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In this study, we investigated the plasmon resonances of split-ring resonators (SRRs) numerically at incident angles of 0 and 45° under illumination with linearly and circularly polarized waves. At 45° incidence, perpendicular polarized waves excited distinct odd plasmon modes; the difference in the reflections of right and left circularly polarized incident radiation was very large. From simulated near field plots, we found that the parallelism of the incident electric field and the induced plasmon current was the key factor affecting excitation. We propose the use of a parallelism factor (P-factor), based on a standing-wave approach, to characterize the ability of incident fields to excite multiple plasmon resonance currents. The mechanism of the field and current parallelism can explain the resonance behavior of SRRs when considering the polarization state, incident angle, and geometry of the SRR.
©2010 Optical Society of America
1. Introduction
Split-ring resonators (SRRs) were proposed as a negative refractive index material [1,2]. Negative permeability is an extraordinary phenomenon and can be produced with the magnetic resonance of SRRs [3]. Interesting applications of SRRs were investigated, such as infrared cloacking [4] and superlensing [5]. As a localized plasmonic sensor, multi plasmon modes provide multi-functional detection [6]. The plasmon resonances of SRR enhance Raman scattering for molecule detection [7]. Localized surface plasmon resonance (LSPR) is the oscillation of charge density confined in metallic nanostructures excited under direct illumination [8]. The magnetic and electric resonances are the earliest discussed plasmon modes in SRR. There are two methods for exciting a magnetic resonance. One is a magnetic coupling of the magnetic field perpendicular to an SRR plane, the other is an electric coupling of the electric field parallel to an SRR gap. In contrast, the electric resonance couples to the electric field perpendicular to an SRR gap [3,9]. Then, the multiple plasmon modes in SRRs at normal incidence were discovered [10,11]. The plasmon modes can be recognized by counting the number of nodes in the near field plot. The LC resonance can be treated as the fundamental mode of an SRR and the electric resonance the second mode. For the polarization state parallel to the gap, the charge distributions of the plasmon modes are in odd symmetry; they are the odd modes. For perpendicular polarization, the plasmon resonance modes are even modes. The antenna resonances [12–15] found in nanowires suggested that the multiple resonances of an SRR could be interpreted using the standing-wave model [13,15,16]. The validity of this model was confirmed by the existence of induced plasmon current nodes in the simulation and by the linear relationship between the resonance wavelength and total length divided by the mode number (L/m) [14,16].
Chiral metamaterials are metamaterials that lack mirror symmetry; they are said to possess intrinsic chirality [17,18]. An optical system is said to be an extrinsic chiral system if the structure of the metamaterial is symmetric but the arrangement of the system is not identical to its mirror image [19,20]. One such example, described herein, is that of SRRs measured at oblique incidence. The distinguishing feature of a chiral optical system is different optical phenomena resulting from irradiation with right- (RCP) and left-circularly-polarized (LCP) electromagnetic (EM) waves—for example, optical activity [21,22], circular dichroism [23], and asymmetric transmission [24]. Those chiral phenomena have possible applications on circular polarizer and waveplate [19,25]. We have found that the reflections of SRRs at 45° incidence are very different for RCP and LCP EM radiation, a phenomenon caused by extrinsic chirality.
From previous research [13], the forbidden even plasmon modes of nanowire were significantly excited under 45° incidence. The mutual orientation between the incident electric field and the induced plasmon current should be the crucial factor affecting exciting plasmon resonances. In this study, we investigated the plasmon resonance in SRR. The orientation of the applied electric field on the SRRs can be varied by changing the polarization state or the incident angle of the incident light. From reflection spectra simulated under various illumination conditions, we observed two special optical phenomena in the spectra recorded at oblique incidences. First, perpendicular polarized EM waves also excited odd plasmon modes, especially for large incident angles. Second, the reflections from RCP and LCP EM waves were different because of the extrinsic chirality of the optical system. These findings can be explained by considering the parallelism between the incident electric field and the induced plasmon current. Using a parallelism factor to quantify this parallelism trend, we found that the reflections of the plasmon resonances follow the same trend as the parallelism factors. The parallelism factor is a useful tool for estimating the strength of plasmon resonance in terms of the polarization state, incident angle, and geometry of the SRR.
