1. Introduction

With the rapid development in ultrafast laser technology [1,2], it has become more attractive to extend the study of the interaction between matter and light into extreme nonlinear regime [3]. The fundamental problem in this field is the interaction of few-cycle pulses with resonant two-level atomic (TLA) ensembles, which lays the foundation for understanding the more complex cases, such as the quantum dot ensembles [4], three-level atoms [5], and specifically designed structures [6].

The transmitted spectrum of few-cycle pulses propagating in vacuum to a TLA material has been studied extensively by the previous works [4–7]. However, few studies have investigated the reflected spectrum. This is because many previous works are based on the envelope form of the Maxwell equations, where forward propagation approximation is implied and the reflected field is ignored [8–10]. Nevertheless, if the density is high enough to allow the backpropagation to make a difference, the full Maxwell-Bloch (MB) equations [11–13] without rotating wave approximation (RWA) and slowly varying envelope approximation (SVEA) have to be integrated. To solve these equations, one usually uses Yee’s finite difference time-domain method (FDTD) assisted by the predictor-corrector method [14,15], or a pseudospectral operator-splitting method combined with a hybrid interface [16].

A dense TLA medium prepared in the ground state can be regarded as a saturable absorbing material. An intense pulse incident upon such a medium can build an absorption front within an optical wavelength [17]. If the induced spatial inhomogeneity is sufficiently sharp, this absorption front will act as an interface for self-reflection [18,19]. Since this reflector moves with the propagating pulse, a doppler redshift appears in the reflection spectrum. This is known as the dynamic nonlinear optical skin (DNOS) effect, which has been used to explain the slight redshift in reflected spectrum for long pulse incident on the interface [17]. However, for the few-cycle pulse, theoretical studies about the behavior of the reflected spectrum are far from complete. The existing theories on the origin of the redshift for the few-cycle case are confined to intrapulse four-wave mixing (FWM) [11, 12]. Whether absorption fronts exist in the medium, whether the Doppler redshift plays a role in the reflected spectrum, and in which way it is manifested still remain unsolved.

In this paper, we study the reflected field of a few-cycle pule propagating through a dense TLA medium by solving the MB equations without the rotating wave and slowly varying envelope approximations. We prove that absorption fronts exist even in the few-cycle cases and that the Doppler redshift induced by the resulting moving front contributes to the formation of the reflected spectrum. We also clarify how the manifestation of Doppler effect in the reflected spectrum depends on the dominant reflection mechanism, which in turn can be modified with laser duration, area, medium density and length.

2. Theoretical model and results

First, we consider the propagation of a few-cycle pulse along z direction through a resonant TLA. The MB equations take the form [20–22]

The macroscopic coherent polarization is P_x(z) = Ndu, where u is the real part of the off-diagonal density matrix element ρ₁₂ = (u + iv)/2. w = ρ₂₂ − ρ₁₁ is the population inversion between the excited state 2 and the ground state 1. The lifetime of the excited state T₁ = 1ps and the dephasing time T₂ = 0.5ps. The Rabi frequency Ω = dE_x/h̄ with dipole moment d = 2 × 10⁻²⁹ A s m. The carrier frequency is ω_p = 2.3fs⁻¹, equal to the transition frequency ω₀ between ground and excited states. The TLA medium is initially prepared in the ground state, w₀ = −1, u = v = 0. The medium length L = 45μm. The collective frequency parameter ω_c = Nd²/ε₀h̄ = 0.05fs⁻¹, where N = 1.1 × 10²⁰ cm⁻³ is the medium density. The incident pulse has a hyperbolic secant envelop, Ω(t = 0, z) = Ω₀ cos[ω_p(z − z₀)/c] sech[1.76(z − z₀)/(cτ_p)], where Ω₀ = 0.7fs⁻¹ is the peak Rabi frequency, τ_p = 5fs the full width at half maximum (FWHM), and z₀ = 26.25μm is used to avoid the initial disturbance from the input pulse. The corresponding envelope area A = Ω₀τ_pπ/1.76 = 2π.

