Degree of Nonlinear Chirality (DNC)

• DNC is defined as a normalized measure of handedness selectivity, often expressed as a difference-over-sum ratio in nonlinear signals.
• It finds applications in harmonic generation, Kerr microresonators, and nonlinear transport, enabling quantitative chiral characterization across diverse systems.
• Different implementations adjust sign and normalization conventions, highlighting the interplay of symmetry, resonance, and interference in determining DNC values.

Degree of Nonlinear Chirality (DNC) denotes a normalized measure of handedness selectivity in a nonlinear response. Across the recent literature, it is not a universally standardized single quantity: several works do not explicitly use the term, but quantify the same idea through nonlinear circular dichroism in harmonic generation, dissymmetry factors, directional chiral dichroism, or normalized population imbalance between opposite chiral channels. In harmonic generation, DNC is most commonly the difference-over-sum between nonlinear signals produced under opposite input helicities or between right- and left-circularly polarized components of the emitted nonlinear field. In broader settings, closely related DNC-like quantities quantify spontaneous CW/CCW imbalance in Kerr microresonators, enantio-sensitive directional emission driven by synthetic chiral light, fluctuation-enabled chiral two-dimensional spectroscopy, or chirality-dependent second-order transport coefficients in chiral crystals (Tonkaev et al., 18 Aug 2025, Lai et al., 8 Sep 2025, Cao et al., 2016, Rego et al., 2022, Pan et al., 2024).

1. Definitions and normalization conventions

The central feature of DNC is normalization by the total nonlinear signal, so that the metric reports selectivity rather than absolute intensity. What varies across subfields is the observable being contrasted and the chosen prefactor. Several papers therefore use mathematically equivalent or linearly related quantities with different names and bounds.

Nonlinear context	Reported metric	Relation to DNC
THG circular dichroism	Difference-over-sum of THG yields	Usually identical to DNC
SHG-CD microscopy	SHG-CD with denominator divided by 2	Equals $2\times$ DNC
SHG dissymmetry factor	$g_{\mathrm{SHG-CD}}$	Often a $[-2,2]$ form
CW/CCW spontaneous chirality	Power imbalance or power ratio	Natural DNC from normalized imbalance
Nonlinear transport	Difference of enantiomeric coefficients	DNC from normalized coefficient contrast

For helicity-resolved third-harmonic generation in linearly achiral membrane metasurfaces, the natural DNC is the nonlinear circular dichroism of the total THG yield,

$$\mathrm{DNC}\equiv \mathrm{CD}^{(3\omega)}= \frac{I_{R\bullet}^{(3\omega)}-I_{L\bullet}^{(3\omega)}}{I_{R\bullet}^{(3\omega)}+I_{L\bullet}^{(3\omega)}},$$

with $$I_{R\bullet}^{(3\omega)}=I_{RR}^{(3\omega)}+I_{RL}^{(3\omega)}$$ and $$I_{L\bullet}^{(3\omega)}=I_{LL}^{(3\omega)}+I_{LR}^{(3\omega)}$$. A channel-specific variant is

$$\mathrm{DNC}_{\beta}= \frac{I_{R\beta}^{(3\omega)}-I_{L\beta}^{(3\omega)}}{I_{R\beta}^{(3\omega)}+I_{L\beta}^{(3\omega)}},$$

which reduces to the total DNC after summing over output helicities $\beta\in\{R,L\}$ (Tonkaev et al., 18 Aug 2025).

In resonant SHG-CD microscopy of collagen, the compact nonlinear-chirality normalization is

$$\mathrm{DNC}(2\omega)=\frac{I_R(2\omega)-I_L(2\omega)}{I_R(2\omega)+I_L(2\omega)},$$

whereas the paper’s reported SHG-CD is

$$\mathrm{SHG\text{-}CD}= \frac{I_{\mathrm{RHC}}(2\omega)-I_{\mathrm{LHC}}(2\omega)} {(I_{\mathrm{RHC}}(2\omega)+I_{\mathrm{LHC}}(2\omega))/2} =2\times \mathrm{DNC}(2\omega).$$

This is a normalization change, not a different physical observable (Chen et al., 2016).

In twisted $g_{\mathrm{SHG-CD}}$0 heterobilayers, nonlinear chirality is quantified by the SHG circular-dichroism dissymmetry factor

$g_{\mathrm{SHG-CD}}$1

which is bounded by $g_{\mathrm{SHG-CD}}$2. In that work, DNC is mapped directly to $g_{\mathrm{SHG-CD}}$3, so its bound is $g_{\mathrm{SHG-CD}}$4 rather than $g_{\mathrm{SHG-CD}}$5 (Zhang et al., 17 May 2026).