2. Simulation structures and methods
The simulated structure was an array of SRRs [Fig. 1(a) ]. Each SRR was made from gold. The geometrical parameters were the period (p), base length (b), arm length (a), wire thickness (t), and wire width (w). The total length of the SRR (L) was equal to 2a + b. L was set to be much longer than t and w because our discussion was based on the wire geometry assumption. In all of the simulations, the wire thickness (t) and the wire width (w) were both 50 nm and the total length (L) was 1100 nm. The choice of the structure dimension was based on the capability of currently nano-fabrication technology and the resulting resonance range from visible to short-wavelength infrared regions. The refractive index of the environment was 1. The SRRs were freestanding structures so that the analysis would focus on the plasmon resonances of the gold structures while neglecting the effects of a substrate. At oblique incidence, the incident plane was perpendicular to the arms of the SRR. The incidence angles (θ) in the simulations were 0 and 45°. Four polarization states were simulated: perpendicular polarization, parallel polarization, RCP, and LCP. The perpendicular and parallel polarizations were defined as incident electric fields perpendicular and parallel to the incident plane, respectively. Any other polarization states can be linear combinations of the above states. A rigorous coupled-wave analysis (RCWA) algorithm [26,27] was the numerical method used in this study. RCWA has been used widely to analyze the optical diffraction of periodic structures. The advantage of using RCWA is that the reflected and transmitted fields from the periodic structures are expressed in terms of Rayleigh expansions, which can be directly employed to calculate the diffraction efficiencies. Its computing time is significantly faster and the required memory size is much smaller as compared with the finite-difference time domain (FDTD) method. The refractive indexes of gold were obtained through interpolation of tabulated values provided in the literature [28].
Fig. 1 (a) Simulated SRR array. (b) Unit cell of the SRR pattern. The blue () and red () arrows are the wave vectors of incidence and reflection; the green arrow () is the normal vector. The incident plane is perpendicular to the arms of the SRR.
3. Results and discussions
3.1. Plasmon resonances at normal incidence
Figure 2 displays the reflection spectra of SRRs at normal incidence. The base length (b) was 350 nm, the arm length (a) of the SRR was 375 nm. The period (d) was 600 nm. The multiple reflection peaks correspond to the plasmon resonance modes [10,11]. The incident EM wave drives the conducting electrons in SRRs and, thus, induces plasmon currents. The electrons reradiate forward and backward. The forward radiation tends to cancel the incident wave and the backward radiation appears as the reflection [29]. Therefore, the reflection peaks indicate the spectral positions of strong plasmon currents.
Fig. 2 Reflections of SRRs at normal incidence. The reflection peaks are labeled with the plasmon mode, with subscripts indicating the polarization states of the incident waves.
The peaks in Fig. 2 are labeled with the plasmon mode, with subscripts indicating the polarization states of the incident waves. We observe that the parallel polarized wave excites odd modes, namely mode 1 at 3110 nm and mode 3 at 1120 nm. The perpendicular polarized wave excites even modes, namely mode 2 at 1420 nm and mode 4 at 800 nm. The reflection spectra of the RCP and LCP EM waves overlap because of the symmetric configuration at normal incidence. The circularly polarized waves comprised equal amounts of parallel and perpendicular polarized waves. It is distinct that the circularly polarized waves could excite all of the plasmon modes from 1 to 4, with the strength of the reflection being exactly the average value of the parallel and perpendicular polarized waves.
