Fig. 1. Numerical solutions of signal gain Is(z)/Is(0) (green solid lines) and idler intensity Ii(z)/Is(0) (red solid lines) and pump depletion ratio ηp (blue lines) versus crystal length z for QPCPA with (a) α=2  cm−1 and (b) α=5  cm−1, respectively. The corresponding analytical solutions calculated with Eqs. (2) and (5) are plotted as dashed and dotted lines, respectively, for comparison.

Fig. 2. Pump depletion curves (blue lines) for (a) OPCPA with α=0 and QPCPAs with (b) α=1.6  cm−1, (c) α=6  cm−1 and (d) α=10  cm−1, respectively. The red lines plot the evolution of idler energy (normalized to the initial pump energy).

Fig. 3. (a), (b), (c) Contour maps of pump depletion ηp versus the crystal length z/z0 and idler-absorption coefficient α/g0 for QPCPAs pumped by a spatiotemporal top-hat pulse. The contour lines for ηp are plotted with a resolution of ∼0.08. (d), (e), (f) The critical idler-absorption coefficients versus pump intensity Ip, seeding ratio Is(0)/Ip(0), and phase mismatch Δk, respectively. The blue solid circles and red open circles plot the results from numerical simulations and the empirical formula of Eq. (7), respectively.

Fig. 4. Pump depletion contour map versus the crystal length z and initial pump intensity Ip for (a) QPCPA with αi(z)=5.3  cm−1 and (b) OPCPA with αi(z)=0, respectively.

Fig. 5. (a) Contour map of the pump depletion ratio versus the crystal length z and idler-absorption coefficient α/g0 calculated for QPCPA pumped by a spatiotemporal Gaussian pulse. For other parameters, see Table 1. (b) Two different idler-absorption spectra and (c) their corresponding efficiency curves. The pump is assumed to be spatiotemporally top-hat in (b) and (c).

Fig. 6. Comparison of (a) idler-absorption coefficient, (b) signal gain, (c) pump depletion, and (d) idler intensity in OPCPA (solid lines) and QPCPAs with a constant absorption (dash dot lines), a step absorption function (dashed lines), and an exponential gradient absorption function (dotted lines), respectively.

Table 1. Default Simulation Parameters in This Letter