L26 · Match phase, exchange energy

Nonlinear Frequency Conversion

Exchange energy between coupled complex fields. Acquire an angle scan, change temperature, and compare pump depletion with finite-beam overlap.

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{\%}

Capture matched conversion, then miss the angle or change temperature. In sum-frequency mixing, a longer crystal can send converted energy back into the inputs.

A lossless, forward-collinear type-I BBO teaching model for second harmonic, sum frequency and seeded difference frequency. Two lower-frequency ordinary waves couple to one higher-frequency extraordinary wave, with all participating fields depleted. The internal propagation direction defines depth; the angle is between wave vector and optic axis, not an external mechanical mount angle. Each process chooses an exactly energy-conserving frequency triplet. A Sellmeier fit supplies principal indices; an illustrative wavelength-independent linear thermo-optic correction uses manufacturer coefficients around a nominal twenty-degree reference. The angle offset is measured from the calculated twenty-degree phase-match angle. This thermal model is not a measured wide-temperature calibration. The supported wavelength window is 220–2200 nanometres; impossible difference-frequency inputs or absent type-I matching return an explicit unsupported state. The effective nonlinear coefficient is a user-set scalar, not a solved tensor or azimuthal dependence. Field amplitudes use photon-weighted megawatt normalization and fourth-order propagation; second harmonic uses the required degenerate-wave factor. Energy and Manley–Rowe residuals are reported without renormalizing the fields. A prescribed common Gaussian-mode overlap reduces coupling as the extraordinary mode walks off; it does not solve transverse deformation, diffraction, focusing, group-velocity mismatch, temporal pulse shapes or spatially resolved conversion. The equivalent uniform mode area is defined by the radius, and the intensity is a diagnostic under that convention. Disabling walk-off gives a plane-wave comparison. Difference frequency starts with a high-frequency pump and one lower-frequency coherent seed; the empty partner is generated classically. This is not spontaneous parametric down conversion or vacuum noise. The phase control changes the partner or signal seed phase; with an initially empty generated wave it changes output phase, not efficiency. Crystal coatings, absorption, thermal gradients, damage and real dichroic throughput are omitted. Data rows record the propagated powers and rerun the same depleted-field model at 61 nearby internal angles; no ideal sinc curve is substituted for acquired data. Internal depth is a model diagnostic, not an insertable physical detector. Apparatus dimensions and wavelength routing are schematic.

Start with the essentials

Focus question: Does a longer crystal always make more new light?
One-sentence intuition: Phase matching lets newly generated amplitudes add coherently, while depletion permits energy to flow back.

Core mathematical model

Choose an allowed frequency triplet

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

Second harmonic uses two identical fundamental photons. Difference frequency starts with wave three and a coherent lower-frequency seed.

Define 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 is extraordinary; the two lower waves are ordinary. The reference angle comes from the BBO Sellmeier fit.

Propagate all 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}

All waves can deplete. For second harmonic, replace the two low-frequency waves by one; the high-wave equation has half the coupling coefficient.

Count energy and photon flow

\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 apply to distinct waves. In second harmonic the invariant is fundamental photon flow plus twice harmonic photon flow.

Separate phase matching from overlap

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

This prescribed common Gaussian overlap is a scalar finite-beam approximation. It is not a solution for diffracting or distorted transverse fields.

Check an independent second-harmonic limit

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

The lossless matched plane-wave limit is checked independently against the numerical propagation. At weak conversion and nonzero mismatch, the familiar squared-sinc limit emerges.

Common difficulties

Color identifies the wavelength

Typical misconceptionThe displayed colors are the actual color of every beam.

Better mental modelColors identify model channels, including infrared. The wavelength readouts define the physical channels.

The angle scan is a measured ideal curve

Typical misconceptionAn ideal phase-matching curve can substitute for detector data.

Better mental modelEach recorded point reruns the depleted-field model. These are simulated powers, not hardware measurements.

Run the experiment

01
Acquire the actual model response
Capture matched second harmonic. Move the depth diagnostic without changing optical settings.
What to observe: The stored trace remains fixed while the live diagnostic follows the same propagated fields.
02
Break coherent accumulation
Compare angle and temperature mismatch, then repeat the scan.
What to observe: An energy-allowed frequency does not guarantee phase matching.
03
Find conversion and back-conversion
Capture sum frequency and compare different crystal lengths; then try seeded difference frequency.
What to observe: All fields exchange energy. A longer crystal can reverse an earlier transfer while the invariants stay fixed.