Figure 3 presents near field plots of the lowest four resonance modes for the parallel and perpendicular polarizations. The plotted field is located at the center of the SRR plane, positioned 25 nm from the upper and lower surfaces of the SRR. We depict only the electric fields inside the SRRs; those outside the SRRs are neglected for clarity. Four plots in a row are provided to illustrate the behavior over time; the times are labeled on the top side, where T is the oscillating time period of the incident EM waves. The thick arrow at the center of each plot indicates the instantaneous polarization direction of the incident EM waves. According to Gauss’s law in Maxwell’s equations, the electric field diverges from the region if it is positively charged; it converges into the region if it is negatively charged [30]. We label the positively and negatively charged regions in the SRRs with positive and negative signs, respectively, correspondingly in the plots of time t 0 + T/4 and t 0 + 3T/4. Currents must exist to create the charge distributions. From previous research [13,15,16], we know that the multiple plasmon resonances can be interpreted as standing-wave resonances. The directions of the standing-wave currents are illustrated with dashed arrows at times t 0 and t 0 + T/2. Because the plasmon current is driven by the incident electric field, it would tend to align with the incident field. In mode 1, the current is excited by the incident field and flows from the left arm to the right arm at time t 0; the charge distribution is built at time t 0 + T/4; the current then flows in the opposite direction to build an opposite charge distribution; finally, the cycle continues. Considering mode 2, the current and charge density oscillate up and down; there is a standing-wave node in the middle of the SRR base. In the higher modes, we observe two standing-wave nodes in mode 3, except at the ends of the arms, and three nodes in mode 4. In summary, if the mode number is m, the number of standing-wave nodes is m – 1.
Fig. 3 Near field plots of the lowest four resonance modes for parallel and perpendicular polarizations. The plotted field is located at the center of the SRR plane, positioned 25 nm from the upper and lower surfaces of the SRR. Four plots in a row provide a time succession; the times are labeled on the top side, where T is the oscillating time period of the incident EM waves. The thick arrow at the center of each plot indicates the instantaneous polarization direction of incident EM waves. The positively and negatively charged regions in the SRRs are labeled with positive and negative signs, respectively, correspondingly at times t 0 + T/4 and t 0 + T3/4. The directions of the standing-wave currents are illustrated with dashed arrows at times t 0 and t 0 + T/2.
3.2. Plasmon resonances at 45° incidence
Figure 4 displays the reflection spectra of the SRRs at 45° incidence. The geometrical parameters are the same as those described in the previous section. Compared with the results obtained at normal incidence, the excitation wavelengths of the plasmon modes underwent only small spectral shifts because the excitation wavelengths were determined mainly by the nanowire geometry. The reflection of the parallel polarization did not change much, except for the appearance of a small peak (4∥). The spectrum of perpendicular polarization features two new odd peaks (1⊥ and 3⊥) in addition to the peaks of the even modes, suggesting that the perpendicular polarized waves excited all of the four plasmon modes. The symmetry of this optical system was destroyed by oblique incidence; thus, it is an extrinsically chiral system, as discussed previously [19,20]. We observe that the reflections of the RCP and LCP radiation waves at 45° incidences were separated because of chirality.
Next, we investigated the reason why the perpendicular polarized wave at 45° incidence could excite odd modes but the perpendicular polarized wave at 0° incidence could not. Figure 5 presents the near field plots for perpendicular polarized incidence at a wavelength of 1090 nm, the spectral position of the signal for 3∥. The plotted field is located at the center of the SRR plane. In Fig. 5(a), the angle of incidence is 0°. The near field plots have mirror symmetry because the SRR and the incident electric field were both symmetric. At 0° incidence, the incident EM wave was uniform across the upper surface of the SRR plane; its instantaneous polarization direction of the incident EM waves is indicated by the thick arrow. In Fig. 5(b), the mirror symmetry is destroyed by oblique incidence. Comparing the charge distribution with the near field plot of 3∥ in Fig. 3 confirms that the plasmon resonance was that of mode 3. At 45° incidence, the electric field of the incident wave on the SRR plane varied with respect to the horizontal position. The thick arrows on the arms of the SRR indicate the electric field of incidence at those locations. The second and fourth plots in Fig. 5(b) reveal that the incident electric fields were zero in the middle of the SRR, but because the incident electric fields on the arms drove the plasmon currents, the excitation occurred in the arms of the SRR. As a result of the oblique incidence, a phase difference of the incident electric field was introduced on the two arms of the SRR and then the odd plasmon modes were excited by the perpendicular polarized waves. Because the excitation wavelength of mode 1 was longer than that of mode 3, the phase difference in mode 1 was smaller and the excitation weaker. We confirmed this phenomenon analytically using the parallelism factor proposed in the next section.