Outside harmonic CD, the same normalization logic appears in Kerr microresonators, where a natural DNC is

$g_{\mathrm{SHG-CD}}$6

and in nonlinear transport, where DNC is defined from the difference between second-order coefficients of opposite enantiomers, normalized by the sum of their magnitudes (Cao et al., 2016, Pan et al., 2024).

The immediate consequence is that DNC values are only directly comparable after the normalization and sign convention are specified. A negative value may mean stronger LCP-driven emission in one paper and stronger RCP-driven emission in another, depending on numerator order.

2. Harmonic-generation realizations

The most developed use of DNC is in nonlinear harmonic generation, especially SHG and THG. In free-standing silicon membrane metasurfaces that are effectively achiral in the linear regime, the unperturbed $g_{\mathrm{SHG-CD}}$7-symmetric sample exhibits dominant cross-polarized THG in the $g_{\mathrm{SHG-CD}}$8 channel and a nonlinear DNC minimum near $g_{\mathrm{SHG-CD}}$9 at pump wavelength $[-2,2]$0 in experiment; simulations give a negative peak of $[-2,2]$1 near resonance. After intentional in-plane symmetry breaking, a co-polarized $[-2,2]$2 channel becomes strongly enabled as the pump wavelength increases toward $[-2,2]$3–$[-2,2]$4, and the DNC reverses sign, reaching $[-2,2]$5 at $[-2,2]$6 experimentally, with a simulated positive peak of $[-2,2]$7. The same samples show only modest linear CD, up to $[-2,2]$8 in the unperturbed case and $[-2,2]$9 in the perturbed case, demonstrating that nonlinear chirality can greatly exceed linear chirality under symmetry-controlled THG selection rules (Tonkaev et al., 18 Aug 2025).

A distinct THG formulation appears in achiral dielectric metasurfaces that generate chiral nonlinear light through resonant modal interference. There the DNC is defined directly from the emitted third-harmonic helicity components,

$$\mathrm{DNC}\equiv \mathrm{CD}^{(3\omega)}= \frac{I_{R\bullet}^{(3\omega)}-I_{L\bullet}^{(3\omega)}}{I_{R\bullet}^{(3\omega)}+I_{L\bullet}^{(3\omega)}},$$0

with DNC $$\mathrm{DNC}\equiv \mathrm{CD}^{(3\omega)}= \frac{I_{R\bullet}^{(3\omega)}-I_{L\bullet}^{(3\omega)}}{I_{R\bullet}^{(3\omega)}+I_{L\bullet}^{(3\omega)}},$$1 for purely RCP emission, DNC $$\mathrm{DNC}\equiv \mathrm{CD}^{(3\omega)}= \frac{I_{R\bullet}^{(3\omega)}-I_{L\bullet}^{(3\omega)}}{I_{R\bullet}^{(3\omega)}+I_{L\bullet}^{(3\omega)}},$$2 for purely LCP emission, and DNC $$\mathrm{DNC}\equiv \mathrm{CD}^{(3\omega)}= \frac{I_{R\bullet}^{(3\omega)}-I_{L\bullet}^{(3\omega)}}{I_{R\bullet}^{(3\omega)}+I_{L\bullet}^{(3\omega)}},$$3 for equal helicity content. Experimentally, the chirality of the TH radiation from a single achiral metasurface was tuned continuously from $$\mathrm{DNC}\equiv \mathrm{CD}^{(3\omega)}= \frac{I_{R\bullet}^{(3\omega)}-I_{L\bullet}^{(3\omega)}}{I_{R\bullet}^{(3\omega)}+I_{L\bullet}^{(3\omega)}},$$4 to $$\mathrm{DNC}\equiv \mathrm{CD}^{(3\omega)}= \frac{I_{R\bullet}^{(3\omega)}-I_{L\bullet}^{(3\omega)}}{I_{R\bullet}^{(3\omega)}+I_{L\bullet}^{(3\omega)}},$$5 by varying only the polarization angle of the incident wave. In simulations, pure LCP and pure RCP emission were obtained at the same TH wavelength for different pump polarization angles, corresponding to DNC $$\mathrm{DNC}\equiv \mathrm{CD}^{(3\omega)}= \frac{I_{R\bullet}^{(3\omega)}-I_{L\bullet}^{(3\omega)}}{I_{R\bullet}^{(3\omega)}+I_{L\bullet}^{(3\omega)}},$$6 and $$\mathrm{DNC}\equiv \mathrm{CD}^{(3\omega)}= \frac{I_{R\bullet}^{(3\omega)}-I_{L\bullet}^{(3\omega)}}{I_{R\bullet}^{(3\omega)}+I_{L\bullet}^{(3\omega)}},$$7 (Lai et al., 8 Sep 2025).

