Degree of chirality of electromagnetic fields and maximally chiral light

Ofer Neufeld, Matan Even Tzur, and Oren Cohen

Phys. Rev. A 101, 053831 – Published 14 May 2020

Abstract

Recently, synthetic chiral light was introduced and theoretically shown to be highly effective for chiral light-matter interactions [D. Ayuso, O. Neufeld, A. F. Ordonez, P. Decleva, G. Lerner, O. Cohen, M. Ivanov, and O. Smirnova, Nat. Photonics 13, 866 (2019)]. This electromagnetic (EM) field possesses a new intrinsic property denoted “local chirality.” Contrary to standard circularly polarized light, it is chiral within the electric dipole approximation, even if the field is spatially uniform. The degree of chirality (DOC) of such light was defined, but has not yet been explored. Here we provide a comprehensive study of the DOC of locally chiral EM fields. We present numerical schemes that are capable of calculating this quantity for arbitrary fields at a low computational cost (we provide open access codes). We utilize these approaches to explore the functional dependence of the degree of chirality of bichromatic laser beams on their various degrees of freedom such as beam intensities, polarization states, and so on. We perform an extensive numerical search for the bichromatic EM field that possesses the maximal possible value of local chirality, and find a field that exhibits a 69.6% degree of chirality (out of the theoretical limit of 200%). This geometry sets the current record for a maximally chiral EM field. The numerical approach and results presented here will be useful for future investigations of chiral light-matter interactions.

Received 9 March 2020
Accepted 17 April 2020

DOI:https://doi.org/10.1103/PhysRevA.101.053831

Physics Subject Headings (PhySH)

Chirality Electromagnetism Laser spectroscopy Light-matter interaction

Atomic, Molecular & Optical

Authors & Affiliations

Ofer Neufeld ^*, Matan Even Tzur , and Oren Cohen

Physics Department and Solid State Institute, Technion–Israel Institute of Technology, Haifa 32000, Israel

