Light circulation and weaving in periodically patterned structures

The circulation and weaving of light in one-dimensionally periodic structures as a result of the excitation of phase resonances are described. It is shown that phase resonances occur for $s$-polarized incident light with many similarities and some important differences compared to well-known $p$-polarized phase resonances. It is shown that surface plasmons are not necessary for either enhanced transmission or phase resonances to occur in one-dimensionally periodic slit arrays. Characteristics of phase resonances of both polarizations, including their production of a sharp minimum on the shoulder of a broader transmission peak are described by analyzing the power flow associated with these modes. Electromagnetic field intensities and phase difference in the coupled cavity modes that comprise phase resonances are discussed.

Figure 1

Three groove structures are studied in this work. All the gratings have Au wires with a dielectric constant obtained from Ref. 17, $h=1\mu \mathrm{m}$ and $\Lambda =3.5\mu \mathrm{m}$. Grating 1 is a single-groove-per-period grating with all the grooves being identical, with ${\epsilon }_1={\epsilon }_2=23$ and $c_1=c_2=0.745\mu \mathrm{m}$; in this case, the period is reduced by half from what is indicated in the figure. Grating 2 is two-groove-per-period structure with ${\epsilon }_1={\epsilon }_2=23$ and $c_1\ne c_2$, with $c_1=0.755\mu \mathrm{m}$ and $c_2=0.735\mu \mathrm{m}$. Grating 3 is also a two-groove-per-period structure with ${\epsilon }_1\ne {\epsilon }_2$ and $c_1\ne c_2$ with ${\epsilon }_1=25$, ${\epsilon }_2=21$, $c_1=0.755\mu \mathrm{m}$, and $c_2=0.735\mu \mathrm{m}$. Grating 1 will exhibit dual polarization transmission, grating 2 (3) will have $s$-polarization ($p$-polarization) $\pi$ resonances.

Figure 2

The transmissivitty for (a) $s$-polarized incident light and (b) $p$-polarized incident light for grating 1 (grating parameters are given in Fig. 1). $K$ is $2\pi ∕\Lambda$, where $\Lambda =3.5\mu \mathrm{m}$. The period $\Lambda$ is deliberately chosen as twice as large as in the one-groove-per-period grating in anticipation of perturbing the width of every other groove to produce a two-groove-per-period grating with twice the period later in this work.

Figure 3

(a) The transmissivity for $s$-polarized incident light on grating 2. (b) The orientation of $\mathbf{H}$ for the $s$-polarized $0.248\mathrm{eV}$; $k_x=0$ $\pi$ resonance in grating 2 shows the $\pi$ phase difference of $\mathbf{H}$ in neighboring grooves. (c) The transmissivity for $p$-polarized incident light on grating 3. The insets show the $\pi$ resonance, showing that the broad transmission bands now have transmission minima on the shoulder of the bands.

Figure 4

For $s$-polarized normal incident light on grating 2, we have (a) the Poynting vector profile on the low energy side of the $\pi$ resonance (i.e., $\hslash \omega =0.2479\mathrm{eV}$). The power flowing down through the left groove is decreasing and not totally offsetting the circulating power flowing down through the right groove and up through the left groove. (b) For an energy slightly larger than the $\pi$ resonance (i.e., $\hslash \omega =0.2484\mathrm{eV}$), the opposite situation occurs where the power flowing down through the right groove is decreasing and not totally offsetting the circulating power flowing down through the left groove and up through the right groove.

Figure 5

The reflectivity and transmissivity for $p$-polarized normal incident light on grating 3. The $p$-polarized $\pi$ resonance with an energy of $0.239\mathrm{eV}$ exhibits strong light circulation only on the low energy side of the peak, as shown in Fig. 6.

Figure 6

The Poynting vector profiles for $p$-polarized normal incident light with energies (a) $0.236\mathrm{eV}$ that is slightly below the $\pi$ resonance (point a in Fig. 5), (b) $0.239\mathrm{eV}$ that is the peak in reflectance of the $\pi$ resonance (point b in Fig. 5), and (c) $0.246\mathrm{eV}$ that is slightly above the $\pi$ resonance (point c in Fig. 5). At the peak of reflectance of the $\pi$ resonance [see (b)], the counterpropagating light circulations largely offset each other in terms of power flow in the grooves. The small amount of power flowing into the two grooves flows into the walls of the groove and is absorbed by the lossy gold wires. For energies slightly less than the peak [see (a)], the high-$Q$ CM in the left-hand groove dominates but the higher energy low-$Q$ CM of the right-hand groove can still present to the light transmitted through the left-hand groove with a strong transmission channel back through to grating, resulting in light circulation. For energies slightly less than the peak [see (c)], the low-$Q$ CM in the right-hand groove dominates but the CM of the left-hand groove generally has too high of a $Q$ to present to the light transmitted through the right-hand groove a strong transmission channel. Hence, little light circulation is observed on this high energy side of the $\pi$ resonance reflectance peak.

Figure 7

For $s$-polarized, $0.24815\mathrm{eV}$ light, incident on grating 2 at a 15° angle of incidence, the $\pi$ resonance causes light to weave up and down through the grooves with a net power flow in the $\widehat{x}$ direction.