Chiral Quantities¶

The optical chirality $\chi$ is a measure of the local density of chirality of the electromagnetic field [1]. It satisfies the continuity equation $\partial_t \chi + \boldsymbol{\nabla} \cdot \boldsymbol{\Sigma} = 0$ in isotropic homogeneous media. For monochromatic, i.e. time-harmonic, fields the optical chirality $\chi$ is proportional to the helicity of light and the optical chirality flux $\boldsymbol{\Sigma}$ is proportional to the spin angular momentum [2].

In its generalized time-harmonic form, the optical chirality density $\mathfrak{X} = \mathfrak{X}_\text{e} + \mathfrak{X}_\text{m}$ and the optical chirality flux density $\boldsymbol{\mathfrak{S}}$ satisfy a continuity equation valid in arbitrary, i.e. bi-anisotropic, space [3] :

$\begin{eqnarray} 2 i \omega (\mathfrak{X}_\text{e} - \mathfrak{X}_\text{m}) + \boldsymbol{\nabla} \cdot \boldsymbol{\mathfrak{S}} = -\frac{1}{4} \left[ \VField{J}^* \cdot \left( \boldsymbol{\nabla} \times \boldsymbol{\VField{E}}\right) + \boldsymbol{\VField{E}} \cdot \left( \boldsymbol{\nabla} \times \VField{J}^*\right) \right], \label{eq:chCont} \end{eqnarray}$

where $\mathfrak{X}_\text{e}$ is the ElectricChiralityDensity, $\mathfrak{X}_\text{m}$ is the MagneticChiralityDensity and $\boldsymbol{\mathfrak{S}}$ is the ElectromagneticChiralityFluxDensity.

ElectricChiralityDensity

The electric part of the optical chirality density $\mathfrak{X}_\text{e}$ is accessible as ElectricChiralityDensity within JCMsuite and defined as

$\begin{eqnarray*} \mathfrak{X}_\text{e} = \frac{1}{8} \left[ \boldsymbol{\VField{D}}^* \cdot \left( \boldsymbol{\nabla} \times \boldsymbol{\VField{E}}\right) + \boldsymbol{\VField{E}} \cdot \left( \boldsymbol{\nabla} \times \boldsymbol{\VField{D}}^*\right) \right]. \end{eqnarray*}$

In isotropic media, this reduces to

$\begin{eqnarray*} \widetilde{\mathfrak{X}_\text{e}} = \frac{1}{8} i \omega \left[ \boldsymbol{\VField{D}}^*\cdot \boldsymbol{\VField{B}} - \left(\varepsilon \boldsymbol{\VField{B}}\right)^* \cdot \boldsymbol{\VField{E}} \right] \end{eqnarray*}$

which is accessible as ElectricChiralityDensity. This quantity $\widetilde{\mathfrak{X}_\text{e}}$ can be computed numerically more accurate and faster. However, it only satisfies the continuity equation $(1)$ in isotropic media.

MagneticChiralityDensity

The magnetic part of the optical chirality density in $(1)$ is accessible as AnisotropicMagneticChiralityDensity and defined as

$\begin{eqnarray*} \mathfrak{X}_\text{m} = \frac{1}{8} \left[ \boldsymbol{\VField{H}}^* \cdot \left( \boldsymbol{\nabla} \times \boldsymbol{\VField{B}}\right) + \boldsymbol{\VField{B}} \cdot \left( \boldsymbol{\nabla} \times \boldsymbol{\VField{H}}^*\right) \right]. \end{eqnarray*}$

In isotropic media, this reduces to

$\begin{eqnarray*} \widetilde{\mathfrak{X}_\text{m}} = \frac{1}{8} i \omega \left[ \boldsymbol{\VField{D}}^* \cdot \boldsymbol{\VField{B}} - \boldsymbol{\VField{H}}^* \left(\mu \boldsymbol{\VField{D}}\right) \right], \end{eqnarray*}$

which is accessible as MagneticChiralityDensity. This quantity $\widetilde{\mathfrak{X}_\text{m}}$ can be computed numerically more accurate and faster. However, it only satisfies the continuity equation $(1)$ in isotropic media.

For homogeneous isotropic lossless media, the real part of the sum of the ElectricChiralityDensity and the MagneticChiralityDensity is the time-harmonic optical chirality density $\mathcal{C} = -\varepsilon_0\omega/2 \operatorname{Im}\left(\boldsymbol{\VField{E}}^* \cdot \boldsymbol{\VField{B}}\right)$ used by many authors [4].

ElectromagneticChiralityFluxDensity

The optical chirality flux density in $(1)$ is accessible as ElectromagneticChiralityFluxDensity and defined as

$\begin{eqnarray*} \boldsymbol{\mathfrak{S}} = \frac{1}{4} \left[ \boldsymbol{\VField{E}} \times \left( \boldsymbol{\nabla} \times \boldsymbol{\VField{H}}^*\right) - \boldsymbol{\VField{H}}^* \times \left( \boldsymbol{\nabla} \times \boldsymbol{\VField{E}}\right) \right]. \end{eqnarray*}$

Due to Maxwell’s equations, this flux density can be rewritten as

$\begin{eqnarray*} \boldsymbol{\mathfrak{S}} = \frac{1}{4} i \omega \left[\boldsymbol{\VField{E}} \times \boldsymbol{\VField{D}}^* - \boldsymbol{\VField{H}}^* \times \boldsymbol{\VField{B}} \right]. \end{eqnarray*}$

Its real part is proportional to the spin angular momentum, whereas its imaginary part has no physical significance.