^*Corresponding author: ofern@tx.technion.ac.il

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Vol. 101, Iss. 5 — May 2020

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Figure 1
Illustration of locally chiral EM fields. (a, b) A locally chiral bichromatic electric field and its mirror twin. The scheme of how such fields are generated by two noncollinear beams of carrier frequencies ω and 2ω is illustrated. The Lissajou plot describes the electric vector field's time-dependent polarization in a given point in space. (c) Illustration of maximal overlap between the two mirror twins in (a, b) that is found for some particular set of space-time angles, ${\vec{μ}}_{min}$ , that minimize the integral in Eq. (1); i.e., the Lissajou plot from (b) is rotated onto the one in (a) and translated in time by a temporal shift. In this particular case the DOC is 56%, and obtained for laser parameters: $ɛ_{1} = 0.3$ , $ɛ_{2} = 0.4$ , $A = 1$ , $α = 20^{\circ}$ , $β_{1} = β_{2} = 0$ , $η = π / 3$ . In all Lissajou plots the arrow indicates the direction of time, and is positioned at $t = 0$ for the electric field.
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Figure 2
Physical dependence of the DOC of bichromatic beams on the laser parameters, α and $A$ . (a) DOC vs the opening angle, α, for several exemplary cases (the legend numbers refer to the following random choice of laser parameters: ${\vec{κ}}_{1} = {- 0.25, - 0.26, 0.11, 0.23, α, 1.33, 0.93}$ , ${\vec{κ}}_{2} = {0.37, 0.41, 0.80, 0.93, α, 1.20, 1.50}$ , ${\vec{κ}}_{3} = {- 0.65, - 0.46, 0.87, 1.05, α, 1.27, 1.87}$ , ${\vec{κ}}_{4} = {- 0.14, - 0.78, 0.88, 0.59, α, 0.99, 1.10}$ , ${\vec{κ}}_{5} = {- 0.27, 0.95, 0.92, 0.83, α, 1.41, 1.85}$ , ${\vec{κ}}_{6} = {- 0.04, 0.31, 0.15, 1.41, α, 1.00, 0.02}$ ). (b) DOC vs the beam amplitude ratios, $A$ , for several exemplary cases (the legend numbers refer to the following random choice of laser parameters: ${\vec{κ}}_{1} = {- 0.61, 0.64, 0.25, 0.27, 0.51, A, 0.93}$ , ${\vec{κ}}_{2} = {0.07, 0.87, 0.78, 1.05, 0.57, A, 0.10}$ , ${\vec{κ}}_{3} = {0.05, 0.66, 0.58, 0.81, 0.37, A, 2.14}$ , ${\vec{κ}}_{4} = {0.73, - 0.76, 0.29, 0.38, 0.21, A, 1.65}$ , ${\vec{κ}}_{5} = {0.92, 0.31, 1.29, 1.34, 0.35, A, 1.55}$ , ${\vec{κ}}_{6} = {- 0.62, 0.50, 0.71, 0.58, 0.15, A, 1.83}$ , ${\vec{κ}}_{7} = {- 0.03, 0.22, 0.73, 1.45, 0.26, A, 1.60}$ , ${\vec{κ}}_{8} = {- 0.39, 0.08, 0.93, 1.38, 0.51, A, 2.32}$ , ${\vec{κ}}_{9} = {- 0.35, 0.81, 1.03, 0.09, 0.05, A, 1.35}$ , ${\vec{κ}}_{10} = {- 0.76, 0.50, 0.67, 0.32, 0.00, A, 2.12}$ ). Values for $\vec{κ}$ are given in units of radians or are unitless.
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Figure 3
Physical dependence of the DOC of bichromatic beams on the laser parameters $ɛ_{1, 2}$ and $β_{1, 2}$ . (a) DOC vs the laser ellipticities, $ɛ_{1, 2}$ , for $β_{1, 2} = 0$ . (b) Same as (a) but for $β_{1} = 0, β_{2} = 45^{\circ}$ . (c) Same as in (a) but for $β_{1} = 0$ , $β_{2} = 90^{\circ}$ . In all cases we used the additional laser parameters $A = 1$ , $α = 10^{\circ}$ , $η = 0$ . Note that the color schemes are not uniformly normalized between plots.
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Figure 4
Physical dependence of the DOC of bichromatic beams on the laser parameters $ɛ_{1, 2}$ and $β_{1, 2}$ . (a) DOC vs the laser major elliptical axes, $β_{1, 2}$ , for ellipticities $ɛ_{1} = 0.5$ , $ɛ_{2} = 0.5$ . (b) Same as (a) but for $ɛ_{1} = 0.5$ , $ɛ_{2} = 0$ . (c) Same as in (a) but for $ɛ_{1} = 0.5$ , $ɛ_{2} = - 0.5$ . In all cases we used the additional laser parameters $A = 1$ , $α = 10^{\circ}$ , $η = 0$ . The axes are given in units of degrees. Note that the color schemes are not uniformly normalized between the plots.
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Figure 5
Physical dependence of the DOC of bichromatic beams on the relative phase, η, for several exemplary cases (the legend numbers refer to the following random choice of laser parameters: ${\vec{κ}}_{1} = {0.88, - 0.79, 1.51, 0.69, 0.40, 1.01, η}, {\vec{κ}}_{2} = {0.57, 0.63, 0.01, 0.44, 0.29, 1.04, η}, {\vec{κ}}_{3} = {0.15, 0.80, 1.09, 1.55, 0.66, 0.63, η}, {\vec{κ}}_{4} = {0.58, - 0.89, 1.43, 1.40, 0.27, 1.42, η}, {\vec{κ}}_{5} = {0.98, - 0.32, 1.11, 1.49, 0.32, 1.11, η}, {\vec{κ}}_{6} = {0.42, - 0.11, 1.27, 1.45, 0.65, 1.48, η}, {\vec{κ}}_{7} = {0.83, 0.04, 0.61, 1.31, 0.21, 1.08, η}, {\vec{κ}}_{8} = {0.22, - 0.12, 0.56, 0.85, 0.04, 0.67, η}$ ). Values for $\vec{κ}$ are given in units of radians or are unitless.
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Figure 6
Illustration of the maximally chiral EM field obtained by our optimization procedure with a DOC of 69.6%. (a,b) Illustration of how this EM field is generated, of its Lissajou plot that describes the electric vector field's time-dependent polarization, and of its mirror twin in (b). (c) Illustration of the maximal overlap between the field and its mirror twin that is found for some particular set of space-time angles, ${\vec{μ}}_{min}$ , that minimize the integral in Eq. (1).
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