Reduced models and design principles for half-harmonic generation in synchronously pumped optical parametric oscillators

We develop reduced models that describe half-harmonic generation in a synchronously pumped optical parametric oscillator above threshold, where nonlinearity, dispersion, and group-velocity mismatch are all relevant. These models are based on (1) an eigenmode expansion for low pump powers, (2) a simultonlike sech-pulse ansatz for intermediate powers, and (3) dispersionless box-shaped pulses for high powers. Analytic formulas for pulse compression, degenerate vs nondegenerate operation, and stability are derived and compared to numerical and experimental results.

Figure 1

Typical synchronously pumped OPO design. A PPLN waveguide OPA with an optical-fiber feedback loop is considered in the text.

Figure 2

Left: Plot of the output signal power $P_{a,\mathrm{out}}$ (in photons per round-trip) for a 2-mm crystal; no dispersion compensation (“free space”). Right: Resonance plots of the power spectrum $P\left(\lambda \right)$ for a 4-cm crystal, no dispersion compensation, with normalized pump values $p\equiv b/b_0=1.16$, 1.10, and 1.04 (top) and the spectrum for the GVD-compensated cavity at $p=1.08$ (bottom).

Figure 3

Top: CW gain $\left|\mathrm{\Delta }\right(\delta \omega \left)\right|$ as a function of $\delta \lambda =(-{\lambda }^2/2\pi c)\delta \omega$. Bottom: Plot of the power spectral density $P\left(\lambda \right)$ (in photons/nm), from simulation. The white curve shows the threshold condition $\left|\mathrm{\Delta }\right|=1$. The dashed line represents Eq. (13). PPLN OPO with $L=4$ cm, free space.

Figure 4

PPLN OPO, 1-m SMF-28e fiber (${\beta }_2=-1.58\times {10}^{-26}$ ${\text{s}}^2/\text{m}, {\beta }_3=1.10\times {10}^{-40} {\text{s}}^3/\text{m}$). This fiber overcompensates the PPLN GVD by a factor $|{\varphi }_2/{\beta }_{2}L|=3.5$

Figure 5

PPLN OPO, GVD-compensating fiber (${\varphi }_2=-{\beta }_{2}L=-4.49\times {10}^{-27} {\text{s}}^2, {\varphi }_3=5.14\times {10}^{-41} {\text{s}}^3$).

Figure 6

Illustration of gain clipping. A train of femtosecond pulses (blue) is amplified by a picosecond pump pulse (green).

Figure 7

Left: Dispersionless round-trip gain ${\mathrm{\Delta }}_{\max }\mathrm{\Gamma }\left(t\right)$ as a function of the pump power and time, given by Eq. (21). Right: Pulse power ${\left|a\left(t\right)\right|}^2$ for a PPLN waveguide; fiber lengths $L_f=0$ and 5 m (which overcompensates the GVD by a factor of 17.6).

Figure 8

Top: Shapes of eigenmodes $a_k\left(t\right)$ as a function of ${\varphi }_0$, PPLN OPO with $p=1.1$ and no fiber. The dark line represents the dispersionless gain $ln\left({\mathrm{\Delta }}_{\max }\right)+G\left(t\right)$. Bottom: Power spectra of eigenmodes $|a_k{\left(\omega \right)|}^2$; the dark line shows the CW gain $ln\left({\mathrm{\Delta }}_{\max }\right)+D\left(\delta \omega \right)$.

Figure 9

Plots of eigenmodes $a_k\left(t\right)$ and eigenvalues $g_k$ at pump $p=1.1$ as a function of the cavity phase ${\varphi }_0$ and fiber length $L$. Pulse widths not to scale between graphs.

Figure 10

OPO steady-state pulse shape just above threshold. Filled blue curve: numerical result. Solid green curve $({\varphi }_0\ne 0)$: Airy-function solution, (24) for the degenerate case and (33) for nondegenerate case. The envelope ${\overline{a}}_k\left(t\right)$ is plotted for the nondegenerate case. Solid red curve: $({\varphi }_0=0)$: hypergeometric result, (26).

Figure 11

Plot of OPO efficiency $\eta =P_{a,\mathrm{out}}/P_{b,\mathrm{in}}$ as a function of $p$ and ${\varphi }_0$, with cavity dispersion provided by various lengths of fiber $L_f$. “GVD-cancelled” refers to a fiber that compensates the dispersion of the ${\chi }^{\left(2\right)}$ medium. Contours are thresholds for the first 20 eigenmodes $a_k\left(t\right)$.

Figure 12

Left: Phase diagram of the two-mode model in terms of normalized parameters $\overline{J}=J/g_0$ and $\overline{g}=g_1/g_0$. Right: Typical phase-space plots corresponding to the four regions in the diagram.

Figure 13

Left: Two-mode-model phase plot for PPLN OPO with a 20-m fiber, ${\varphi }_0=0$. Right: Photon number plot for numerical simulation.

Figure 14

Photon number as a function of the pump amplitude. Darker-colored lines are eigenmode models with increasing $N$. The black line represents the numerical result.

Figure 15

Signal pulse shape, computed numerically (filled blue curve) and with the eigenmode theory (black line).

Figure 16

Resonance diagrams, computed numerically (final column) and with eigenmode theories of increasing $N$ (first four columns).

Figure 17

Plot of the simulton solution ${\left(2\tau \right)}^{-1/2}\text{sech}(t/{\tau }_{\mathrm{sech}})$ versus the Airy [Eq. (24)] and hypergeometric [Eq. (26)] eigenfunctions.

Figure 18

Near-threshold pulse shape computed numerically (filled curve), compared to the steady-state sech solution (black line).

Figure 19

Plot of the simulated pulse shape (filled curve) compared to the simulton steady state of Eqs. (90, 91, 92) (black line). The dashed black line shows relation (100).

Figure 20

Evolution of the pulse photon number $N_{\mathrm{ph}}\left(t\right)$ and centroid $T\left(t\right)$ for the sech-pulse model (dashed lines) and full numerics (solid lines). Four detuning values are shown. PPLN OPO, no fiber.

Figure 21

Plots of the steady-state photon number (left), centroid (center), and pulse width (right) as a function of the detuning $\ell$ and pump $p$. Numerical simulations (top) are compared with the sech-pulse model (bottom). The dashed black line represents Eq. (101). Simulations are for PPLN OPO, 4-m fiber.

Figure 22

Left: Plot of the pulse shape for the signal and its box-pulse approximation via Eq. (108), for ${\varphi }_0=0, p\in [1.0,1.5]$. The pump is shown as the bottom trace, in green. Right: Power spectrum for the same data, on a log scale.

Figure 23

Resonance diagrams for the box-pulse model [Eq. (111)] compared to the numerical result.

Figure 24

Time-domain pulse shapes as a function of the phase and power; numerical results.