When nonlinear optical media are exposed to intense laser fields, strong nonlinear optical phenomena occur, such as strong field multi-photon absorption,^[1,2] high-order harmonic generation,^[3–5] supercontinuum generation,^[6,7] optical power limiting,^[8–15] self-induced transparency,^[16–18] complete population inversion,^[19–21] and even creation of particles.^[22] It is well known that laser pulse can be characterized by its polarization, field amplitude, carrier temporal profile, pulse duration, carrier frequency, and carrier-envelope phase. Nonlinear optical processes can be coherently controlled by optimizing these parameters.^[23–30] For example, shaped pulses have been used to enhance the transient population of excited states and to obtain maximum population inversion;^[23–26] the anti-Stokes and Stokes generations can be tailored into a variety of forms in Raman-resonant four-wave-mixing process by manipulating the phase relationship among relevant electromagnetic fields;^[27,28] the propagation dynamics of a femtosecond chirped Gaussian pulse and the population distribution in a dense three-level Λ-type atomic medium can be manipulated by changing the sign and size of the chirp coefficient;^[29] the robust population inversion against pulse area variation in a V-shaped three-level system can be created by resonant excitation with cross-polarized laser pulse pairs.^[30]

The effect of temporal pulse shape on the strong field quantum dynamics involving ultrashort few-cycle pulses has always been one of the focuses of attention of scientists.^[31–33] The attosecond electron dynamics and the ultrafast population transfer processes in few-state atomic and molecular systems can be coherently manipulated according to the requirements by using the ultrashort pulses of different field envelopes.^[34–36] Conover^[37] investigated the effects of pulse shape on strongly driven two-level system and demonstrated experimentally that relatively small changes in the pulse shape of nonresonant radiation can have a dramatic effect on the intensity dependence of the transition probability in a two-level system. Novitsky^[38] studied the propagation of subcycle pulses in a two-level medium and found the clear breakdown of the McCall–Hahn area theorem^[16,17] for the subcycle pulses of large enough areas, which depends also strongly on the pulse shape. Kumar et al.^[39] studied optical dipole force and coherence creation on a beam of neutral three-level Λ-like atomic system induced by femtosecond super-Gaussian pulses, and showed that with pulses of high order profiles one can achieve better control of the population transfer process over a large part of the cross sectional plane of the pulse.

In this article, strong nonlinear optical interaction between different orders of femtosecond hyper-Gaussian pulses and a cascade three-level molecular medium is studied. The propagation dynamic behaviors of the ultrashort hyper-Gaussian pulses are investigated by solving the full-wave Maxwell–Bloch equations with the finite-difference time-domain and predictor–corrector numerical method. By tuning the carrier-wave frequency of the field in two-photon resonance with transition between the ground and the uppermost state, optical power limiting behaviors depending on the temporal shape of the pulses are observed for the strong field hyper-Gaussian pulses. Due to the breakdown of the slowly varying amplitude approximation and the invalidity of the monochromatic field condition, the two-photon area theorem^[9,10,40,41] is found to be not accurate anymore. The full-wave Maxwell’s and optical Bloch equations that describe the interaction of a three-level system with strong laser fields are presented in Subsection 2.1. The definition of two-photon area and the two-photon area theorem are given in Subsection 2.2. The results of numerical simulations by solving the full-wave Maxwell–Bloch equations are discussed in Section 3, and our findings are summarized in Section 4.