The FDTD method assisted by the predictor-corrector method is used to solve the full MB equations [14, 15]. The reflected field and the corresponding spectrum obtained with the above parameters are shown in Fig. 1. The reflected field in Fig. 1(a) consists of a leading part produced by front surface reflection, and a tail generated by backpropagation waves. The reflected fields from the back surface and the remaining atomic oscillation are irrelevant to the Doppler redshift in question. Hence, the reflections corresponding to these two parts are omitted. Figure 1(b) shows that, besides the double-peak structure, a low-frequency spike (LFS) centered at 0.215ω₀ with an amplitude of 0.7×10⁻⁴fs⁻¹ appears at the red edge of the reflected spectrum. To find out the origin of the LFS, we take the respective Fourier transformations to the tail and leading parts. The results show that, this LFS and the double-peak structure correspond to the tail and leading part of the reflected fields, respectively. The spectrum of the reemitted field produced by macroscopic coherent polarization at z = 1.5μm is shown in Fig. 1(b) [23]. Apparently, it covers most of the frequency components in reflected spectrum except for the LFS. That is to say, nonlinear polarization is responsible for the spectral broadening of the double-peak structure, but has no contribution to the generation of the LFS.

 

Fig. 1 (a) The reflected field of a few-cycle pulse propagation through a dense TLA medium, which consists of a leading (thick line) and tail parts (thin line). (b) The corresponding reflected (solid line) and reemitted field (dashed line) spectra. (c) The velocity variations of the moving absorption front versus time and the corresponding frequencies obtained with $\omega /{\omega }_0=\frac{c-\upsilon }{c+\upsilon }$. The horizontal dashed line is the central frequency of the LFS, ω/ω₀ = 0.215, and the two horizontal solid linea represent its FWMH τ = 0.063ω₀.

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Nonlinear polarization may include many nonlinear effects, such as intrapulse FWM and self-phase modulation (SPM). According to the previous works on few-cycle pulse propagation, intrapulse FWM, which is ignored in the framework of RWA, is significant for very short pulses. It is widely accepted as the nonlinear effect that determines the spectral transformation due to the ultrashort feature of the few-cycle pulse [11]. Moreover, SPM may also contribute to the spectral widening. Refractive index does not change under RWA, but it changes with the carrier in the few-cycle case. Thus, SPM, which is an ignored nonlinear process in the former case, can be very important for the short pulse propagation. However, which nonlinear effects play the dominant role in the spectral broadening is determined by the laser and medium parameters. As for the parameters in Fig. 1, the SPM may be the dominant nonlinear effect that results in a symmetrical broadening, while for the parameters used in Fig. 5, as in [11], intrapulse FWM probably is the dominant nonlinear effect that results in redshift peaks excluding the Doppler redshift.

According to the dynamic nonlinear optical skin (DNOS) effect [17], when a long pulse is incident on a saturable medium, an absorption front is built up near the interface and its motion within the nonlinear skin depth produces a slight redshift that appears in the reflected spectrum. In contrast, for few-cycle pulses, the absorption fronts can travel beyond the nonlinear skin depth with a greater speed, as shown in the following. When the spatial structure of the self-reflected waves are short enough to yield constructive interference in the backwards direction, low-frequency backpropagation waves are generated. Therefore, velocity-dependent low-frequency components are expected in the reflected spectrum. To validate this physical picture, we have determined the absorption front velocity from the numerical simulation of the population inversion. The locations z_i at which the population inversion occurred at t_i are recorded with a time interval Δt = t_i+1 − t_i = 12.5fs. The average velocity is then obtained using ${\upsilon }_i=\frac{z_{i+1}-z_i}{\mathrm{\Delta }t}$. Figure 1(c) shows that the velocities of the absorption front of few-cycle pulse are relatively stable and distribute within 1.79 ∼ 2.1×10⁸m/s. The relation between velocity υ and frequency ω of the self-reflected waves fulfills $\omega =\frac{c-\upsilon }{c+\upsilon }{\omega }_0$, with which we get a frequency band [0.178ω₀, 0.253ω₀] corresponding to the velocity range. Surprisingly, the frequencies of backpropagation waves are exactly within the frequency band of the LFS, which justifies the our assumption that a Doppler effect inducing redshift in backwards propagation waves is the source of the LFS, as shown in Fig. 1(c). Thus, besides the FWM, the Doppler effect also plays an important role in the reflected spectra of the few-cycle pulses. In the following, we discuss how the laser and medium parameters influence the self-reflections, and how the Doppler effect manifests itself in the reflected spectrum.

