Maximum Chirality in Electromagnetic Fields

• Maximum chirality is an extreme electromagnetic condition defined by full handedness selectivity and helicity preservation in reciprocal systems.
• It is quantified by circular dichroism (CD = ±1) and achieved when one polarization is fully transmitted or suppressed through balanced electric and magnetic interactions.
• Applications include chiral sensing, resonant mode engineering, and enantio-selective photonics by suppressing one helicity channel while optimizing the other.

Maximum chirality denotes an extremal regime of electromagnetic handedness selectivity. In scattering theory, it refers to the upper-bound case in which an object is transparent to all fields of one helicity and, for reciprocal systems, preserves helicity upon interaction with the other (Fernandez-Corbaton et al., 2015). In chiral metasurfaces, the same limit is عادة quantified by circular dichroism reaching $\mathrm{CD}=\pm1$, meaning that one circular polarization is fully transmitted, reflected, or absorbed while the opposite handedness is unperturbed or strongly suppressed (Gorkunov et al., 2020). In structured-light and near-field formulations, maximum chirality is the saturation of the universal helicity-density bound $|h|=u/\omega$, achieved when the local fields satisfy $\mathbf E=\pm i\,\eta_0\,\mathbf H$ (Hanifeh et al., 2018). The term therefore spans operator-based scattering measures, resonant mode engineering, multipolar interference, and helicity-maximized electromagnetic fields.

1. Definitions and figures of merit

Several non-equivalent but closely related observables are used to quantify chirality at its maximum. In metasurfaces and resonant planar photonics, the most common quantity is circular dichroism. A standard transmission definition is

$$\mathrm{CD}(\omega)=\frac{|t_{\rm RCP}(\omega)|^2-|t_{\rm LCP}(\omega)|^2}{|t_{\rm RCP}(\omega)|^2+|t_{\rm LCP}(\omega)|^2},$$

so that $\mathrm{CD}=\pm1$ corresponds to complete helicity selectivity. In fully reflecting plasmonic systems, CD may instead be defined as the absolute reflectance difference,

$\mathrm{CD}=|R_{\rm LCP}-R_{\rm RCP}|,$

because no transmitted light appears when the bottom Au film is thicker than the skin depth (Niu et al., 27 Feb 2025, Wu et al., 2021).

A more general, direction-independent measure is electromagnetic chirality defined from the helicity-basis scattering or $T$-matrix. Writing the helicity-resolved blocks as $T^{\sigma,\sigma'}$, the normalized electromagnetic chirality $\overline{\chi}$ lies in $[0,1]$ and reaches unity only when the object is invisible to one helicity and interacts exclusively with the other (Santiago et al., 2021). The foundational operator formulation expresses the same idea by comparing the singular-value spectra of the helicity-resolved scattering blocks and proves an absolute upper bound $|h|=u/\omega$0 (Fernandez-Corbaton et al., 2015).

A third definition concerns the field itself rather than the scatterer. For monochromatic fields, the time-averaged helicity density is

$|h|=u/\omega$1

with time-averaged energy density

$|h|=u/\omega$2

Maximum field chirality is the condition $|h|=u/\omega$3 (Hanifeh et al., 2018). In paraxial singular-optics formulations, the equivalent time-averaged helicity density is written as

$|h|=u/\omega$4

which makes explicit the equivalence between helicity density and chirality density in that setting (Babiker et al., 2023).

2. Upper bounds, reciprocity, and symmetry conditions

The extremal character of maximum chirality is controlled by strict symmetry and conservation constraints. For linear interactions with finite cross-sections, electromagnetic chirality obeys

$|h|=u/\omega$5

For reciprocal objects, attaining the bound requires transparency to one helicity together with helicity preservation, i.e. electromagnetic duality symmetry. In the dipolar and constitutive-tensor formulations, this requirement translates into specific equalities among electric, magnetic, and magnetoelectric response tensors rather than geometric handedness alone (Fernandez-Corbaton et al., 2015).

For local fields, the universal bound is

$|h|=u/\omega$6

with equality only under the balanced or dual condition

$|h|=u/\omega$7

This condition simultaneously fixes the amplitude ratio $|h|=u/\omega$8 and the $|h|=u/\omega$9 phase shift needed to maximize helicity at fixed energy density (Hanifeh et al., 2018, Hanifeh et al., 2019).

