Second-Harmonic Generation & Phase Matching

Nonlinear Crystal (χ²)

Pump strength (κ · mm⁻¹) 0.40

Crystal length L (mm) 6.0

Phase mismatch Δk (rad/mm) 0.00

Live Readouts

SH conversion η

0.0%

Fundamental left

100.0%

Coherence length L_c

∞

κL

2.40

How it works

In a medium with second-order nonlinear susceptibility χ², an intense fundamental beam at frequency ω drives a nonlinear polarization at 2ω, radiating a second-harmonic (SH) field. The coupled-wave equations for the fundamental amplitude A₁ and SH amplitude A₂ along the propagation axis z are:

dA1/dz = -kappa* A2 A1* exp(-i*Delta_k*z)
dA2/dz =  kappa  A1^2  exp(+i*Delta_k*z)

where Δk = 2k(ω) - k(2ω) is the phase mismatch between the fundamental and SH wavevectors, set by the crystal's dispersion (tuned in practice via crystal angle or temperature).

Two exact limits are shown here:

Perfect phase matching (Δk = 0): the exact energy-conserving solution is η(z) = tanh²(κz) — conversion grows monotonically and saturates as the pump depletes.
Nonzero mismatch: in the low-conversion (undepleted-pump) limit, η(z) ∝ (κz)² sinc²(Δkz/2) — energy flows back and forth between the two beams with period 2π/Δk, the classic phase-matching interference pattern.

This simulator blends the two via η(z) = tanh²(κz·sinc(Δkz/2)), which reduces exactly to each limit above and stays energy-bounded for any pump strength — a standard way to visualize both effects at once. The coherence length L_c = π/|Δk| is the crystal length beyond which conversion starts reversing; real SHG crystals (e.g. KTP or BBO converting a 1064 nm Nd:YAG laser to 532 nm green) are angle- or temperature-tuned to keep Δk ≈ 0 across the whole crystal.

κ slider — pump intensity × crystal nonlinearity (χ², d_eff); higher κ converts light faster per millimetre.
L slider — physical crystal length; only the beam profile up to z=L is amplified, it is frozen (locked-in) beyond the crystal exit.
Δk slider — phase mismatch; 0 is ideal, larger |Δk| shortens the coherence length and caps the peak conversion.

Fundamental (ω, e.g. 1064 nm) Second harmonic (2ω, e.g. 532 nm) χ² nonlinear crystal

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