Nonlinear crystal
A periodically poled crystal, MgO:PPLN or PPKTP, that doubles the frequency of light polarised along its z axis. The second harmonic leaves as a beam of its own, its power from Boyd and Kleinman's theory for the focusing, the poling period and the temperature, and the fundamental leaves depleted.
The model
Phase matching
The crystal is quasi-phase-matched, type 0: the fundamental and the harmonic are both polarised along the crystal’s z axis, which lies out of the bench plane (s converts) or in it (p converts). A crystal rolled about the beam by its Roll \(\rho\) turns its axis with it: the light along the turned axis converts, the rest passes, and the harmonic leaves polarised along it. So rolled \(90°\), an s crystal converts p light; rolled \(45°\), half the power of s light is along its axis. Light crossing the poled grating at an angle \(\theta\) inside it (the crystal turned, or tipped with the 3D view) goes \(L/\cos\theta\) through it and sees the grating’s wave vector shortened to \(K\cos\theta\); quasi-phase-matching forgives small angles, so a few tenths of a degree barely detune it. Tipped, its harmonic leaves from where the light leaves the crystal, along it, raised as a tipped plate raises it. Its extraordinary index against wavelength and temperature is from:
- MgO:LiNbO₃: Gayer et al., Appl. Phys. B 91, 343 (2008);
- KTiOPO₄: Kato and Takaoka, Appl. Opt. 41, 5040 (2002), with Emanueli and Arie’s temperature terms, Appl. Opt. 42, 6661 (2003).
The poling period \(\Lambda\) cancels the mismatch \(\Delta k = k_{2\omega} - 2k_\omega - 2\pi/\Lambda\) at one wavelength and temperature. Leave the period at 0 and it is chosen for the beam through it. Set it, and the temperature tunes the phase matching.
Conversion
A Gaussian beam’s harmonic is Boyd and Kleinman’s (J. Appl. Phys. 39, 3597, 1968):
\[P_2 = \frac{16\pi^2 d_\text{eff}^2\, L\, h}{\varepsilon_0 c\,\lambda^3 n_1 n_2}\, P_1^2, \qquad h(\sigma, \xi, \mu) = \frac{1}{4\xi}\left\lvert\int e^{i\sigma\tau}\,\frac{\mathrm d\tau}{1 + i\tau}\right\rvert^2,\]with the integral over the crystal. Here:
- \(\xi = L/b\) is the focusing, \(b\) the confocal parameter inside the crystal;
- \(\sigma = b\,\Delta k/2\) is the phase mismatch the poling leaves;
- \(\mu\) is where the waist sits.
At best focusing, \(\xi = 2.84\), \(h = 1.068\): aim for a crystal about 2.84 confocal parameters long. \(d_\text{eff}\) is 14 pm/V for MgO:PPLN and 9.5 pm/V for PPKTP.
The fundamental is used up as
\[P_2 = P_1 \tanh^2\!\left(\sqrt{\eta P_1}\right),\]with \(\eta\) the low-power efficiency above, so the conversion saturates and the fundamental leaves with what remains.
The beams
The crystal is a block of index \(n\) with two faces, anti-reflection coated (0.1 %) or bare, and a plane in its middle where the harmonic is made. The harmonic leaves as a beam of its own, at half the wavelength, from that plane. Separate it from the fundamental with a short-pass filter or a dichroic.
Settings
| Field | Default | Meaning |
|---|---|---|
length | 3 | mm: the aperture across the beam. |
nlLen | 20 | mm along the beam (10 to 50 is usual). |
material | "ppln" | "ppln" (MgO:PPLN, d_eff 14 pm/V) or "ppktp" (d_eff 9.5 pm/V). |
period | 0 | poling period, µm; 0 (the default) picks the best period for the beam through it. |
temp | 40 | °C: tunes the phase matching (PPLN is usually run hot, 40 to 200). |
axis | "s" | "s" (z axis out of the bench plane: s light converts) or "p". |
faces | "ar" | "ar" (end faces anti-reflection coated, 0.1%; the default) or "bare". |
From the tool's own tables, file format version 1: the names are those in a .jekray file.
Assumptions
- The crystal is thin to the rays: its index does not move the focus. A focus aimed at its middle in air lands further in, so aim \(L/(2n)\) past its entrance face.
- The light’s own absorption, walk-off and heating, and photorefractive damage, are left out.
- Only second-harmonic generation is modelled: no sum- or difference-frequency mixing, and no parametric oscillation.