3. Behaviors of Doppler shift with different laser and medium parameters

3.1. Influence of pulse duration

To further prove that the physical mechanism behind the LFS is unified with the DNOS effect, we study the reflected spectrum with the parameters in section 2, except for a variable duration τ_p. A pulse with duration τ_p = 10fs is used to study the influence of pulse duration on the Doppler redshift. The results are shown in the top of Fig. 2. It can be seen from Fig. 2(a) that the LFS produced by backpropagation waves disappears. When a 2π pulse propagates through a TLA medium, the energy transfer from the pulse leading edge to the trailing edge due to Rabi flop, which results in a slowdown of the propagating pulse. Obviously, the longer the incident pulse is, the more slowly it propagates [24]. Since the amount of Doppler redshift is proportional to the propagation velocity, the redshift of the LFS for a 10fs pulse decreases compared with that for a 5fs pulse. Moreover, as the pulse area is fixed, increasing duration τ_p is equivalent to reducing peak intensity. Since the condition for reflection to play a significant role is Ω ∼ ω_cu [25], 10fs pulse is more vulnerable to reflections. The dramatic reflection loss causes deceleration and prohibits pulse propagating through the medium. As shown in Fig. 2(b), both the energy and propagation velocity of the main pulse decrease until settling around 135μm. Compared with the case of 5fs pulse in Fig. 1(c), the absorption front of 10fs pulse travels more slowly and decelerates from 9.09×10⁷m/s to 8.16×10⁶m/s within 625fs, as shown in Fig. 2(c). According to the relation between velocity and frequency, the corresponding frequency band [0.53ω₀, 0.95ω₀] is obtained. This shows that the Doppler redshift no longer manifests itself as an isolated LFS but is merged with the spectrum of the reemitted field.

 

Fig. 2 (a)(d) The reflected spectrum. (b)(e) The spatial evolution of the absorption front. (c)(f) The time evolution of the absorption front’s velocity (circles) and the backpropagation wave’s frequency (squares). The top and bottom correspond to τ_p = 10fs and τ_p = 40fs, respectively. The dashed line in (d) is the spectrum of the incident pulse.

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As shown in Fig. 2(f), the propagation velocity of a 40fs pulse further decreases from 9.4×10⁶m/s to 1.44×10⁶m/s within 150fs. The corresponding frequency obtained by velocity-dependent theory ranges from 0.94ω₀ to 0.99ω₀. This frequency band is consistent with the LFS obtained by numerical simulation in Fig. 2 (d), in terms of its center frequency and FWHM. For a 40fs pulse, the nonlinear reflection near the front surface becomes so intense that the pulse barely propagates within the medium. This is demonstrated in in Fig. 2 (e), where the penetration depth is less than 2 μm. Therefore, when it comes to the reflection mechanism of a 40fs pulse, the bulk generation of backpropagation waves is trivial, while the nonlinear reflection near the interface becomes dominant, i.e., DNOS effect.