The strong field nonlinear optical interaction between ultrashort hyper-Gaussian pulses and a cascade three-level system is studied by solving the Maxwell–Bloch equations (2)–(3) with an iterative predictor-corrector finite-difference time-domain numerical scheme. The temporal shape of the hyper-Gaussian laser field is written as F(t) = F₀ exp(−ln 2/2(t‒t₀),²ⁿ/(τ/2)²ⁿ), where F₀ is the peak value of the field amplitude, τ is the full-width at half-maximum (FWHM) of the temporal profile of the light intensity I(t) = cε₀F²(t)/2, and n is the order of the hyper-Gaussian pulse which determines the temporal shape of the pulse. The pulse has a Gaussian shape with n = 1 and the pulse shape tends to be rectangular with the increase of n. The carrier-wave frequency of the laser pulse is tuned in two-photon resonance with the two-photon absorption state, i.e. ω = ω₃₁/2. The energy levels and the dipole moments for the three-level system are taken from 4,4’-bis(dimethylamino) stilbene molecule^[11,44] with ω₂₁ = 3.46 eV, ω₃₁ = 4.27 eV, d₂₁ = 10.433 deb (1 deb = 3.33564 × 10⁻³ C·m), d₃₂ = 18.497 deb, and d_ii = 0. The molecules are in the ground state before the incidence of the laser pulse, namely . The relaxation rates γ₂₂ = 10⁹/s and γ₃₃ = 10¹²/s, and the rate of decoherence γ_ij = 10¹²/s. The molecular number density N = 7.0× 10¹⁹/cm³.

The propagation dynamic behaviors of the hyper-Gaussian pulses of different temporal profile orders are numerically simulated by solving the Maxwell–Bloch equations (2) and (3). Figure 1 shows the evolution of the output field intensity at the propagation distance z = 7.0 μm as a function of the input field intensity for three different orders of the hyper-Gaussian pulses (n = 1, 2, 3) with an equal initial FWHM τ = 5 fs. One can see obviously the two-photon absorption induced optical power limiting behaviors for all the three orders of the hyper-Gaussian pulses. By comparing the three lines shown in Fig. 1, one can also see that with the increase of the input field intensity, the optical limiting behavior breaks down earlier for the lower order pulse. On the other hand, the lower order pulse has a smaller output intensity of optical limiting. For n = 1, the optical limiting effect breaks down with a saturation output intensity when the input field intensity increases to 3.0 × 10¹² W/cm². For n=2, the optical limiting effect breaks down with a saturation output intensity when the input field intensity increases to 3.12 × 10¹² W/cm². For n=3, the optical limiting effect breaks down with a saturation output intensity when the input field intensity increases to 3.3 × 10¹² W/cm². The output intensities of optical limiting increase with the increase of n, but their differences decrease with the increase of n, and become invisible for high order pulses. That is because the rate of change of the pulse shape with respect to the pulse order n slows down with the increase of n.

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. (color online) Optical power limiting behaviors of strong hyper-Gaussian laser pulses with FWHM τ = 5.0 fs for three values of order n of incident hyper-Gaussian pulses: 1 (rectangle), 2 (circle), and 3 (triangle).

The optical power limiting behaviors of strong hyper-Gaussian laser pulses with FWHM τ = 7.5 fs and 10 fs are presented in Figs. 2 and Fig. 3, respectively. They have very similar behaviors. Both cases show that with the increase of the input field intensity, the optical limiting effect breaks down first for pulses of a small value of n, but with a small output intensity. The difference of output intensity of optical limiting between and is smaller than the difference between and . By comparing Figs. 1, 2, and 3, one can see that both the output saturation values and the damage thresholds of optical power limiting are smaller for pulses of a longer duration.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. (color online) Optical power limiting behaviors of strong hyper-Gaussian laser pulses with FWHM τ = 7.5 fs for three values of order n of the incident hyper-Gaussian pulses: 1 (rectangle), 2 (circle), and 3 (triangle).

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. (color online) Optical power limiting behaviors of strong hyper-Gaussian laser pulses with FWHM τ = 10 fs for three different values of order n of the incident hyper-Gaussian pulses: 1 (rectangle), 2 (circle), and 3 (triangle).

The above results show that the two-photon absorption induced optical power limiting behaviors of the ultrashort hyper-Gaussian pulses are consistent qualitatively with the two-photon area theorem (Eq. (5)). For example, the optical power limiting effect breaks down with initial two-photon areas around 1.5π. Since the two-photon area of a pulse is proportional to its pulse width τ and field intensity I( z,t ) according to Eq. (6), the output saturation intensity and the damage threshold intensity are smaller for pulses of a bigger τ. And for pulses of the same peak intensity I₀ and pulse duration τ, the pulse with a large n has a small input two-photon area θ₀, therefore, the intensity range of optical power limiting is broad for the pulse of a high order. However, due to invalidity of the slowly varying amplitude approximation and monochromatic field hypothesis, the two-photon area theorem is not accurate for the quantitative description of the evolution of ultrashort hyper-Gaussian pulses. For example, after the breakdown of the optical power limiting effect, the speed rate of field intensity growth is much slower for the ultrashort pulses than that given by the two-photon area theorem due to the generation of new field components during pulse propagation.

