Energy level diagram of the CARS process. The left panel corresponds to the resonant signal generation: pump and Stokes pulses generate coherence between two vibrational levels, when they have a frequency difference which coincides with the Raman resonance Ω R . The probe pulse then induces the anti-Stokes signal. The right panel corresponds to the non-resonant background contribution: the non-resonant background is produced via an intermediate virtual state that does not reflect the resonant molecular energy level.
The resonant signal and non-resonant background with different phase shaping schemes for the probe pulse while keeping the pump and Stokes pulses unshaped in three-pulse CARS. The arctan(ω pr /Γ) phase (red solid lines), π-step phase (blue dashed-dotted lines), and TLP (black dashed lines) shaping schemes are shown together for comparison. The bandwidths of the pump, Stokes, and probe pulses are the same: Δ p = Δ s = Δ pr = Δ = 50 cm−1.
Optimal phase function (via the Covariance Matrix Adaptation - Evolutionary Strategy (CMA-ES) optimization method42,43) for the pump pulse in two-pulse CARS. The parameters: Δ p = Δ s = 50 cm−1,Γ = 4.8 cm−1. In the top panel, the red solid line is the optimal phase function of the pump pulse for maximal resonant signal intensity, the magenta dotted line corresponds to the arctan(ω p /Γ)/2 phase, and the blue dashed line is a linear phase profile. The bottom panel shows the outcome of optimal pulse (red solid line) and TLP (black dashed line) in the time domain.
The resonant signal and non-resonant background spectra with different phase shaping schemes for the pump pulse in two-pulse CARS. The optimal phase (red solid lines), phase(blue dotted lines), π step phase (magenta dashed-dotted lines), and TLP (black dashed lines) schemes are shown together for comparison. The parameters are the same as in Fig. 3.
Numerical optimal phase functions for achieving a maximal resonant signal with three-pulse CARS using different pulse bandwidths: Δ p = 125 cm−1, Δ s = 100 cm−1, Δ pr = 80 cm−1. The top and bottom panels correspond to the frequency and time domains, respectively. The phase (black solid lines) is also shown for comparison.
(Top panel): Numerical optimal phase function of the probe pulse for |P r |2 − k|P nr |2 with different weights k. The color of the lines indicates the value of k, which is represented in the color bar on the right corresponding to log10(k + 0.1). All the phase functions in this figure could significantly suppress the background. (Bottom panel): The Pareto surface for the optimization of signal enhancement and background suppression. With different weights k, the value of |P r |2 is bounded in [0.765, 0.828], while |P nr |2 is always much smaller than |P r |2.
(Top panel): The resonant signal (red dashed line), background (blue dotted line), and the whole CARS signal (black solid line) with the π step phase scheme; (Bottom panel): The π step phase profile of the probe pulse which steps about ω pr = 0. Parameters: Δ = 50 cm−1, Γ = 4.8 cm−1.
(Top panel): The resonant signal intensity |P r | and non-resonant background |P nr | intensity with the multi-π step phase scheme (red lines) and the time delay scheme (blue lines). (Bottom panel): The multi-π step phase profile (red solid line) and the time delay phase profile (blue dashed line).
Optimal phase profile (with the CMA-ES algorithm) for the maximization of I r − I nr . Because the numerical optimal phase function for the pump and Stokes pulses are zero phase, i.e., TLP, they are not shown in this figure. The top and bottom panels show the phase function and amplitude for the probe pulse (red solid lines) in the frequency and time domain, respectively. The unshaped TLP (blue dashed line) and quasi-optimal time delay (black dash dotted lines) schemes are also shown for comparison. Parameters: Δ = 50 cm−1 and Γ = 4.8 cm−1.
The resonant signal and non-resonant background with different phase shaping schemes for the probe pulse, while keeping pump and Stokes pulses unshaped. The optimal shaping scheme for maximal I r − I nr (red solid lines), quasi-optimal time delay scheme (black dashed-dotted lines), and TLP scheme (blue dashed lines) are shown together for comparison. Parameters are the same as in Fig. 9.
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