Second-harmonic implementations exhibit the same structure with different conventions. In collagenous tissue, resonant SHG-CD imaging revealed a clear enhancement at $$\mathrm{DNC}\equiv \mathrm{CD}^{(3\omega)}= \frac{I_{R\bullet}^{(3\omega)}-I_{L\bullet}^{(3\omega)}}{I_{R\bullet}^{(3\omega)}+I_{L\bullet}^{(3\omega)}},$$8 excitation, tied to a linear absorption peak near $$\mathrm{DNC}\equiv \mathrm{CD}^{(3\omega)}= \frac{I_{R\bullet}^{(3\omega)}-I_{L\bullet}^{(3\omega)}}{I_{R\bullet}^{(3\omega)}+I_{L\bullet}^{(3\omega)}},$$9. Image-averaged SHG-CD at room temperature is $$I_{R\bullet}^{(3\omega)}=I_{RR}^{(3\omega)}+I_{RL}^{(3\omega)}$$0 at resonance and $$I_{R\bullet}^{(3\omega)}=I_{RR}^{(3\omega)}+I_{RL}^{(3\omega)}$$1 at non-resonant wavelengths under the paper’s normalization, so the corresponding DNC at resonance is $$I_{R\bullet}^{(3\omega)}=I_{RR}^{(3\omega)}+I_{RL}^{(3\omega)}$$2. Heating from $$I_{R\bullet}^{(3\omega)}=I_{RR}^{(3\omega)}+I_{RL}^{(3\omega)}$$3 to $$I_{R\bullet}^{(3\omega)}=I_{RR}^{(3\omega)}+I_{RL}^{(3\omega)}$$4 suppresses the non-resonant background toward zero while leaving the resonant peak elevated, and heating to $$I_{R\bullet}^{(3\omega)}=I_{RR}^{(3\omega)}+I_{RL}^{(3\omega)}$$5–$$I_{R\bullet}^{(3\omega)}=I_{RR}^{(3\omega)}+I_{RL}^{(3\omega)}$$6 then causes the resonant peak to decay with triple-helix denaturation. The work thus separates anisotropy-dominated background from molecular-chirality-dominated nonlinear response (Chen et al., 2016).

Twisted heterobilayers provide an extreme SHG limit. In $$I_{R\bullet}^{(3\omega)}=I_{RR}^{(3\omega)}+I_{RL}^{(3\omega)}$$7, the sign of the SHG dissymmetry factor is governed by structural handedness, and its magnitude reaches $$I_{R\bullet}^{(3\omega)}=I_{RR}^{(3\omega)}+I_{RL}^{(3\omega)}$$8 near a $$I_{R\bullet}^{(3\omega)}=I_{RR}^{(3\omega)}+I_{RL}^{(3\omega)}$$9 twist angle under $$I_{L\bullet}^{(3\omega)}=I_{LL}^{(3\omega)}+I_{LR}^{(3\omega)}$$0-nm excitation, approaching the theoretical limit of $$I_{L\bullet}^{(3\omega)}=I_{LL}^{(3\omega)}+I_{LR}^{(3\omega)}$$1. A right-handed sample at $$I_{L\bullet}^{(3\omega)}=I_{LL}^{(3\omega)}+I_{LR}^{(3\omega)}$$2 shows $$I_{L\bullet}^{(3\omega)}=I_{LL}^{(3\omega)}+I_{LR}^{(3\omega)}$$3, and a left-handed sample at $$I_{L\bullet}^{(3\omega)}=I_{LL}^{(3\omega)}+I_{LR}^{(3\omega)}$$4 shows $$I_{L\bullet}^{(3\omega)}=I_{LL}^{(3\omega)}+I_{LR}^{(3\omega)}$$5. Because the chosen normalization allows one helicity channel to vanish while the other remains finite, these values are near-complete nonlinear helicity selectivity (Zhang et al., 17 May 2026).

Single-particle plasmonic systems also support large nonlinear chirality. For L-handed helicoid-III gold nanoparticles, measured SHG $$I_{L\bullet}^{(3\omega)}=I_{LL}^{(3\omega)}+I_{LR}^{(3\omega)}$$6-factors reach $$I_{L\bullet}^{(3\omega)}=I_{LL}^{(3\omega)}+I_{LR}^{(3\omega)}$$7 near $$I_{L\bullet}^{(3\omega)}=I_{LL}^{(3\omega)}+I_{LR}^{(3\omega)}$$8 and $$I_{L\bullet}^{(3\omega)}=I_{LL}^{(3\omega)}+I_{LR}^{(3\omega)}$$9 near $$\mathrm{DNC}_{\beta}= \frac{I_{R\beta}^{(3\omega)}-I_{L\beta}^{(3\omega)}}{I_{R\beta}^{(3\omega)}+I_{L\beta}^{(3\omega)}},$$0. Under the exact mapping

