On this page, we calculate the circular dichroism (CD) of a gammadion shaped structure, then optimize the structure dimensions to maximize the CD at 1.1 um.

We consider a gammadion shaped periodic structure shown in Fig. 1. In this structure, an aluminum layer is sandwiched by silver layers and the excitation of surface plasmon leads to the enhancement of the circular dichroism.

(a) Geometry of one unit cell | (b) Top view | (c) Side view |

Figure 1: Gammadion shaped planar chiral material

## Circular dichroism for four-fold symmetrical structure

The circular dichroism CD is defined by

$$ CD=\left|T_{R}-T_{L}\right| $$ (1)

where TR and TL is a transmittance when the right- and the left- circularly polarized plane wave is incident on the device, respectively. To get the transmittance for circularly polarized incidence, we have two alternatives as follows.

1. Use two plane wave sources to generate circularly polarized illumination, as described in the circular polarization page. In this case, two FDTD simulations will be required to get the CD; One for right-circular polarization and the other for left-circular polarization. This approach is not used in the associated example simulation file.

2. Use one plane wave source (as in the example simulation file *gammadion_dichroism.fsp*. By taking advantage of the four-fold rotational symmetry of the structure, the transmittance can be obtained from a single simulation, as explained below:

The field distributions F (E or H) for circular illumination can be obtained from a single linearly polarized simulation by

$$ \begin{array}{l}{F_{R}(x, y, z)=F_{x}(x, y, z)+F_{y}(x, y, z) e^{j \pi / 2}} \\ {F_{L}(x, y, z)=F_{x}(x, y, z)+F_{y}(x, y, z) e^{-j \pi / 2}}\end{array} $$ (2)

where F_{R} (F_{L}) is the field distribution for right- (left-) circularly polarized incident wave, and Fx (Fy) is the field distribution for a x (y) linearly polarized plane wave. If we assume four-fold rotational symmetry of the structure, the field distribution for y-polarized plane wave is incident on the structure, Fy, is given by that for x-polarized incident plane wave, Fx, as

$$ \begin{array}{l}{F_{x_{-y}}(x, y, z)=-F_{y_{y} x}\left(y_{z}-x_{z}\right)} \\ {F_{y_{-} y}(x, y, z)=F_{x_{-} x}\left(y_{z}-x_{z}\right)} \\ {F_{z_{-} y}(x, y, z)=F_{z_{-} x}\left(y_{z}-x_{z}\right)}\end{array} $$ (3)

where FU_V (u=x, y, z; v=x, y) is the u-component of the field distribution for v-polarized incident wave.

Once we get the field distribution FU (U=R or L) for circularly polarized plane wave using the relation Eqs. (2) and (3) from one FDTD simulation (simulation for x- or y- polarized incident wave), the power traveling down in the substrate over a unit cell is given by

$$ P_{U}=\int_{S_{-} \ldots \infty} \frac{1}{2} \operatorname{Re}\left(E_{U} \times H_{U}^{*}\right) \mathrm{d} \mathrm{S} $$ (4)

If we normalize the power P by incident power using script function "sourcepower", we obtain transmittance T as

$$ T_{v}=P_{v} / \text { sourcepower } $$ (5)

The simulation file *gammadion_dichrosim.fsp* uses a single x-polarized source. The field distribution on a plane under the gammadion structure (in the substrate) is recorded in a power monitor named "T", within the analysis group named "CD". The script in the "Analysis" tab = "Script" tab of this analysis group calculates the transmittance of circular polarizations following the way mentioned above (method 2). From the CD analysis object, you can plot the CD vs wavelength by use of the "visualizer". In the figure below, we can see a peak in the CD around 1.1 um.

### Related publications

Do-Hoon Kwon et al., "Optical planar chiral metamaterial designs for strong circular dichroism and polarization rotation," Opt. Express 16, 11802 (2008).