Terahertz Fingerprint: Phase Space & Cole-Cole View

Sample Molecule

THz Probe

Frequency ω/2π 1.00 THz

Field amplitude E₀ 1.00

Sample temperature 220 K

Live Readout

Frequency1.00 THz

Absorption α(ω)0.00

Transmission100%

How it works

This is a 2D dynamical-systems counterpart of the 3D fingerprinting simulator: same physics, same numbers, a different — genuinely 2D — way of seeing it. Each molecule still carries three THz-active vibrational modes, each a driven, damped harmonic oscillator:

x_i'' + γ_i x_i' + ω0_i² x_i = S_i E₀ cos(ωt)

Left disk — phase space (x vs v): instead of animating dipoles in a 3D slab, each mode's own equation of motion is integrated live and plotted on its own position–velocity plane. Off resonance the trajectory collapses to a tiny loop; sweep the probe near a mode's ω0 and the loop swells into a wide limit-cycle ellipse — the phase-space signature of resonance, autoscaled per mode so both weak and strong responses stay visible.

Right disk — Cole-Cole plot: the same Lorentz model has an exact closed-form complex susceptibility

χ_i(ω) = S_i / (ω0_i² − ω² + iγ_i ω)
χ'(ω)  = S(ω0²−ω²) / D,   χ''(ω) = Sγω / D,   D = (ω0²−ω²)² + (γω)²
α(ω) = ω · χ''(ω)   (identical to the 3D engine's absorption sum)

Tracing (χ', χ'') as ω sweeps 0.1–3.2 THz draws the classic arcs real dielectric spectrometers plot — one loop per resonance, its diameter set by the mode's coupling strength S and its width by damping γ. The bright dot marks the current probe frequency's exact position on that curve, so you can watch it round each resonance arc as you sweep.

Sample molecule — swaps in a different (ω0, γ, S) triple per mode, reshaping both the phase portraits and the Cole-Cole arcs.
Frequency slider / Auto Sweep — scans ω; both panels update from the same live number.
Field amplitude — scales the phase-space loop size (linear response); the Cole-Cole curve itself is amplitude-independent, exactly as the underlying math predicts.
Temperature — raises γ_i, shrinking and widening each Cole-Cole arc and damping the phase-space loops faster — the real reason cooled samples give sharper THz fingerprints.

Resonance frequencies are illustrative, literature-typical values per substance class, not a specific lab measurement — identical parameters to the 3D version, so the two are directly comparable.

Phase space (x vs v)Cole-Cole: χ'(ω) vs χ''(ω)

Mode 1 Mode 2 Mode 3 Current probe ω