$$\mathrm{DNC}_{\beta}= \frac{I_{R\beta}^{(3\omega)}-I_{L\beta}^{(3\omega)}}{I_{R\beta}^{(3\omega)}+I_{L\beta}^{(3\omega)}},$$1

these correspond to DNC values of approximately $$\mathrm{DNC}_{\beta}= \frac{I_{R\beta}^{(3\omega)}-I_{L\beta}^{(3\omega)}}{I_{R\beta}^{(3\omega)}+I_{L\beta}^{(3\omega)}},$$2 and $$\mathrm{DNC}_{\beta}= \frac{I_{R\beta}^{(3\omega)}-I_{L\beta}^{(3\omega)}}{I_{R\beta}^{(3\omega)}+I_{L\beta}^{(3\omega)}},$$3, respectively (Spreyer et al., 2022).

3. Symmetry, resonances, and channel selection

The reported DNC mechanisms are governed less by bulk chirality alone than by the interplay of symmetry, resonance, and interference. In silicon membrane metasurfaces, up–down mirror symmetry suppresses linear chiral transmission, and in-plane $$\mathrm{DNC}_{\beta}= \frac{I_{R\beta}^{(3\omega)}-I_{L\beta}^{(3\omega)}}{I_{R\beta}^{(3\omega)}+I_{L\beta}^{(3\omega)}},$$4 rotational symmetry forbids linear cross-polarization in transmission while also dictating THG selection rules. In the fully $$\mathrm{DNC}_{\beta}= \frac{I_{R\beta}^{(3\omega)}-I_{L\beta}^{(3\omega)}}{I_{R\beta}^{(3\omega)}+I_{L\beta}^{(3\omega)}},$$5 case, only cross-polarized THG channels are symmetry-allowed and co-polarized THG channels are forbidden. Breaking the in-plane $$\mathrm{DNC}_{\beta}= \frac{I_{R\beta}^{(3\omega)}-I_{L\beta}^{(3\omega)}}{I_{R\beta}^{(3\omega)}+I_{L\beta}^{(3\omega)}},$$6 symmetry lifts that selection-rule prohibition, opens co-polarized THG channels, and redistributes the nonlinear power between $$\mathrm{DNC}_{\beta}= \frac{I_{R\beta}^{(3\omega)}-I_{L\beta}^{(3\omega)}}{I_{R\beta}^{(3\omega)}+I_{L\beta}^{(3\omega)}},$$7, $$\mathrm{DNC}_{\beta}= \frac{I_{R\beta}^{(3\omega)}-I_{L\beta}^{(3\omega)}}{I_{R\beta}^{(3\omega)}+I_{L\beta}^{(3\omega)}},$$8, $$\mathrm{DNC}_{\beta}= \frac{I_{R\beta}^{(3\omega)}-I_{L\beta}^{(3\omega)}}{I_{R\beta}^{(3\omega)}+I_{L\beta}^{(3\omega)}},$$9, and $\beta\in\{R,L\}$0, which is the direct reason the DNC changes sign between the unperturbed and perturbed samples (Tonkaev et al., 18 Aug 2025).

A related but more explicitly geometric control appears in metasurfaces governed by meta-atom rotation. Free-standing membrane metasurfaces composed of periodic lattices of tilted elliptic holes preserve out-of-plane mirror symmetry $\beta\in\{R,L\}$1 while breaking all in-plane mirror symmetries through in-plane rotation of the elliptical meta-atoms. With $\beta\in\{R,L\}$2, the $\beta\in\{R,L\}$3 THG channel dominates and $\beta\in\{R,L\}$4 around $\beta\in\{R,L\}$5–$\beta\in\{R,L\}$6; with the complementary rotation $\beta\in\{R,L\}$7, the $\beta\in\{R,L\}$8 channel dominates and $\beta\in\{R,L\}$9. The paper identifies a striking swapping of nonlinear chiral channels for complementary rotation angles and an approximate antisymmetry,

$$\mathrm{DNC}(2\omega)=\frac{I_R(2\omega)-I_L(2\omega)}{I_R(2\omega)+I_L(2\omega)},$$0

with sign inversion across $$\mathrm{DNC}(2\omega)=\frac{I_R(2\omega)-I_L(2\omega)}{I_R(2\omega)+I_L(2\omega)},$$1 (Hariharan et al., 16 Apr 2026).

