Harmonics as a Geometry Test

Intuition

A symmetric response treats opposite field directions as opposites

Suppose the net material response reverses when the driving electric field reverses:

Then its simplest nonlinear expansion contains only odd powers:

Drive it with

The cubic term contains

So a third harmonic appears naturally while the second harmonic is absent from this symmetric electric-dipole response.

Break the symmetry and even harmonics can appear

If inversion symmetry is broken, an even term is allowed:

Because

the quadratic response produces a component at \(2\omega\).

The geometry question is deeper

A finite organised structure can contain many local responses. Some can be out of phase and cancel in the far field even if each local region contains the harmonic.

Titraj proposes that the actual constrained geometry of participating charge is the mechanism behind that addition and cancellation. If so, changing the geometry should change the harmonic fingerprint in a calculable way.

The Argument

The standard symmetry statement

Write a nonlinear polarisation schematically as

For an inversion-symmetric material in the electric-dipole approximation, reversing both material coordinates and field requires

which eliminates the even-order electric-dipole susceptibilities:

Odd-order nonlinearities remain allowed. Real finite systems can still produce even harmonics through surfaces, magnetic-dipole or electric-quadrupole terms, spatial dispersion, broken geometry or other symmetry-breaking mechanisms.

Local generation and net radiation are not the same

Let local harmonic sources be \(p_n(\mathbf r)\). The detected field at harmonic order \(n\) depends on their vector and phase sum:

A symmetry operation can pair regions whose even-order contributions cancel while odd-order contributions add. Thus

This distinction is especially useful for a mechanism built around organised many-body geometry.

The Titraj interpretation

If the organised charge path is a real microscopic object of the model, its symmetry should determine the relative phase and orientation of local current or dipole contributions.

A closed symmetric mode would then be expected to suppress some net even-order outputs, while truncation, asymmetric confinement, unequal local environments or oriented boundaries could release them.

That qualitative behaviour matches known symmetry physics. The mechanism is not yet distinctive until it predicts more detail.

The stronger prediction

Define a harmonic fingerprint

together with the polarisation and angular distribution of each component.

A completed Titraj geometry should map

without selecting the geometry after the spectrum is known.

The experimentally useful test is therefore to change one geometrical symmetry while keeping composition, band gap and excitation as controlled as possible, and predict in advance which harmonic channels open, close or rotate.

Deep Notes

Harmonic generation is an unusually useful test because it converts hidden internal symmetry into measurable frequencies, polarisations and directions. But the basic selection rules are not new. Standard nonlinear optics already connects harmonic order to inversion and rotational symmetry, and experiments on nanostructures show that changing finite geometry can enable or suppress harmonic channels.

The opportunity for Titraj is therefore not to rediscover that symmetry matters. It is to provide a concrete real-space mechanism for why the contributions cancel or survive, and eventually to derive the measured harmonic spectrum from the proposed organised charge motion.

That turns a qualitative antenna analogy into a quantitative obligation.

Odd powers from inversion symmetry

For a scalar illustration, assume

The Taylor expansion around zero field must then be odd:

For a sinusoidal drive, every odd power can be decomposed into odd multiples of the fundamental. For example,

This gives the familiar sequence \(\omega,3\omega,5\omega,\ldots\) in the simplest symmetric nonlinear response.

Even powers reveal broken inversion at electric-dipole level

When the response is no longer odd in \(E\), a quadratic term can appear:

Higher even powers similarly contain even harmonics. In real materials the full tensor symmetry, surface structure, multipole contributions and propagation geometry matter, so “centrosymmetric means absolutely no second harmonic” is too strong. The correct statement is narrower: bulk electric-dipole \(\chi^{(2)}\) vanishes under inversion symmetry, while other mechanisms can still generate SHG.

Why finite geometry is especially relevant

A finite nanostructure can break symmetries that exist in the underlying infinite crystal. Conversely, a symmetric finite object can cause locally generated even-order components to cancel in a chosen radiation direction.

This is already observed and modelled in nonlinear nanophotonics. It is therefore a strong empirical constraint on any Titraj path geometry: the proposed motion has to reproduce the same symmetry dependence.

From susceptibility tensors to a path model

The standard description packages the nonlinear response into tensors such as \(\chi^{(2)}\) and \(\chi^{(3)}\). A microscopic Titraj account would instead have to start from a constrained charge distribution and derive the current

then obtain its Fourier components

and finally the radiated field from their spatial phase sum.

Only then would statements about odd harmonics, polarisation and angular lobes follow from the mechanism instead of being added as verbal expectations.

A clean geometry experiment

Prepare structures of the same material with a controlled symmetry pair — for example a nearly inversion-symmetric geometry and a deliberately asymmetric variant — while characterising their linear spectrum and field enhancement.

Measure

for several harmonic orders. Standard nonlinear optics supplies the baseline symmetry and susceptibility prediction. Titraj must predict the same established channels and, if it is to add physics, a quantitative residual tied to its particular path geometry.

The falsifiable form

Let

A distinct Titraj prediction would have to specify

from the path geometry before fitting the data.

The boundary

The current portal can justify the qualitative expectation that organised symmetry affects which contributions cancel. It cannot yet calculate the absolute harmonic ratios. That quantitative derivation remains an explicit obligation of the local-path conjecture.