For pulses with a fixed area, the longer the pulse is, the more severely it will suffer from reflections. As the duration increases from 5fs to 40fs, both the nonlinear reflections near front surface and backwards propagation inside the medium increase. Consequently, the propagation velocity of the main pulse decreases, so does the velocity of the moving absorption front following it. According to the velocity-frequency relation, the frequency of the LFS increases until it merges into the redshift directly produced by nonlinear effects. At this point, the LFS disappears and the dominant reflection mechanism changes to be the nonlinear reflection near the front surface instead of the bulk generation of backward propagation waves. Thus, the underlying physics of the LFS is essentially consistent with that of the DNOS effect. That is, the Doppler redshift induced by a moving absorption front manifests itself in two forms: one is the LFS for short pulses, and the other is the DNOS effect for long pulses.

3.2. Influence of pulse area

Besides pulse duration, pulse area also has a significant influence on the behavior of the Doppler redshift. Here, we give a discussion about the behavior of the Doppler redshift with respect to different pulse areas. For a 3π pulse, the reflected spectrum, as shown in Fig. 3(a), consists of a strong spike centered at ω₀ and a LFS located at 0.054ω₀. Compared with that of a 2π pulse, the LFS of a 3π pulse has a lower frequency and a larger amplitude of 5.06×10⁻⁴fs⁻¹. According to the area theory, when a 3π pulse propagates through a TLA medium, its area decreases to 2π through splitting. Since the 2π pulse obtained in this way is much more intense and shorter than the incident 2π pulse, the former can propagate much faster than the latter, as shown in Fig. 3(c). Accordingly, the absorption front generated by the 3π pulse has a higher speed compared to that of the 2π pulse, which results in a larger redshift of the LFS. Figure 3(b) shows that, the velocity of the moving front ranges from 2.656×10⁸m/s to 2.7×10⁸m/s, and the corresponding frequency is within [0.052ω₀, 0.06ω₀]. The obtained frequencies are consistent with the frequency band of the LFS shown in Fig. 3(a). Meanwhile, since the amplitude of the reflection is proportional to the injected energy, the amplitude of the LFS is expected to increase. Moreover, after a complete Rabi flop, the remaining energy produces a small area pulse, which is supposed to propagate forwards following the main 2π pulse [22]. However, for the dense medium in question, the small area pulse suffers from dramatic reflections and propagates backwards instead of forwards, which generates the strong spike with center frequency, as shown in Fig. 3(a).

 

Fig. 3 The reflected (solid line) and reemitted spectra (dashed line) (a)(e). Time evolution of velocities (b)(f) and corresponding frequencies (b)(g). The top and bottom correspond to 3π and 4π pulses, respectively. Subscripts 1 and 2 in (f)(g) stand for the first and second moving fronts respectively. (c) The instantaneous fields of 2π (dotted line), 3π (dashed line), and 4π (solid line) incident pulses. The insert in (a) is the enlarged view to the LFS for a 3π incident pulse.

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As for a 4π pulse, both the amount of redshift and amplitude of the LFS increase. The LFS shown in Fig. 3(e) has a center frequency of 0.027ω₀ and an amplitude of 5.6×10⁻⁴fs⁻¹. This pulse splits into two 2π pulses with different propagation velocities, and two accompanied absorption fronts are induced. It can be seen in Fig. 3(f) that the first split pulse travels with a stable velocity much faster than the second. Hence, the backwards propagation waves produced by these two fronts have distinct frequency bands. From Figs. 3(f) and 3(g) we can see that, for the first pulse, the velocity of the moving front and frequency of the backpropagation waves are stabilized around 2.84×10⁸m/s and 0.027ω₀, respectively. The velocity of the second front continuously decline from 1.992×10⁸m/s to 1.868×10⁸m/s, generating backpropagation waves with frequency within [0.202ω₀, 0.233ω₀]. Therefore, the LFS appeared in Fig. 3(e) is essentially the Doppler redshift induced by the first moving front, while the second moving front is responsible for the redshift near 0.2ω₀. Since the first 2π pulse split from 4π pulse is more intense and shorter than that split from the 3π pulse, a LFS with a larger redshift is expected in the reflected spectrum of a 4π pulse. Thus, the redshift of the LFS is proportional to the pulse area, if the other parameters fixed.

