Intensity of parametric fluorescence pumped by ultrashort pulses

We investigate intensity of the parametric down conversion of ultrashort, ultraviolet pulse both in a low- and high-conversion regime. In the first regime, we develop a simple analytical expressions for the photon flux of the fluorescence. Numerical simulations using three-dimensional stochastic Wigner method provide results in the latter regime. We find that the perturbative approximation for the photon flux as a function of the wavelength of the signal and the observation angle, quantitatively describes even the nonperturbative regime. For short pump pulses the intensity of the fluorescence is highest at frequencies where the group velocities of the pump and the down-conversion are equal. For longer pump pulses this requirement is gradually relaxed. Additionally, for small pump beams, the intensity strongly depends on the spatial divergence between interacting fields.

Figure 1

Scheme of a measurement of the photon flux $\overline{n}(\lambda ,\alpha )$ of the spontaneous parametric fluorescence. The nonlinear crystal is pumped along the $z$ axis. Out of the emerging parametric fluorescence we observe a small portion at the wavelength $\lambda$ propagating at an angle $\alpha$ to the $z$ axis in the $x\text{−}z$ plane.

Figure 2

(Color online) Observation of a plane-wave monochromatic component of the fluorescence field, described by wave vector ${\kappa }_s$, leads to projection of the conjugate idler field into a pulsed form, described by wave vectors ${\kappa }_i$. The idler wave packet takes up all the frequency and wave vector span of the pump pulse, described by wave vectors ${\kappa }_p$.

Figure 3

(Color online) Angle ${\alpha }_s=\arcsin [c{k}^0\left(\omega \right)∕\omega ]$ at which perfectly phase-matched fluorescence at the wavelength $\lambda =2\pi c∕\omega$ emerges outside the crystal. Plot corresponds to type-I phase matching in BBO pumped at $2\pi c∕\left(2{\omega }_0\right)=400\mathrm{nm}$ for crystals cut at angles $\theta =29°$ (solid line), 31.3° (dashed line), 35° (dotted line), and 40° (dash-dot line).

Figure 4

(Color online) Square idler spatial displacement coefficient $\Delta {\rho }_{x^{\prime }}^2+\Delta {\rho }_{y^{\prime }}^2$ in BBO for four different crystal cut angles $\theta =29°$ (solid line), $\theta =31.3°$ (dashed line), $\theta =35°$ (dotted line), and $\theta =40°$ (dot-dashed line).

Figure 5

(Color online) Inverse of the group velocities of pump (dotted line), signal (solid line), and idler (dashed line) for following angles of the crystal setting: (a) $\theta =29°$, (b) $\theta =31.3°$, (c) 35°, and (d) 40°.

Figure 6

(Color online) Photon flux of the fluorescence $\overline{n}\mathbf{(}{\omega }_{\mathrm{obs}},k^0\left({\omega }_{\mathrm{obs}}\right),0\mathbf{)}$ along the perfect phase matching line plotted in Fig. 3 calculated using Gaussian approximation for $2\text{−}\mathrm{mm}$-long BBO crystal. Plots are made for four different crystal cut angles (a) $\theta =29°$, (b) 31.3°, (c) 35°, and (d) 40°. Various lines indicate different pump pulse parameters: solid lines correspond to ${\tau }_p=60\mathrm{fs}$, $w_p=80\mu \mathrm{m}$, dashed lines correspond to larger pump beam, ${\tau }_p=60\mathrm{fs}$, $w_p=160\mu \mathrm{m}$ and dotted lines correspond to longer pump pulses ${\tau }_p=120\mathrm{fs}$, $w_p=80\mu \mathrm{m}$.

Figure 7

Graphical illustration of the Wigner method.

Figure 8

The contour plot of the averaged photon flux of the fluorescence as a function of the angle $\alpha =\arcsin (c\mid k_x\mid ∕\omega )$ and the wavelength $\lambda =2\pi c∕\omega$ for a $2\text{−}\mathrm{mm}$-long BBO. Each row corresponds to different crystal cuts: $\theta =29°$ (first row), $\theta =31.3°$ (second row), $\theta =35°$ (third row), and $\theta =40°$ (fourth row). Each column corresponds to different pump parameters: column A is for ${\tau }_p=60\mathrm{fs}$, $w_p=80\mu \mathrm{m}$; column B is for ${\tau }_p=60\mathrm{fs}$, $w_p=160\mu \mathrm{m}$; and column C is for ${\tau }_p=120\mathrm{fs}$, $w_p=80$ $\mu \mathrm{m}$. The contour lines were drawn at 0.25, 0.5, and 0.75 of the maximum of each plot.