A common misconception is that intrinsic structural chirality guarantees maximal or even nonzero optical chirality for all illumination directions. Quasi-normal-mode analysis shows instead that the optical response depends on the polarization singularities of the radiated far field. For a reciprocal scatterer illuminated along a direction $\mathbf E=\pm i\,\eta_0\,\mathbf H$0, the circular dichroism equals the third Stokes parameter of the time-reversed radiation direction,

$\mathbf E=\pm i\,\eta_0\,\mathbf H$1

Hence C-points with $\mathbf E=\pm i\,\eta_0\,\mathbf H$2 yield ideal extrema $\mathbf E=\pm i\,\eta_0\,\mathbf H$3, whereas L-points with $\mathbf E=\pm i\,\eta_0\,\mathbf H$4 yield ideal minima $\mathbf E=\pm i\,\eta_0\,\mathbf H$5. An intrinsically chiral structure can therefore exhibit zero chirality or even opposite-handed chirality for different incidence directions (Chen et al., 2021).

3. Bound states in the continuum and planar dielectric routes

Bound states in the continuum and quasi-BIC resonances provide one of the most explicit routes to maximum optical chirality in planar dielectric metasurfaces. In rotationally symmetric chiral metasurfaces, a true BIC is first formed by a symmetry-protected dark mode with vanishing radiative coupling. A weak perturbation then opens a radiative channel while keeping the resonance arbitrarily sharp. In coupled-mode form, maximum chirality requires one helicity to be completely decoupled, $\mathbf E=\pm i\,\eta_0\,\mathbf H$6, together with critical coupling $\mathbf E=\pm i\,\eta_0\,\mathbf H$7. For the dielectric-bar-pair realization, the geometric condition

$\mathbf E=\pm i\,\eta_0\,\mathbf H$8

enforces the required helicity decoupling. Finite-element simulations for $\mathbf E=\pm i\,\eta_0\,\mathbf H$9 gave an optimum at $$\mathrm{CD}(\omega)=\frac{|t_{\rm RCP}(\omega)|^2-|t_{\rm LCP}(\omega)|^2}{|t_{\rm RCP}(\omega)|^2+|t_{\rm LCP}(\omega)|^2},$$0 nm with $$\mathrm{CD}(\omega)=\frac{|t_{\rm RCP}(\omega)|^2-|t_{\rm LCP}(\omega)|^2}{|t_{\rm RCP}(\omega)|^2+|t_{\rm LCP}(\omega)|^2},$$1 nm and $$\mathrm{CD}(\omega)=\frac{|t_{\rm RCP}(\omega)|^2-|t_{\rm LCP}(\omega)|^2}{|t_{\rm RCP}(\omega)|^2+|t_{\rm LCP}(\omega)|^2},$$2, where $$\mathrm{CD}(\omega)=\frac{|t_{\rm RCP}(\omega)|^2-|t_{\rm LCP}(\omega)|^2}{|t_{\rm RCP}(\omega)|^2+|t_{\rm LCP}(\omega)|^2},$$3, $$\mathrm{CD}(\omega)=\frac{|t_{\rm RCP}(\omega)|^2-|t_{\rm LCP}(\omega)|^2}{|t_{\rm RCP}(\omega)|^2+|t_{\rm LCP}(\omega)|^2},$$4, and thus $$\mathrm{CD}(\omega)=\frac{|t_{\rm RCP}(\omega)|^2-|t_{\rm LCP}(\omega)|^2}{|t_{\rm RCP}(\omega)|^2+|t_{\rm LCP}(\omega)|^2},$$5 (Gorkunov et al., 2020).

A related but experimentally distinct route unlocks the out-of-plane degree of freedom of all-dielectric quasi-BIC metasurfaces. In that framework, the circular-polarization couplings satisfy

$$\mathrm{CD}(\omega)=\frac{|t_{\rm RCP}(\omega)|^2-|t_{\rm LCP}(\omega)|^2}{|t_{\rm RCP}(\omega)|^2+|t_{\rm LCP}(\omega)|^2},$$6