Fig. 5 Near field plots for perpendicular polarized incidence at a wavelength of 1090 nm. The plotted field is located at the center plane of the SRR. Angle of incidence: (a) 0°; (b) 45°.
3.3. Parallelism factor
There are two conditions for exciting a strong plasmon resonance in an SRR. First, the frequency must be close to the resonant frequency of the plasmon mode. It is mostly related to the total length (L). Second, the induced plasmon current must be aligned with the polarization direction. Here, we propose the use of a parallelism factor (P-factor) to estimate the ability of the incident EM wave to excite a plasmon current.
The electric fields of EM waves are represented as
where k is the wave vector and ω is the angular frequency. Figure 6 displays the configuration of the SRR in Cartesian coordinates; the incident angle is θ and the x–y plane is parallel to the incident wavefront. The induced plasmon currents in SRRs exist in the form of standing waves. Therefore, we can simplify the currents to be represented in the form of sinusoidal waves:where m is the plasmon mode; the number of the current node is equal to m – 1; and ϕ is the phase difference between the induced plasmon current and the applied electric field. The current would automatically adjust its phase to tend to align with the incident electric field. The descriptor represents the path of integration along the structure. The total length of the SRR is L. Integration of the dot product of the electric field and the current is performed as follows:where pi is a function of ϕ and is normalized to 1; i represents the polarization state. We define the P-factor as the maximum value of the integrationFig. 6 Configuration of an SRR in Cartesian coordinates. The path of integration is represented by the blue dotted arrows along the structure. The incident wavefront is parallel to the x–y plane. The incident angle is θ.
The part in Eq. (3) is the power dissipation per length [31]. Equation (3) can be regarded as the normalized time-averaged power dissipation of the entire SRR. Hence, the P-factor is actually the power transferred from the electric field to the plasmon current, but it is normalized to unity.
Figure 7 plots the calculated P-factors with respect to incident angles varying from 0 to 45°. We substitute the excitation frequencies of the plasmon modes at normal incidence into the calculations because the resonant frequencies of the modes are quite consistent under the various incident conditions. The odd modes in Figs. 7(a) and 7(c) represent the cases for modes 1 and 3, respectively. At normal incidence, the P-factor of parallel polarization is the largest and that of perpendicular polarization is zero, indicating that the parallel polarized wave excites odd modes, but the perpendicular polarized wave cannot. Checking with the near field plots in Fig. 3, we find that the odd modes arise for parallel polarization because the induced current in the base is aligned with the direction of the electric field. The electric field on the arms is perpendicular to the currents and has no influence on them. In contrast, for perpendicular polarization, the electric field on the base is perpendicular to the current and the contributions of the electric field on the currents of each arm cancel each other. Therefore, the odd modes disappear for perpendicular polarization. A similar consideration can be used to check the behavior of even modes for both polarizations. At oblique incidence, the mirror symmetry of SRR geometry is destroyed with respect to the incident beam. All modes appear for both linear polarizations. There are distinct odd modes for perpendicular polarization because the electric fields on the two arms point in opposite directions. As we discussed in the previous section, the contributions of the electric field to the currents of each arm cannot cancel under such conditions. The asymmetric configuration separates P R from P L and also spectrums of RCP and LCP. Comparing the P-factors in Fig. 7 with the reflections in Figs. 2 and 4, the reflections of the plasmon resonance peaks correlate positively with the P-factors; i.e., the strength of the plasmon resonances is proportional to the power transferred from the incident electric fields to the plasmon currents.
