2D companion to the 3D upconversion model: the identical Yb³⁺/Er³⁺ steady-state rate-balance equations, read off four analytic plots instead of a rendered nanocrystal cluster:
Yb³⁺ sensitizer population N₁ (²F₅/₂):
0 = σP·(N₀−N₁) − k₁N₁ − 2W·N₁²
Er³⁺ emitting population N₂ (⁴S₃/₂ / ⁴F₉/₂):
0 = W·N₁² − k₂N₂
⇒ I_UC = k₂N₂ = W·N₁²
- Tissue cross-section (top-left) — a side-on view of the pump beam entering tissue and attenuating exponentially, driving a nanocrystal cluster at the chosen depth; cluster colour and brightness track the live green/red emission mix.
- Power-law curve (top-right) — log-log I_UC vs. pump power for the current depth/doping, with dashed guides at the ideal two-photon slope n≈2 (low power) and the saturated slope n≈1 (high power, N₁→N₀). The live marker and the "power-law slope n" readout sit exactly on this curve.
- Depth attenuation (bottom-left) — detected green and red intensity vs. tissue depth at the current pump/doping settings. Red (⁴F₉/₂→⁴I₁₅/₂, ~660 nm) is attenuated by tissue far more slowly than green (⁴S₃/₂→⁴I₁₅/₂, ~540 nm) — the reason NIR-in/red-or-NIR-out UCNP probes are preferred for deep-tissue imaging.
- Sensitizer/activator populations (bottom-right) — live bars for N₁ (Yb sensitizer) and N₂ (Er emitter), plus the resulting green:red emission branching ratio, showing the mechanism the other three panels are computed from.
σ is the NIR absorption cross-section, P the local irradiance, k₁/k₂ spontaneous decay rates, and W the Yb→Er energy-transfer rate. Because N₂ depends on N₁², emitted intensity scales as I ∝ Pn with n ≈ 2 at low pump power, falling toward n ≈ 1 as the sensitizer population saturates at high pump power. Denser Er doping (lower Yb:Er ratio) increases Er–Er cross-relaxation, which preferentially feeds the red channel. An 808 nm pump is absorbed less by water (lower tissue heating) at the cost of a lower effective cross-section without Nd³⁺ co-sensitization, which is why it trades peak brightness for safety in living-tissue work.