Choosing

$$\mathrm{CD}(\omega)=\frac{|t_{\rm RCP}(\omega)|^2-|t_{\rm LCP}(\omega)|^2}{|t_{\rm RCP}(\omega)|^2+|t_{\rm LCP}(\omega)|^2},$$7

suppresses one helicity channel completely while keeping the opposite channel bright. The demonstrated amorphous-Si metasurfaces used a square lattice with $$\mathrm{CD}(\omega)=\frac{|t_{\rm RCP}(\omega)|^2-|t_{\rm LCP}(\omega)|^2}{|t_{\rm RCP}(\omega)|^2+|t_{\rm LCP}(\omega)|^2},$$8 nm, rod length $$\mathrm{CD}(\omega)=\frac{|t_{\rm RCP}(\omega)|^2-|t_{\rm LCP}(\omega)|^2}{|t_{\rm RCP}(\omega)|^2+|t_{\rm LCP}(\omega)|^2},$$9 nm, heights $\mathrm{CD}=\pm1$0 nm and $\mathrm{CD}=\pm1$1 nm, and opening angle $\mathrm{CD}=\pm1$2. The measured resonances occurred at $\mathrm{CD}=\pm1$3 nm for the left-handed structure under LCP illumination and at $\mathrm{CD}=\pm1$4 nm for the right-handed structure under RCP illumination, both with $\mathrm{CD}=\pm1$5 and $\mathrm{CD}=\pm1$6 (Kühner et al., 2022).

Maximum chirality can also arise from strong coupling between accidentally degenerate quasi-BICs with orthogonal polarization states. In the square-lattice silicon-dimer metasurface of period $\mathrm{CD}=\pm1$7 nm, controlled in-plane asymmetry lifts the TE–TM degeneracy and creates two split hybrid modes. Beyond the exceptional-point transition, the higher-frequency mode radiates pure LCP and the lower-frequency mode pure RCP, with suppressed cross-polarization scattering and $\mathrm{CD}=\pm1$8. Full-wave simulations at the design point $\mathrm{CD}=\pm1$9 nm and $\mathrm{CD}=|R_{\rm LCP}-R_{\rm RCP}|,$0 nm showed nearly unit CD at the split resonances, while quasinormal-mode perturbation theory reproduced the resonance frequencies, $\mathrm{CD}=|R_{\rm LCP}-R_{\rm RCP}|,$1 factors, and averaged Stokes parameter within $\mathrm{CD}=|R_{\rm LCP}-R_{\rm RCP}|,$2 near $\mathrm{CD}=|R_{\rm LCP}-R_{\rm RCP}|,$3 nm (Niu et al., 27 Feb 2025).

4. Mode hybridization, nominally achiral bilayers, and multipolar interference

A distinct research line shows that maximum or near-maximum chirality does not require intrinsically chiral meta-atoms. In free-standing silicon membrane metasurfaces with rotated $\mathrm{CD}=|R_{\rm LCP}-R_{\rm RCP}|,$4-symmetric apertures, a single layer retains a horizontal mirror plane $\mathrm{CD}=|R_{\rm LCP}-R_{\rm RCP}|,$5 and therefore exhibits $\mathrm{CD}=|R_{\rm LCP}-R_{\rm RCP}|,$6. Adding a thin PMMA overlayer removes $\mathrm{CD}=|R_{\rm LCP}-R_{\rm RCP}|,$7, reduces the point group to $\mathrm{CD}=|R_{\rm LCP}-R_{\rm RCP}|,$8, and mixes modes that previously had opposite vertical parity. The resulting bilayer exhibits anti-crossings and pronounced mode-resolved chirality; simulations yielded $\mathrm{CD}=|R_{\rm LCP}-R_{\rm RCP}|,$9, while experiments showed clear peaks near $T$0, $T$1, and $T$2m with $T$3. The coupled-mode expression

$T$4

identifies the strong-coupling, balanced-loss limit as the route to maximum chirality (Kumar et al., 12 May 2026).

The more general bilayer-mode-coupling analysis of Kumar et al. distinguishes several scenarios. In a $T$5 metasurface, co-polarized transmission CD is the relevant observable,

$T$6

For a lossless reciprocal $T$7 structure, $T$8; absorption must therefore be introduced to make the chiral eigenmodes visible in transmission. With four-petal holes defined by $T$9, $T^{\sigma,\sigma'}$0, $T^{\sigma,\sigma'}$1, a high-index bottom membrane, and a lossy top membrane with $T^{\sigma,\sigma'}$2, the optimized bilayer reached $T^{\sigma,\sigma'}$3 and $T^{\sigma,\sigma'}$4 for the two branches of one avoided crossing at $T^{\sigma,\sigma'}$5, while another crossing gave $T^{\sigma,\sigma'}$6 with intrinsic $T^{\sigma,\sigma'}$7 (Kumar et al., 23 Jul 2025).