In twisted heterobilayers, the controlling symmetry is instead the threefold rotational symmetry of each monolayer and the twist-dependent nonlinear Pancharatnam–Berry phase. For circularly polarized excitation, the SH field acquires a geometric phase $$\mathrm{DNC}(2\omega)=\frac{I_R(2\omega)-I_L(2\omega)}{I_R(2\omega)+I_L(2\omega)},$$2, with $$\mathrm{DNC}(2\omega)=\frac{I_R(2\omega)-I_L(2\omega)}{I_R(2\omega)+I_L(2\omega)},$$3 for LCP/RCP pump. The total SHG field is the coherent sum of the two layer-resolved fields,

$$\mathrm{DNC}(2\omega)=\frac{I_R(2\omega)-I_L(2\omega)}{I_R(2\omega)+I_L(2\omega)},$$4

so DNC is determined by helicity-dependent interference between the two monolayer contributions. The compact analytical expression for $$\mathrm{DNC}(2\omega)=\frac{I_R(2\omega)-I_L(2\omega)}{I_R(2\omega)+I_L(2\omega)},$$5 contains $$\mathrm{DNC}(2\omega)=\frac{I_R(2\omega)-I_L(2\omega)}{I_R(2\omega)+I_L(2\omega)},$$6 and $$\mathrm{DNC}(2\omega)=\frac{I_R(2\omega)-I_L(2\omega)}{I_R(2\omega)+I_L(2\omega)},$$7, which explains the $$\mathrm{DNC}(2\omega)=\frac{I_R(2\omega)-I_L(2\omega)}{I_R(2\omega)+I_L(2\omega)},$$8 periodicity, the sign reversal under $$\mathrm{DNC}(2\omega)=\frac{I_R(2\omega)-I_L(2\omega)}{I_R(2\omega)+I_L(2\omega)},$$9, and the reversal upon incidence-direction reversal (Zhang et al., 17 May 2026).

In achiral resonant metasurfaces that emit chiral TH light, the decisive element is modal interference at the nonlinear frequency. The TH field is expanded as

$$\mathrm{SHG\text{-}CD}= \frac{I_{\mathrm{RHC}}(2\omega)-I_{\mathrm{LHC}}(2\omega)} {(I_{\mathrm{RHC}}(2\omega)+I_{\mathrm{LHC}}(2\omega))/2} =2\times \mathrm{DNC}(2\omega).$$0

where the $$\mathrm{SHG\text{-}CD}= \frac{I_{\mathrm{RHC}}(2\omega)-I_{\mathrm{LHC}}(2\omega)} {(I_{\mathrm{RHC}}(2\omega)+I_{\mathrm{LHC}}(2\omega))/2} =2\times \mathrm{DNC}(2\omega).$$1 are eigenmode amplitudes and $$\mathrm{SHG\text{-}CD}= \frac{I_{\mathrm{RHC}}(2\omega)-I_{\mathrm{LHC}}(2\omega)} {(I_{\mathrm{RHC}}(2\omega)+I_{\mathrm{LHC}}(2\omega))/2} =2\times \mathrm{DNC}(2\omega).$$2 is an uncoupled contribution. Rotating the pump’s linear polarization angle changes the amplitude and phase with which the nonlinear current drives these channels. When the summed $$\mathrm{SHG\text{-}CD}= \frac{I_{\mathrm{RHC}}(2\omega)-I_{\mathrm{LHC}}(2\omega)} {(I_{\mathrm{RHC}}(2\omega)+I_{\mathrm{LHC}}(2\omega))/2} =2\times \mathrm{DNC}(2\omega).$$3- and $$\mathrm{SHG\text{-}CD}= \frac{I_{\mathrm{RHC}}(2\omega)-I_{\mathrm{LHC}}(2\omega)} {(I_{\mathrm{RHC}}(2\omega)+I_{\mathrm{LHC}}(2\omega))/2} =2\times \mathrm{DNC}(2\omega).$$4-components of the emitted TH field have equal magnitudes and a $$\mathrm{SHG\text{-}CD}= \frac{I_{\mathrm{RHC}}(2\omega)-I_{\mathrm{LHC}}(2\omega)} {(I_{\mathrm{RHC}}(2\omega)+I_{\mathrm{LHC}}(2\omega))/2} =2\times \mathrm{DNC}(2\omega).$$5 phase difference, the output becomes purely circular. This suggests that strong nonlinear chirality can be engineered in systems that are achiral in the linear regime, provided the nonlinear emission channels support the appropriate resonant phase relations (Lai et al., 8 Sep 2025).