The evolutions of the two-photon area θ(z) (Eq. (6)) along the pulse propagation for different orders of the 5.0 fs incident hyper-Gaussian pulses are shown in Fig. 4. The initial two-photon areas are respectively 0.5π (Fig. 4 (a)) and π (Fig. 4 (b)). At the early stage of pulse propagation, the areas of the pulses drop drastically due to strong two-photon absorption, and after a short propagation distance of about z_c ∼ 4.0 μm the pulse areas then change slowly. For different orders of pulses, the areas have similar dependence on the propagation distance z during the fast decrease stage, but the differences become visible around and after the characteristic distance z_c. It is worth noting that although the results given by the two-photon area theorem Eq. (5) show the same trend of evolution, the rates of change of the pulse areas for the ultrashort hyper-Gaussian pulses as a function of propagation distance z are slower than that given by the two-photon area theorem, and the theorem gives much smaller area after the characteristic distance z_c. For example, the areas are about 0.04 for both initial areas of 0.5π and 1.0π at z = 17.5 μm by the area theorem. But the ultrashort hyper-Gaussian pulses give very different values, respectively 0.1π and 0.22π at z=17.5 μm for initial areas of 0.5π and 1.0π with n = 1, and these values increase with the increase of pulse order n. That is because the two-photon area theorem is derived for single-color field, and it neglects the relaxation dynamic behavior by the use of amplitude equations. But for the ultrashort laser pulses, we solve directly the full-wave Maxwell–Bloch equations (2)–(3) and take into account the phenomenological relaxation effect. Due to the strong nonlinear optical effects, new field components such as hyper-Ramans, odd harmonics and resonant fields starting from spontaneous emissions are generated during pulse propagation, and the consideration of these new fields neutralizes the two-photon area theorem and slows down the decrease of the pulse area. And for the pulses of a large n, the new fields are generated earlier than those of a small n, therefore there are stronger deviations from the two-photon area theorem.

Fig. 4.

Figure Option ViewDownloadNew Window

Fig. 4. (color online) Evolutions of the two-photon areas of the hyper-Gaussian pulses withτ = 5.0 fs. (a) θ0 = 0.5π, peak field intensities for n = 1 (rectangle), 2 (circle), and 3 (triangle) are respectively I0 = 0.808 × 1012 W/cm2, 0.866 × 1012 W/cm2, and 0.872 × 1012 W/cm2. (b) θ0 = π, peak field intensities for n = 1 (rectangle), 2 (circle), and 3 (triangle) are respectively I0 = 1.616 × 1012 W/cm2, 1.731 × 1012 W/cm2, and 1.744 × 1012 W/cm2. Results given by two-photon area theorem (Eq. (5)) are shown with star-dotted lines.

We have studied here the optical power limiting behaviors of strong femtosecond hyper-Gaussian pulses induced by two-photon absorption in a cascade three-level system. It is found that the optical power limiting behavior breaks down earlier for pulses of a lower order, but with a smaller output intensity of optical limiting. Since the rate of change of the pulse shape with respect to the pulse order n slows down with the increase of n, the difference between the output saturation intensities of optical limiting also decreases with the increase of n, and becomes invisible for high order pulses. With the decrease of the pulse duration, both the output saturation intensity and the power range of optical limiting increase. Although the two-photon absorption induced optical power limiting behaviors of the ultrashort hyper-Gaussian pulses can be described qualitatively with the two-photon area theorem, due to the generation of the new fields during propagation of the hyper-Gaussian pulse, the two-photon area theorem is not strict anymore, and the decrease of the pulse area along the propagation is much slower than that obtained from the two-photon area theorem.