Plasmonic metasurfaces show the same dependence on symmetry breaking and resonance alignment, although typically with smaller reported maxima. Wu et al. studied a chiral Au metasurface based on rectangular holes on a hexagonal lattice and found that mirror or six-fold symmetry at $T^{\sigma,\sigma'}$8 forces $T^{\sigma,\sigma'}$9, whereas off-symmetry rotations produce strong handedness selectivity. Their strongest response occurred for $\overline{\chi}$0 nm, $\overline{\chi}$1 nm, $\overline{\chi}$2 nm, $\overline{\chi}$3, and $\overline{\chi}$4 nm, where the reported circular dichroism reached $\overline{\chi}$5. Electric-field maps showed strong RCP hot spots at the rectangle corners and short sides, but much weaker LCP enhancement, explaining the large reflectance asymmetry (Wu et al., 2021).

A complementary interpretation is given by generalized multipole decomposition. In that framework, maximum structural chirality arises when the generalized chiral electric dipole, magnetic dipole, electric quadrupole, and magnetic quadrupole contributions interfere constructively for one handedness and destructively for the other. For the tilted double-layer cylinder metasurface with $\overline{\chi}$6 nm, $\overline{\chi}$7 nm, $\overline{\chi}$8 nm, $\overline{\chi}$9, $[0,1]$0, and $[0,1]$1, the third resonance at $[0,1]$2 nm achieved $[0,1]$3, $[0,1]$4, and maximum spin–orbit conversion through simultaneous phase alignment in one helicity channel and generalized-anapole cancellation in the other (Zhao et al., 2024).

5. Maximally electromagnetically chiral scatterers, helices, nanowires, and cavities

Three-dimensional scatterers remain the canonical platform for approaching the absolute upper bound of electromagnetic chirality. In optimized silver helices, the normalized electromagnetic chirality exceeded $[0,1]$5 for design wavelengths $[0,1]$6m, decreased to $[0,1]$7 at $[0,1]$8 nm, $[0,1]$9 at $|h|=u/\omega$00 nm in water, and $|h|=u/\omega$01 at $|h|=u/\omega$02 nm. The optimization used four geometric parameters—helix radius, wire thickness, pitch per turn, and number of turns—together with finite-element $|h|=u/\omega$03-matrix extraction, shape derivatives, and Gaussian-process optimization. At $|h|=u/\omega$04m, a dilute slab model predicted that an $|h|=u/\omega$05m thick suspension with filling factor $|h|=u/\omega$06 could absorb $|h|=u/\omega$07 of one helicity and only $|h|=u/\omega$08 of the opposite helicity (Santiago et al., 2021).

Free-form thin metallic nanowires generalize the helix concept by optimizing the full spine curve and twist rather than a fixed helical ansatz. In the asymptotic thin-wire formulation, the relevant objective functions were the normalized chirality measures $|h|=u/\omega$09 and $|h|=u/\omega$10. For silver, full free-form optimization below resonance achieved $|h|=u/\omega$11 up to $|h|=u/\omega$12 and $|h|=u/\omega$13, whereas traditional helices in the optical band achieved $|h|=u/\omega$14–$|h|=u/\omega$15. Gold showed strong degradation at higher frequencies because the imaginary part of the permittivity increases (Fernandez-Corbaton et al., 2022).

Exact maximum electromagnetic chirality is also possible in homogeneous chiral spheres. For every multipole order $|h|=u/\omega$16, the exact Mie-coefficient condition

$|h|=u/\omega$17

guarantees transparency to helicity $|h|=u/\omega$18 and dual, helicity-preserving scattering for the opposite helicity. The associated spectral trajectories in $|h|=u/\omega$19 space are given by a pair of Riccati–Bessel transcendental equations, and the first Kerker condition reduces to a single scalar equation,

$|h|=u/\omega$20

These results are exact for arbitrary multipolar order, refractive-index contrast, optical size, and intrinsic chirality parameter (Olmos-Trigo et al., 2023).