4. Measurement, retrieval, and modeling

The experimental extraction of DNC in harmonic generation is typically helicity-resolved and wavelength-resolved. For THG circular dichroism in membrane metasurfaces, the procedure is explicit: generate RCP and LCP pump beams with a linear polarizer and achromatic quarter-wave plate; measure the transmitted THG in each detected circular polarization channel $$\mathrm{SHG\text{-}CD}= \frac{I_{\mathrm{RHC}}(2\omega)-I_{\mathrm{LHC}}(2\omega)} {(I_{\mathrm{RHC}}(2\omega)+I_{\mathrm{LHC}}(2\omega))/2} =2\times \mathrm{DNC}(2\omega).$$6, $$\mathrm{SHG\text{-}CD}= \frac{I_{\mathrm{RHC}}(2\omega)-I_{\mathrm{LHC}}(2\omega)} {(I_{\mathrm{RHC}}(2\omega)+I_{\mathrm{LHC}}(2\omega))/2} =2\times \mathrm{DNC}(2\omega).$$7, $$\mathrm{SHG\text{-}CD}= \frac{I_{\mathrm{RHC}}(2\omega)-I_{\mathrm{LHC}}(2\omega)} {(I_{\mathrm{RHC}}(2\omega)+I_{\mathrm{LHC}}(2\omega))/2} =2\times \mathrm{DNC}(2\omega).$$8, and $$\mathrm{SHG\text{-}CD}= \frac{I_{\mathrm{RHC}}(2\omega)-I_{\mathrm{LHC}}(2\omega)} {(I_{\mathrm{RHC}}(2\omega)+I_{\mathrm{LHC}}(2\omega))/2} =2\times \mathrm{DNC}(2\omega).$$9; form the totals $g_{\mathrm{SHG-CD}}$00 and $g_{\mathrm{SHG-CD}}$01; and compute

$g_{\mathrm{SHG-CD}}$02

Under fixed pump power, this ratio is robust; for variable pump power, the intensities are normalized to pump power and detection efficiencies before forming the ratio. The reported setup used a femtosecond OPO, wavelength tuning from $g_{\mathrm{SHG-CD}}$03 to $g_{\mathrm{SHG-CD}}$04, pump power fixed at $g_{\mathrm{SHG-CD}}$05 before the $g_{\mathrm{SHG-CD}}$06 objective for spectral scans, and circular-polarization analysis of the THG with a quarter-wave plate and linear polarizer (Tonkaev et al., 18 Aug 2025).

In achiral resonant metasurfaces, the emitted-field formulation is especially transparent. The measured or simulated TH fields are converted from the linear basis to the circular basis through

$g_{\mathrm{SHG-CD}}$07

with $g_{\mathrm{SHG-CD}}$08 and $g_{\mathrm{SHG-CD}}$09. The DNC is then the degree of circular polarization of the nonlinear emission. The accompanying modified temporal coupled-mode theory includes three explicitly accounted TH eigenmodes, four TH radiation channels, and an overlap integral between each eigenmode and the nonlinear current density $g_{\mathrm{SHG-CD}}$10, thereby connecting pump polarization to the chirality of the emitted TH field (Lai et al., 8 Sep 2025).

For twisted heterobilayers, the retrieval is conceptually simpler but phase-sensitive. The directly measured observables are the SHG intensities under LCP and RCP excitation, and the key modeling object is the complex effective nonlinear susceptibility of each layer, written as $g_{\mathrm{SHG-CD}}$11. The analytical dissymmetry factor emerges after substituting the coherent sums for the two excitation helicities. The amplitude ratio $g_{\mathrm{SHG-CD}}$12 and the intrinsic phase difference $g_{\mathrm{SHG-CD}}$13 determine whether interference is destructive for one pump helicity and constructive for the other, which is the condition for approaching $g_{\mathrm{SHG-CD}}$14 (Zhang et al., 17 May 2026).

In SHG-CD microscopy of collagen, the workflow is spatially resolved rather than channel resolved. RCP and LCP SHG images are acquired in rapid succession and combined per pixel through the paper’s SHG-CD normalization. Polarization purity is maintained with a Soleil–Babinet compensator adjusted at each wavelength to keep the ellipticity below $g_{\mathrm{SHG-CD}}$15 at focus, and excitation power is stabilized at $g_{\mathrm{SHG-CD}}$16 at the objective back aperture across wavelengths. The key strategy is spectral discrimination: resonant SHG-CD near $g_{\mathrm{SHG-CD}}$17 tracks molecular chirality, whereas broadband non-resonant SHG-CD tracks macroscopic anisotropy (Chen et al., 2016).