The concept of P-factor can be explained by taking the following discussion into consideration. Figure 8 presents schematic representations of the incident electric fields of RCP and LCP waves and the induced plasmon currents of mode 1. The wavefront is parallel to the x–y plane; the dotted red arrows are the plasmon currents; the solid arrows are the incident electric fields. At oblique incidence, the wavefront initially propagates to the left arm of the SRR, passes the base portion of the SRR, and then reaches the right arm. The phases of the EM wave are different in the arms and in the base. In Fig. 8(a), the incident electric field of the RCP wave in the middle of the SRR base is aligned in the + y direction; its projection to the SRR base is parallel to the direction of the plasmon current in the base. The projection of the electric field on the left arm is in + x direction; the projection on the right arm is in the –x direction. Because these projections are aligned anti-parallel to the direction of the currents, they suppress the currents in the arms. When we integrate Eq. (3), we find that the integration along the arms is negative and cancels out the integration along the base. In Fig. 8(b), the electric field of the LCP wave on the arms enhances the currents. Because the LCP wave is parallel to the current of mode 1, P L is higher than P R, and the mode 1 resonance under LCP wave excitation is stronger than that of RCP wave excitation at oblique incidence.
Fig. 8 Schematic representations of the incident electric fields and the induced plasmon currents of mode 1 at oblique incidences. Polarization state of the incident EM wave: (a) RCP (dotted red arrows are the currents; solid arrows are the incident electric fields) and (b) LCP.
Figure 7(c) displays the P-factors of mode 3. The value of P L is zero at an incident angle of 27°. The phase ϕ max changes by π at that point; i.e., the incident wave reverses the induced current direction when the incident angle is larger than 27°. This preliminary result obtained from Eq. (4) must be checked in an RCWA simulation. Figure 9(a) presents the near field plots in the mode 3 resonance under LCP wave incident angles of 0 and 45°; the blue thick arrows represent the directions of the incident electric fields. The incident electric field was uniform over the entire SRR plane in the case of normal incidence. At 45° incidence, the incident field varied with respect to the horizontal position. The blue arrows reveal the direction of the incident field in the middle of the SRR; the dashed arrows reveal that the currents under 0 and 45° incidences were in opposite directions. The near field plots prove that a phase change occurred for the plasmon current. Figures 9(b) and 9(c) are schematic representations of the incident electric fields and the induced plasmon currents of mode 3 of the LCP waves; they are 3D sketches of the first column in Fig. 9(a). The incident angles were 0 and 45°, respectively; the wavefront was aligned parallels to the x–y plane. In Fig. 9(b), in the case of normal incidence, the electric field drove the current in the base. The electric fields on the arms did not enhance or suppress the current. In Fig. 9(c), in the case of 45° incidence, the excitation on the arms was sufficiently strong to reverse the direction of the current in the base, making it oppose the electric field in the base and tend to align parallel to the electric fields on the arms. There must be a phase change point between 0 and 45°, confirming the prediction made using the P-factor. As the incident angle increases from zero, the electric field on the arms increases its component in the x direction. When the incident angle is sufficiently large, the electric fields on the arms are sufficiently strong to reverse the plasmon current.
Fig. 9 (a) Near field plots for the mode 3 resonance under LCP wave incident angles of 0 and 45°. (b, c) Schematic representations of the incident electric fields of LCP waves and the induced plasmon currents of mode 3 at (b) normal and (c) 45° incidence.