Highly chiral scatterers can then be assembled into more complex photonic environments. A particularly explicit example is the infrared Fabry–Pérot cavity formed by diffracting lattices of optimized silver helices. The isolated helix was optimized at $|h|=u/\omega$21 THz with $|h|=u/\omega$22 nm, $|h|=u/\omega$23 nm, pitch $|h|=u/\omega$24 nm, and $|h|=u/\omega$25, reaching $|h|=u/\omega$26. In the cavity, two nearly pure helicity modes with $|h|=u/\omega$27 emerged; at resonance the $|h|=u/\omega$28 mode achieved $|h|=u/\omega$29, while the opposite-handed mode reached only $|h|=u/\omega$30. The total cavity dissymmetry peaked at $|h|=u/\omega$31m with $|h|=u/\omega$32, which was described as an unprecedented intracavity dissymmetry (Rebholz et al., 14 Jul 2025).

6. Helicity-maximized fields, near-field platforms, and applications

Maximum chirality can be engineered directly in the electromagnetic field. Optimally chiral structured light is realized when the incident beam satisfies $|h|=u/\omega$33 everywhere. A practical implementation is the azimuthally and radially polarized beam superposition, for which the complex amplitudes obey $|h|=u/\omega$34. Under illumination by such a beam, an achiral dielectric nanoantenna with balanced polarizabilities,

$|h|=u/\omega$35

develops induced dipoles obeying $|h|=u/\omega$36, so that even the scattered near field remains optimally chiral. For a silicon sphere illuminated by an ARPB with waist $|h|=u/\omega$37, full-wave analysis found a total helicity-density enhancement of approximately $|h|=u/\omega$38 on the $|h|=u/\omega$39 axis at the particle surface for $|h|=u/\omega$40 nm, while exact saturation of the bound occurred near $|h|=u/\omega$41 nm (Hanifeh et al., 2019).

Planar arrays of achiral high-index nanoparticles provide a larger-area near-field realization. When the effective array polarizabilities satisfy the array-generalized Kerker condition

$|h|=u/\omega$42

the total near field above the array approaches the optimal bound. For crystalline Si nanospheres with radius $|h|=u/\omega$43 nm, period $|h|=u/\omega$44 nm gave nearly zero reflectance at $|h|=u/\omega$45 nm and $|h|=u/\omega$46–$|h|=u/\omega$47 over more than $|h|=u/\omega$48 of the unit cell, with hot-spot $|h|=u/\omega$49; reducing the period to $|h|=u/\omega$50 nm increased the hot-spot enhancement to $|h|=u/\omega$51 while keeping $|h|=u/\omega$52. Because the array itself is achiral and approximately Kerker-balanced, it contributes no CD background when used as a substrate for thin chiral films (Hanifeh et al., 2019).

Paraxial singular optics supplies a different route to unbounded enhancement in principle. For order-$|h|=u/\omega$53 higher-order Poincaré modes, the total helicity per unit length is

$|h|=u/\omega$54

This yields linear scaling in $|h|=u/\omega$55 on the equator of the order-$|h|=u/\omega$56 Poincaré sphere and quadratic scaling at the poles. The formal limit $|h|=u/\omega$57 therefore implies unlimited super-chirality, although the paper identifies practical restrictions from the paraxial approximation, mode-generation fidelity, and diffraction-aberration sensitivity. For typical parameters with $|h|=u/\omega$58, $|h|=u/\omega$59, and $|h|=u/\omega$60, enhancements of order $|h|=u/\omega$61 over conventional circularly polarized Laguerre–Gaussian beams were reported (Babiker et al., 2023).

The application space of maximum chirality is correspondingly broad. The operator-based theory explicitly proposed twofold resonantly enhanced, background-free circular-dichroism measurements and angle-independent helicity-filtering glasses (Fernandez-Corbaton et al., 2015). Planar dielectric metasurface studies identify chiral sensing, emission, and lasing as immediate targets (Niu et al., 27 Feb 2025). The chiral cavity work further connects near-maximal intracavity dissymmetry to enantio-selective sorting and chiral chemistry, emphasizing that molecular infrared dissymmetry factors of order $|h|=u/\omega$62 remain a central systems-level constraint even when the photonic environment approaches the absolute helicity-selective limit (Rebholz et al., 14 Jul 2025).

Taken together, these results establish maximum chirality as a multi-scale extremal condition rather than a single device metric. At the operator level it is bounded by helicity transparency and duality; at the resonator level it is realized through BICs, quasi-BICs, mode hybridization, and exceptional-point-adjacent strong coupling; at the multipolar level it is enforced by constructive and destructive interference; and at the field level it is the saturation of $|h|=u/\omega$63. This suggests that the central design problem is not merely to create handed geometry, but to suppress one helicity channel while preserving or critically coupling the other with as little parasitic conversion as possible.