3.3. Influence of medium density

So far, we have investigated the influences of the laser parameters on the manifestation of the Doppler redshift. Next, the influence of medium parameters, such as medium density and length, are discussed in the following sections. First, we take 2π and 4π pulses as examples to analyze the influence of medium density. The changes in frequency and amplitude of the LFS with respect to density for 2π and 4π pulses are shown in Figs. 4(a) and 4(b), respectively. With the increase of the density, the changes of amplitude and redshift of the LFS have an upward trend and a downward trend, respectively. As the density increases, the reflections enhance while the propagation velocity of the main pulse reduces. Thus, the LFS has a larger amplitude but a smaller redshift. Figure 4(c) describes the influence of the density on the reflected spectrum of a 2π pulse. As ω_c increases from 0.05fs⁻¹ to 0.2fs⁻¹, the amplitude of the LFS increases as a result of the enhancement in reflections. Meanwhile the amount of redshift decreases due to the slowdown of the moving absorption front. Moreover, the LFS broadens as it moves towards the center frequency ω₀. This is because, the enhanced reflections prohibit forward propagation and result in deceleration of the moving absorption front. Consequently, the wider velocity range of the absorption front results in a broader frequency band of the reflected field. Therefore, for a relatively dilute medium, the Doppler effect manifests itself as a sharper LFS with a lower frequency and a lower amplitude. However, with the increase of the medium density, the LFS broadens and strengthens as it moves towards ω₀. When the medium is dense enough to forbid propagating beyond the nonlinear skin depth, as the case of ω_c =0.3fs⁻¹, the LFS is replaced by the redshift induced by the DNOS effect. Note that, the spectrum of ω_c =0.3fs⁻¹ in Fig. 4(c) is similar to that shown in Fig. 2(d), implying that the underlying physics is actually the same. That is, the process of increasing density is essentially another case where the nonlinear reflections near the surface gradually take over from the bulk generation of backpropagation waves.

 

Fig. 4 Dependence of amplitude (blue squares) and frequency (red circles) of the LFS on medium density for 2π (a) and 4π (b) pulses, respectively. (c)The variation of the reflected spectra with increasing medium density for a 2π pulse.

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Fig. 5 The left and right are the reflected and transmitted spectra, respectively. The top and bottom are that for ω_c = 0.1fs⁻¹ and ω_c = 1.0fs⁻¹, respectively.

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3.4. Influence of medium length

Now let us investigate the influence of the medium length. Considering the case where a 4π pulse propagates through a TLA system with a density ω_c = 0.1fs⁻¹, we change the medium length from L = 45μm to L = 150μm. The reflected and transmitted spectra are shown in Figs. 5(a) and 5(b), respectively. It can be seen that, when the medium length changes to 150μm, the blueshift in the transmitted spectrum increases by 0.05ω₀. This z-dependent maximum shift is produced by intrapulse FWM and can be explained with the phase-matching condition [11]. However, regarding the redshift shown in the reflected spectrum Fig. 5(a), it barely moves as medium length increases. Thus, it is fair to suggest that the intrapulse FWM, which induces the blueshift in transmitted spectrum, is not responsible for the z-independent redshift. This redshift is actually the Doppler redshfit produced by the moving absorption front.