The theoretical frameworks mirror these measurement geometries. Helicity-resolved THG in silicon membranes is modeled in COMSOL Multiphysics using the Wave Optics module, Floquet boundary conditions, a PML-treated substrate, and a domain polarization source built from the silicon $g_{\mathrm{SHG-CD}}$18 tensor. Achiral THG metasurfaces are modeled with eigenmode expansion and modified temporal coupled-mode theory. Twisted heterobilayers use a layer-resolved coherent-interference model. The common structure is that DNC is never extracted from total nonlinear power alone; it is extracted after decomposing the nonlinear response into symmetry-resolved or helicity-resolved channels (Tonkaev et al., 18 Aug 2025, Lai et al., 8 Sep 2025, Zhang et al., 17 May 2026).

5. Extensions beyond standard harmonic circular dichroism

The DNC concept extends well beyond SHG and THG circular dichroism. In an ultrahigh-$g_{\mathrm{SHG-CD}}$19 whispering-gallery microresonator, chirality is not measured from optical helicity but from spontaneous Kerr-induced imbalance between clockwise and counterclockwise waves. The reported experimental chirality metric is the power ratio $g_{\mathrm{SHG-CD}}$20, and the natural DNC is the corresponding normalized imbalance. The antisymmetric standing-wave mode loses stability at the Kerr-induced bifurcation threshold $g_{\mathrm{SHG-CD}}$21, which occurred at an input of about $g_{\mathrm{SHG-CD}}$22 for $g_{\mathrm{SHG-CD}}$23. Before the onset of four-wave mixing, the maximum observed output ratio was approximately $g_{\mathrm{SHG-CD}}$24, corresponding to

$g_{\mathrm{SHG-CD}}$25

The sign and magnitude of DNC in this setting quantify spontaneous parity breaking rather than circular polarization selectivity (Cao et al., 2016).

Synthetic chiral light introduces a different generalization, in which the central object is the local electric-field trajectory rather than the polarization state of a monochromatic beam. In non-collinear $g_{\mathrm{SHG-CD}}$26–$g_{\mathrm{SHG-CD}}$27 fields, the local handedness is quantified by chiral correlation functions such as

$g_{\mathrm{SHG-CD}}$28

and global chirality and polarization of chirality are quantified by spatial moments $g_{\mathrm{SHG-CD}}$29 and $g_{\mathrm{SHG-CD}}$30. In this framework, an intensity-type DNC is the dissymmetry factor

$g_{\mathrm{SHG-CD}}$31

which reaches the ultimate limit $g_{\mathrm{SHG-CD}}$32 when the achiral and chiral pathways are balanced and interfere constructively for one enantiomer while canceling for the other. A direction-type DNC is encoded in the enantio-sensitive average emission angle and is maximized when the field is chirality-polarized rather than globally chiral (Rego et al., 2022). Closely related work on polarization of chirality defines a DNC-like response directly from left-versus-right emission,

$g_{\mathrm{SHG-CD}}$33

and reports chiral dichroism in emission direction reaching its highest possible value of $g_{\mathrm{SHG-CD}}$34 for even harmonics (Ayuso et al., 2020).

In fluctuation-enabled nonlinear spectroscopy, DNC can probe temporal chirality rather than static structure. For a perylene bisimide dyad that is achiral at equilibrium, the linear chiral response vanishes upon angular averaging, but certain two-dimensional nonlinear chiral signals remain finite because they depend on products such as $g_{\mathrm{SHG-CD}}$35, which are controlled by variance and two-time correlations rather than mean geometry. A practical DNC is then defined as the ratio of the purely chiral 2D signal to a standard achiral 2D signal,

$g_{\mathrm{SHG-CD}}$36

with the short-time amplitude scaling as $g_{\mathrm{SHG-CD}}$37 and the waiting-time dependence decaying approximately as $g_{\mathrm{SHG-CD}}$38. Here DNC is a fluctuation-correlation observable (Mann et al., 2014).

Nonlinear transport in chiral solids provides yet another extension. In two-dimensional Te, a DNC can be defined from the second-order longitudinal nonreciprocal coefficient $g_{\mathrm{SHG-CD}}$39 or from the nonlinear planar Hall coefficient $g_{\mathrm{SHG-CD}}$40,

$g_{\mathrm{SHG-CD}}$41

The reported symmetry relation is $g_{\mathrm{SHG-CD}}$42 and $g_{\mathrm{SHG-CD}}$43, so the ideal DNC approaches unity in magnitude. In topological chiral semimetals, a formally similar DNC is defined from the intrinsic nonlinear planar Hall conductivity $g_{\mathrm{SHG-CD}}$44 of opposite enantiomers, and in the single-node-dominated limit the sign of DNC tracks the topological charge $g_{\mathrm{SHG-CD}}$45 through $g_{\mathrm{SHG-CD}}$46 (Niu et al., 2022, Pan et al., 2024).

