# Chiral Quantities¶

The *optical chirality* is a measure of the local density of chirality of the electromagnetic field [1].
It satisfies the continuity equation in isotropic
homogeneous media. For monochromatic, i.e. time-harmonic, fields the optical chirality is proportional
to the *helicity* of light and the optical chirality flux is proportional to the *spin angular momentum* [2].

In its generalized time-harmonic form, the optical chirality density and the optical chirality flux density satisfy a continuity equation valid in arbitrary, i.e. bi-anisotropic, space [3] :

where is the ElectricChiralityDensity, is the MagneticChiralityDensity and is the ElectromagneticChiralityFluxDensity.

**ElectricChiralityDensity**

The electric part of the optical chirality density in is accessible as

AnisotropicElectricChiralityDensityand defined asIn

isotropicmedia, this reduces towhich is accessible as

ElectricChiralityDensity. This quantity can be computed numerically more accurate and faster. However, it only satisfies the continuity equation in isotropic media.

**MagneticChiralityDensity**

The magnetic part of the optical chirality density in is accessible as

AnisotropicMagneticChiralityDensityand defined asIn

isotropicmedia, this reduces towhich is accessible as

MagneticChiralityDensity. This quantity can be computed numerically more accurate and faster. However, it only satisfies the continuity equation 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 used by many authors [4].

**ElectromagneticChiralityFluxDensity**

The optical chirality flux density in is accessible as

ElectromagneticChiralityFluxDensityand defined asDue to Maxwell’s equations, this flux density can be rewritten as

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 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 is due to scattering and conversion which takes place in volumes or at interfaces [3].

The conservation of optical chirality reads as

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 -times the imaginary part:*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 in 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 OutputQuantityScatteredElectromagneticChiralityFlux. The physically relevant part in 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 OutputQuantityExtinctionElectromagneticChiralityFlux. The physically relevant part in 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 and the chirality conversion obtained from volumes and interfaces . 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. |