3.4. P-Factor as a function of geometry
The plasmon response is a function of nanowire geometry. Figures 10(a) and 10(b) display the reflection spectra of five kinds of SRRs at normal incidence. The total length (L) was maintained at 1100 nm while the base length (b) was varied from 150 to 550 nm (with 100 nm spacing). The geometry of the single SRR having a base length 350 nm was the same as the geometry of the SRR in the previous sections. The period of the structure was extended to 800 nm to maintain enough space to elongate the base. The period does not affect the behavior of the LSPRs in SRRs, but the reflection is lowered because of a smaller filling factor, as has been mentioned previously [32]. Odd modes were excited in parallel polarized light [Fig. 10(a)] and even modes were excited in perpendicular polarized light [Fig. 10(b)]. The resonance frequencies of the standing-wave currents were mostly dependant on the total length of the SRR. Variation of the base length resulted in small perturbations of the spectral positions of the plasmon modes. Figure 10(c) provides the calculated P-factors of each mode and the peak reflectance values from the spectra. As mentioned in Section 3.3, the excitations of mode 1 under parallel polarization were contributed mainly by the electric fields acting on the currents in the SRR base. Therefore, the strength of the resonance increased for a longer SRR base. The excitations of mode 2 under perpendicular polarization were contributed mainly by the electric fields acting on the currents of each SRR arm; the strength of resonance decreased for longer SRR bases, in contrast to the case of parallel polarization. In mode 3, the P-factor was maximized at a value of b of L/3 (367 nm); thus, the plasmon current nodes should occur at the turning corners of the SRR. This prediction agrees with the reflectance peak values of mode 3, which featured a maximum reflectance corresponding to a value of b of 350 nm. The trend in the simulated spectra agrees with that of the P-factor. For mode 4, the two peaks having values of b of less than 350 nm are quite small in the reflectance spectra and hard to recognize. Nevertheless, the rest three data points, with b greater than or equal to 350 nm, exhibit an increasing trend—and so does the P-factor. In summary, this result reveals that the trend of the plasmon response is consistent with the prediction of the P-factor. With a predicted P-factor, we can easily design a geometry for the SRR to enhance or suppress a particular plasmon mode, and then further confirm the specific results through RCWA analysis.
Fig. 10 (a, b) Reflection spectra under (a) parallel and (b) perpendicular polarization. (c) P-Factor and peak values of the four plasmon modes. All data were simulated at normal incidence
4. Conclusions
We have investigated the plasmon resonances of SRRs using reflectance spectra and studied the near field plots at normal and oblique incidences through RCWA analyses. At normal incidence, the reflection spectra revealed the presence of four lowest plasmon modes in the wavelength range from 500 to 4000 nm. Parallel and perpendicular polarized EM waves excite odd and even plasmon modes, respectively, consistent with the results of previous studies. We confirmed the standing-wave behavior of multiple resonances from near field plots of the plasmon modes. Because RCP and LCP EM waves comprise equal amounts of parallel and perpendicular polarized waves, they can excite all four of the plasmon modes. At 45° incidence, the reflection spectrum under perpendicular polarization revealed two distinct odd mode peaks in addition to the original even mode peaks. As a result of the destruction of the mirror symmetry of the SRR geometry with respect to the incident beam, the phases of the incident electric field are different on the two arms of the SRR; odd plasmon modes are then excited by the perpendicular polarized waves. The reflections of the RCP and LCP waves were separated at oblique incidences because of the extrinsic chirality of the optical system.
From the simulated near field plots, we found that parallelism of the incident electric field and the induced plasmon current is the key point affecting excitation. We propose the use of a P-factor to characterize the ability of the incident fields to excite the plasmon currents. The analytical form of the P-factor can be regarded as a measure of the power transferred from the incident wave to the resonance current. Our results revealed that the reflection of the plasmon modes was in positive proportion to the P-factor. The mechanism of the field and current parallelism can be used to explain the resonance behavior of SRRs in terms of the polarization state, incident angle, and geometry of the SRR. The P-factor can be an effective index to estimate the strength of excited plasmon resonances. The analysis of P-factors should be effective not only for SRRs but also for the other wire-like structures.
Acknowledgment
This study was supported by the National Science Council of Taiwan grants NSC 98-2221-E-006-018- and NSC 98-2218-E-009-001.
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