6. Interpretation, limits, and recurrent misconceptions

A recurrent misconception is that large nonlinear chirality requires large linear chirality. The opposite conclusion is repeatedly demonstrated. In linearly achiral silicon membrane metasurfaces, strong nonlinear DNC is obtained despite modest linear CD because out-of-plane mirror symmetry and in-plane rotational symmetry suppress linear chiroptics while leaving nonlinear selection rules highly asymmetric (Tonkaev et al., 18 Aug 2025). In achiral resonant metasurfaces, linearly vanishing circular dichroism and circular conversion dichroism coexist with DNC spanning nearly the full $g_{\mathrm{SHG-CD}}$47 range in TH emission (Lai et al., 8 Sep 2025). In planar metasurfaces governed by meta-atom rotation, strong nonlinear chirality arises without invoking intricate three-dimensional architectures, because resonant mode symmetry and channel selection alone can enforce large co-handed THG asymmetry (Hariharan et al., 16 Apr 2026).

Another misconception is that DNC is intrinsically a structural property. In fact, the cited literature shows three distinct origins: structural chirality, as in twisted heterobilayers and chiral plasmonic nanoparticles; dynamically generated chirality, as in Kerr microresonators and polarization-shaped THG metasurfaces; and locally structured chirality of the driving field, as in synthetic chiral light. This suggests that DNC is better understood as a property of a nonlinear process and its symmetry channels than as a property of geometry alone (Zhang et al., 17 May 2026, Cao et al., 2016, Rego et al., 2022).

Interpretation of the sign requires care. In the THG membrane literature, negative DNC indicates that the total THG yield is larger for LCP pumping than for RCP pumping when the output is summed over detected polarizations. In emitted-field formulations, positive DNC means the nonlinear emission itself is more RCP than LCP. In SHG dissymmetry-factor conventions, the same physical asymmetry may appear with an extra factor of $g_{\mathrm{SHG-CD}}$48 or the opposite sign because the numerator is written as $g_{\mathrm{SHG-CD}}$49 instead of $g_{\mathrm{SHG-CD}}$50. Cross-paper comparison therefore requires explicit conversion of normalization and sign convention (Tonkaev et al., 18 Aug 2025, Chen et al., 2016, Zhang et al., 17 May 2026, Spreyer et al., 2022).

The achievable magnitude of DNC depends on the chosen normalization. Difference-over-sum forms based on two positive intensities are bounded by $g_{\mathrm{SHG-CD}}$51, and several experiments approach that bound: $g_{\mathrm{SHG-CD}}$52 and $g_{\mathrm{SHG-CD}}$53 in silicon membrane THG, $g_{\mathrm{SHG-CD}}$54 to $g_{\mathrm{SHG-CD}}$55 in resonant achiral THG metasurfaces, and approximately $g_{\mathrm{SHG-CD}}$56 in spontaneous microresonator chirality (Tonkaev et al., 18 Aug 2025, Lai et al., 8 Sep 2025, Cao et al., 2016). Dissymmetry-factor forms with an explicit prefactor of $g_{\mathrm{SHG-CD}}$57 are bounded by $g_{\mathrm{SHG-CD}}$58, and values as large as $g_{\mathrm{SHG-CD}}$59 have been reported in twisted heterobilayers (Zhang et al., 17 May 2026). Directional chiral dichroism defined as $g_{\mathrm{SHG-CD}}$60 reaches $g_{\mathrm{SHG-CD}}$61 when one side of the far field is completely extinguished for one enantiomer (Ayuso et al., 2020).

The main experimental limitations are likewise system-dependent but structurally similar. Helicity-resolved harmonic DNC requires accurate polarization calibration, stable pump power, and correction for detection efficiencies; small residual symmetry breaking, substrate effects, or fabrication imperfections can generate finite nominally forbidden channels (Tonkaev et al., 18 Aug 2025). In SHG-CD microscopy, anisotropy, birefringence, and orientation distributions can produce substantial non-resonant background, which is why spectral scanning and denaturation controls are necessary to isolate molecular chirality (Chen et al., 2016). In twisted-layer systems, the extreme phase sensitivity that enables $g_{\mathrm{SHG-CD}}$62 also makes the response vulnerable to twist-angle disorder, local strain, and spatial inhomogeneity (Zhang et al., 17 May 2026). In Kerr microresonators, four-wave mixing at higher power degrades the chiral power ratio and reduces DNC (Cao et al., 2016).

Taken together, these works establish DNC as a family of normalized nonlinear-chirality observables rather than a single immutable formula. The invariant content across implementations is the same: DNC measures how strongly a nonlinear process discriminates between opposite handedness channels after appropriate normalization. What differs are the channels themselves—pump helicities, emitted helicities, spatial emission directions, propagation directions, fluctuation-correlated response functions, or enantiomer-resolved transport tensors—and the symmetries that control them.