L26 · Phase matching and energy exchange

Nonlinear Frequency Conversion

Does a longer crystal always make more new light? No. Phase mismatch can make light generated at different depths cancel. Even at matched angles, sum-frequency mixing can send energy back into the input waves. In matched second-harmonic generation from a single input, conversion approaches a limit; finite-beam walk-off can stop extra length from helping.

Watch input light fade as a new frequency grows inside the crystal. Compare the automatically acquired depth trace and angle scan, then try sum-frequency back-conversion or seeded difference mixing.

Selected process $\text{—}$

Ordinary wave 1 wavelength $\text{—}$

Ordinary wave 2 wavelength $\text{—}$

Extraordinary wave 3 wavelength $\text{—}$

Generated channel wavelength $\text{—}$

Actual internal optic-axis angle $\text{—}$

Wave-vector mismatch $\text{—}$

Coherence length $\text{—}$

Prescribed walk-off angle $\text{—}$

Exit-mode transverse displacement $\text{—}$

Selected diagnostic depth $\text{—}$

Live wave 1 power $\text{—}$

Live wave 2 power $\text{—}$

Live wave 3 power $\text{—}$

Generated output power $\text{—}$

Generated power / input pump power $\text{—}$

Generated-wave phase at depth $\text{—}$

Equivalent-area pump intensity $\text{—}$

Maximum energy-balance error $\text{—}$

Maximum photon-flow invariant error $\text{—}$

Captured angle scan points $\text{—}$

Captured generated output $\text{—}$

Model status $\text{—}$

L26

Does a longer crystal always make more new light?

Process: second harmonic / sum / difference

m

0

Pump wavelength

\lambda_p

800\,\mathrm{nm}

Partner or signal seed wavelength

\lambda_s

1064\,\mathrm{nm}

Input pump power

P_p

10\,\mathrm{MW}

Partner or signal seed power

P_s

3\,\mathrm{MW}

Crystal length

L

2\,\mathrm{mm}

Internal angle offset from reference

\delta\theta

0\,\mathrm{{}^\circ}

Crystal temperature

T

20\,\mathrm{{}^\circ C}

Equivalent mode radius

w

0.5\,\mathrm{mm}

Assumed effective nonlinear coefficient

d_{\rm eff}

2\,\mathrm{\frac{pm}{V}}

Walk-off: plane wave / finite overlap

\eta_\rho

1

Partner or signal seed phase

\phi_s

0\,\mathrm{{}^\circ}

Diagnostic depth fraction

z/L

100\,\mathrm{\%}

Start with the automatically recorded matched conversion, then detune its internal angle or change the temperature in Advanced controls. In sum-frequency mixing, a longer crystal can send converted energy back into the input waves.

This is a teaching model of a lossless type-I BBO crystal, with all beams traveling forward along one line. It covers second harmonic, sum frequency and seeded difference frequency. Two lower-frequency ordinary waves couple to one higher-frequency extraordinary wave, and every participating field can be depleted. Depth is measured along the propagation direction inside the crystal. The angle is the one between the wave vector and the optic axis, not the angle of an external mount. Each process uses three frequencies that conserve energy exactly. A Sellmeier fit gives the principal refractive indices. An illustrative linear thermo-optic correction, the same at every wavelength, uses manufacturer coefficients around a nominal reference of twenty degrees. The angle offset is measured from the phase-matching angle calculated at twenty degrees. This thermal model is not a calibration measured over a wide temperature range. Wavelengths from 220 to 2200 nanometres are supported; if a difference-frequency input is impossible or no type-I matching exists, the Lab reports an unsupported state. You set the effective nonlinear coefficient as a single number; the tensor and its azimuthal dependence are not solved. Field amplitudes are normalized to megawatts with photon weighting and propagated with a fourth-order method; second harmonic includes the factor required for degenerate waves. Energy and Manley–Rowe residuals are reported without renormalizing the fields. A prescribed Gaussian-mode overlap reduces the coupling as the extraordinary mode walks off. It does not model transverse deformation, diffraction, focusing, group-velocity mismatch, pulse shapes in time or spatially resolved conversion. The radius defines an equivalent uniform mode area, and the intensity is a diagnostic based on that convention. Switching off walk-off gives a plane-wave comparison. Difference frequency starts from a high-frequency pump and one coherent lower-frequency seed, and the initially empty partner wave is generated classically. The model does not include spontaneous parametric down-conversion or vacuum noise. The phase control sets the phase of the partner or signal seed. When the generated wave starts empty, this changes the output phase but not the efficiency. Crystal coatings, absorption, thermal gradients, damage and the real throughput of dichroic optics are left out. The data rows record the propagated powers and rerun the same depleted-field model at 61 nearby internal angles, so the scan is not an ideal sinc curve. The internal depth readout is a model diagnostic and does not stand for a detector placed inside the crystal. Apparatus dimensions and wavelength routing are schematic. Continuous wave powers are rendered with slow flow markers; these are not temporal pulse solutions. Field radius follows the selected equivalent mode radius, and color labels the wavelength channel. Optical changes acquire a fresh trace and scan automatically.

