoholes is occurred for the sconfig., shown in Fig. 5(b2), where the two nanoholes are aligned along the direction of the incident magnetic field, H inc = H y aˆ y .


Fig. 5. The schematic representation of the coupling mechanism between two nanoparticles (right-column) and their complement nanoholes (left-column). The filed components of the normal incident light (E)inc, (H)inc), and the corresponding electric and magnetic dipoles for the s-config. [(a1), (b2)] and p-config. [(a2), (b1)] are denoted by the arrows.

The circumstance of the strong interaction of the two magnetic dipoles and their similarities and differences with the interaction of the electric dipoles of the complement structure of nano-particles is studied with details of the spectral characteristics of the dipoles modified due to the strong optical interaction. The magnetic dipole model for two nanoholes arranged in the s-config. is depicted in Fig. 6(a). Using the MCDA method, the modified magnetic polarizabilities ( α ym ) of the two strongly interacting rectangular nanoholes (a × b) of Fig. 6(a), with a center-to-center separation distance s, can be obtained:

i

i

(

)

 4π s 3 4π s 3 + 2α ym (ω )eiks  j ,  (4π ) 2 s 6 − 4α ymj (ω )α ymi (ω )   

α ym (ω ) = α ym (ω ) 

(14)

where i ≠ j = 1, 2, and s = Δ + b in which Δ is the edge-to-edge distance of the nanoholes [see Fig. 6(a)]. It should be noted that although for the nearfield interactions ks 0 . As the distance increases, the value of Δλ decreases and the second mode resonant wavelength reaches to the transmission peak position of the single nanohole, associated with its LSPR wavelength, emphasizing on the property of the decoupled nanoholes. In addition, increasing Δ leads to an inconsiderable blue shift for the first mode of the s-config. Similar to the second mode of the s-config., by increasing Δ the first mode attains to the single nanohole resonance. Furthermore, as deduced from Fig. 7(b), the resonance wavelength of the p-config. experiences a small blue shift while increasing the Δ. 280

-30 2nd mode of s-config. 1st mode of s-config.

200

-10

0 5

10

Δ (nm)

Δ λ (nm)

Δ λ (nm)

p-config.

150 15

Fig. 8. The resonance shift, Δλ , of the modes versus the separating distance Δ for the sconfig. (blue-solid and red-dashed curves) and p-config. (green dash-dot curve).


4. Conclusions The quasi-static approximation is extended to analyze the strong optical interaction of a pair of rectangular nanoholes in a gold film. By modeling a single nanohole using a magneticdipole, the interaction of the pair nanohole is described using the coupling of magnetic dipoles in the proposed MCDA method. It is shown that the optical transmission spectrum of the nanostructure exhibits the two resonant peaks which are related to the excited and hybridized in-phase and anti-phase magnetic dipoles, when the magnetic field component of the normal incident light is parallel to the axis of the pair of nanoholes. Although for the perpendicular case, the spectrum shows a single resonant peak. Thus, the results are remarkably affected by the direction of the incident magnetic-field. In contrast to the case of metal nanoparticles, the simulation and theoretical results for this complement plasmonic nanostructure indicate the dominant role of the magnetic-field component of the incident light. Moreover in the s-configuration, by decreasing edge-to-edge distance of the nanoholes from 15 to 5 nm, the lower-energy plasmonic mode experiences a substantial red shift of about 135 nm. However, the corresponding higher-energy transmission mode demonstrates a negligible blue shift, reaffirming the hybridized nature of the excited modes. Finally, the results of the proposed alternative method of computing the magnetic polarizabilities of a nanohole support the electromagnetic simulations.