Integrated Chiral Quantities¶

In order to obtain measurable quantities, the continuity equation for chiral quantities $(1)$ can be integrated. This yields a conservation law analogous to Poynting’s theorem which states the conservation of electromagnetic energy. The extinction of energy occurs due to scattering and absorption. The extinction of optical chirality $\boldsymbol{\mathfrak{S}}_\text{ext}^{(\partial\Omega)}$ is due to scattering $\boldsymbol{\mathfrak{S}}_\text{sca}^{(\partial\Omega)}$ and conversion $\mathfrak{X}_\text{conv} =\mathfrak{X}_\text{conv}^{(\Theta)} + \mathfrak{S}_\text{conv}^{(\partial\Theta)}$ which takes place in volumes $\mathfrak{X}_\text{conv}^{(\Theta)}$ or at interfaces $\mathfrak{S}_\text{conv}^{(\partial\Theta)}$ [3].

The conservation of optical chirality reads as

$\setcounter{equation}{1} \begin{eqnarray} \boldsymbol{\mathfrak{S}}_\text{ext}^{(\partial\Omega)} = \boldsymbol{\mathfrak{S}}_\text{sca}^{(\partial\Omega)} + \mathfrak{X}_\text{conv}^{(\Theta)} + \mathfrak{S}_\text{conv}^{(\partial\Theta)}. \label{eq:chCons} \end{eqnarray}$

Conversion

Volume Conversion: The conversion of optical chirality in volumes is accessible by performing the PostProcess DensityIntegration with the OutputQuantity ElectricChirality and MagneticChirality. Analogous to the case of energy absorption, the physically relevant part is $\omega$ -times the imaginary part:

$\begin{eqnarray*} \mathfrak{X}_\text{conv}^{(\Theta)} = -2 \omega \operatorname{Im}\left(\mathfrak{X}_\text{e}^{(\Theta)} - \mathfrak{X}^{(\Theta)}_\text{m}\right). \end{eqnarray*}$
ElectromagneticChiralityConversionFlux: In contrast to energy, optical chirality is not conserved in media with spatial dependent material parameters, especially on interfaces of piecewise-constant materials. The conversion of optical chirality occurring at interfaces is accessible by performing the PostProcess FluxIntegration with the OutputQuantity ElectromagneticChiralityConversionFlux. The physically relevant part $\mathfrak{S}_\text{conv}^{(\partial\Theta)}$ in $(2)$ is the real part of this quantity.

Note

The InterfaceType has to be set to DomainInterfaces.

ScatteredElectromagneticChiralityFlux

The optical chirality flux of the scattered field is accessible by performing the PostProcess FluxIntegration with the OutputQuantity ScatteredElectromagneticChiralityFlux. The physically relevant part $\boldsymbol{\mathfrak{S}}_\text{sca}^{(\partial\Omega)}$ in $(2)$ is the real part of this quantity.

Note

The InterfaceType is automatically set to ExteriorDomain.

ExtinctionElectromagneticChiralityFlux

The extinction of optical chirality is accessible by performing the PostProcess FluxIntegration with the OutputQuantity ExtinctionElectromagneticChiralityFlux. The physically relevant part $\boldsymbol{\mathfrak{S}}_\text{ext}^{(\partial\Omega)}$ in $(2)$ is the real part of this quantity.

Note

The InterfaceType is automatically set to ExteriorDomain.

Up to numerical inaccuracies, this quantity should equal the sum of the scattered chirality $\boldsymbol{\mathfrak{S}}_\text{sca}^{(\partial\Omega)}$ and the chirality conversion obtained from volumes $\mathfrak{X}_\text{conv}^{(\Theta)}$ and interfaces $\mathfrak{S}_\text{conv}^{(\partial\Theta)}$ . If this is not the case, increase the Precision and/or FiniteElementDegree and/or MaximumSideLength of your simulation.

Bibliography

[1]	Yiqiao Tang and Adam E. Cohen. Optical chirality and its interaction with matter. Physical review letters, 104(16):163901, 2010.

[2]	Konstantin Y. Bliokh and Franco Nori. Characterizing optical chirality. Physical Review A, 83(2):021803, 2011.

[3]	(1, 2) Philipp Gutsche, Lisa V. Poulikakos, Martin Hammerschmidt, Sven Burger, and Frank Schmidt. Time-harmonic optical chirality in inhomogeneous space. In SPIE OPTO, Vol.9756, pages 97560X. International Society for Optics and Photonics, 2016.

[4]	Martin Schäferling, Daniel Dregely, Mario Hentschel, and Harald Giessen. Tailoring enhanced optical chirality: design principles for chiral plasmonic nanostructures. Physical Review X, 2(3):031010, 2012.