Start with the essentials

Focus question: Does a longer crystal always make more new light?
One-sentence intuition: Phase matching lets generated fields add in step; in sum-frequency mixing, depletion can lead to back-conversion.

Core mathematical model

Frequency sum and wave-vector mismatch

\begin{aligned}\omega_3&=\omega_1+\omega_2\\\Delta k&=k_3-k_1-k_2\end{aligned}

In second-harmonic generation, the two lower-frequency photons are identical fundamental photons. Difference-frequency generation starts from wave three plus a coherent lower-frequency seed.

Extraordinary index at the internal optic-axis angle

\frac{1}{n_e(\theta)^2}=\frac{\cos^2\theta}{n_o^2}+\frac{\sin^2\theta}{n_e^2}

The high-frequency wave travels as an extraordinary wave and the two lower-frequency waves as ordinary waves. The reference angle comes from the Sellmeier fit for BBO.

Coupled equations for the three complex fields

\begin{aligned}a_1\prime&=i\kappa W a_2^*a_3e^{-i\Delta kz}\\a_2\prime&=i\kappa W a_1^*a_3e^{-i\Delta kz}\\a_3\prime&=i\kappa W a_1a_2e^{i\Delta kz}\end{aligned}

Every wave can be depleted. For second harmonic, the two low-frequency waves become a single wave, and the equation for the high-frequency wave takes half the coupling coefficient.

Power and the Manley–Rowe relations

\begin{aligned}P_j&=(\omega_j/\omega_r)|a_j|^2\\|a_1|^2+|a_3|^2&=C_1\\|a_2|^2+|a_3|^2&=C_2\end{aligned}

The two Manley–Rowe invariants hold when the waves are distinct. For second harmonic, the conserved quantity is the fundamental photon flow plus twice the harmonic photon flow.

Overlap loss from walk-off

\begin{aligned}\delta x(z)&=z\tan\rho\\W(z)&=\exp\!\left[-\frac{2\delta x(z)^2}{3w^2}\right]\end{aligned}

As the extraordinary beam walks off sideways, a prescribed Gaussian overlap factor reduces the coupling. This is a scalar approximation for finite beams; it does not solve for diffracting or distorted transverse fields.

Plane-wave limit for second harmonic

\begin{aligned}\eta&=\tanh^2\!\left(\frac{\kappa a_0L}{\sqrt2}\right)\\\Delta k&=0,\quad W=1\end{aligned}

For lossless, phase-matched plane waves, this closed form gives an independent check on the numerical propagation. With weak conversion and nonzero mismatch, the familiar sinc-squared curve appears.

Common difficulties

Colors are labels; readouts give wavelengths

Typical misconceptionEach beam really has the color shown on screen.

Better mental modelThe colors only label the model's channels, and some of those channels are infrared. The wavelength readouts tell you what each beam physically is.

Each angle-scan point reruns the full model

Typical misconceptionAn ideal phase-matching curve is as good as detector data.

Better mental modelEach recorded point reruns the model with depleting fields. The values are simulated powers, not measurements from real hardware.

Run the experiment

01
Compare a depth and its recorded scan
Choose Matched second harmonic. A depth trace and angle scan are acquired automatically; open Advanced controls to move the depth diagnostic.
What to observe: The diagnostic reads a different point on the same propagated solution. The stored trace and scan retain their original values.
02
Detune the internal angle
Drag the internal-axis dial or choose Angle mismatch. Temperature mismatch is in Advanced controls and additional demonstrations.
What to observe: The generated output falls and the depth trace can oscillate. The optical frequencies still conserve energy; the fields no longer add in step along the crystal.
03
Find energy flowing back
Choose Sum frequency and increase Crystal length from two to four millimetres. Then try Seeded difference frequency.
What to observe: Sum-frequency output can fall as energy returns to the lower-frequency waves. Seeded difference mixing instead drains the higher-frequency pump and amplifies the seed as it generates its partner.

Nonlinear Frequency Conversion

Light trades energy inside a crystal