To prove this, we investigate the variation of the redshift versus medium length, in terms of the maximum redshift and its amplitude. The medium density is set as ω_c = 1.0fs⁻¹. By increasing the medium length from λ₀/4 to 45μm, the reflected spectrum is obtained, as shown in Fig. 5(c). The maximum redshift in the reflected spectrum increases dramatically within a subwavelength scale then levels off after L = λ₀. This variation of the maximum redshift against the medium length can be predicted by the velocity-dependent theory. At the beginning of the propagation, the nonlinear reflection near the front surface plays a dominant role. At this stage, the main pulse propagates with a relatively low velocity, so does the induced absorption front near the surface. According to the velocity-dependent theory, a slight Doppler redshift appears in the self-reflected waves. However, once the main pulse crosses the surface, it experiences a dramatic acceleration within a subwavelength scale, which results in the increase of the corresponding redshift. After then, the main pulse propagates with a relatively stable velocity, so does the induced absorption front in the medium. The dominant reflection mechanism changes to be the bulk generation of backpropagation waves. In this case, the stable velocity of the absorption front indicates a constant frequency of the backpropagation wave, forming a LFS in the reflected spectrum. This is why the maximum redshift is insensitive to the change of the medium length after a subwavelength scale.

The amplitude of the maximum redshift is expected to increase as the propagation length increases. However, the fact is that the growth rate of the amplitude is gradually vanishing as the propagation length gets longer. As shown in Fig. 5(c), the growth rate of the amplitude is 178% from z = λ₀/4 to z = 4λ₀, while only 22.2% from z = 4λ₀ to z = 54λ₀(45μm). This is because, within a few wavelength scale, the resonance frequency ω₀ of the main pulse is still available to support the increase of the LFS amplitude, as shown in Fig. 5(d). While for a longer distance, the transmitted pulse is blueshifted and gradually centered on 1.5ω₀ due to the nonlinear propagation effects. Since the large detuning prohibits the energy exchange between atoms and the electric field, the atoms prepared in ground states are rarely excited and the absorption front then disappears. This leads to the decrease of the growth rate of the LFS’s amplitude along the propagation path.

Thus, only if the medium is long enough to allow the absorption front to reach the stable velocity, the LFS, if it exists for the medium considered, can be observed in the spectrum. Otherwise, the bulk generation of backpropagation is missed due to the limited medium length, leaving a slight redshift produced by the nonlinear reflection near the front surface. Moreover, with the increase of the length, the LFS’s amplitude keeps increasing until the center frequency of the main pulse is largely detuned.

4. Conclusion

In conclusion, we have made a systematic analysis of the Doppler effect of the absorption front with variable laser and medium parameters. The manifestation of the Doppler effect in the reflected spectrum depends sensitively on the laser and medium parameters, and generally takes two forms: a broad redshift near carrier-wave frequency of the incident pulse and a sharp LFS isolated from the main redshift. The former is produced by the nonlinear reflection near the front surface of the medium, while the latter corresponds to the bulk generation of the backwards propagation waves inside the medium. The speed of the laser pulse near the front surface is much lower than that inside the medium, and the amount of the Doppler redshift is dependent on the velocity. Thus, the redshift produced by the nonlinear reflection near the surface is much smaller than that by the backpropagation waves. Moreover, for the case where nonlinear reflections near the front surface is dominant, the absorption front experiences a deceleration and only penetrates a nonlinear skin depth. The corresponding frequency of the self-reflected field is distributed in a relatively broad band smaller than the carrier-wave frequency. The Doppler redshift manifests itself as the DNOS effect. By reducing pulse duration, increasing pulse area, reducing the medium density, or increasing the medium length, the dominant reflection mechanism gradually changes from the nonlinear reflection near the front surface to the bulk generation of backpropagation waves. When the latter plays the dominant role, the velocity of the moving absorption front is relatively stable and close to vacuum light speed c. In this case, the Doppler redshift manifests itself as a sharp LFS located at the red edge of the reflected spectrum. Our study unifies the underlying physics between the DNOS effect and the observed LFS. Moreover, we investigate how the laser and medium parameters affect the performance of the Doppler effect in the reflected spectrum, and how the behavior of Doppler redshift changes against parameters’ variation. These results are crucial to a further understanding to the reflected spectrum of a few-cycle pulse propagating through dense media, and provide a basis for assessing the performance of theories based on the forward propagation approximation.