Subfemtosecond-resolved modulation of superfluorescence from ionized nitrogen molecules by 800-nm femtosecond laser pulses

An Zhang; Mingwei Lei; Jingsong Gao; Chengyin Wu; Qihuang Gong; Hongbing Jiang

doi:10.1364/OE.27.014922

1. Introduction

Nonlinear interaction between intense femtosecond laser and atom / molecule has caused some well-known strong field phenomena, such as above threshold ionization [1], high order harmonic generation [2], and laser-induced electron diffraction [3,4]. Yao et al. observed directional and polarized coherent emission when they investigated the propagation of intense mid-infrared femtosecond laser pulses in air [5]. The novel phenomenon has attracted great interest because of the potential remote sensing application [6–8]. The generation of the coherent emission around 391 nm, corresponding to the transition of $N_{2}^{+} (B^{2} Σ_{u}^{+}, ν^{″} = 0 \to X^{2} Σ_{g}^{+}, ν = 0),$ has been extensively investigated [9]. For simplicity, We label states $N_{2}^{+} (X^{2} Σ_{g}^{+}),$ $N_{2}^{+} (A^{2} Π_{u}),$ and $N_{2}^{+} (B^{2} Σ_{u}^{+})$ as X, A, and B in our following text. It is found that the origin of the coherent emission depends on the pump laser wavelength [10,11]. When the laser wavelength is in the mid-infrared range, the seed laser around 391 nm is attenuated when it is injected into the ionized nitrogen molecules. There is no population inversion between B (ν” = 0) and X (ν = 0). The coherent emission origins from the resonantly enhanced nonlinear interaction, especially the stimulated Raman processes [12,13]. When the laser wavelength is 800 nm, the seed laser around 391 nm is greatly amplified. However, the origin of the optical gain is still under a hot debate [14–20]. Various mechanisms have been proposed to explain the optical gain. The nature of the 391nm coherent emission has been identified as superfluorescence corresponding to the transition of B (ν” = 0)-X (ν = 0) [10,14]. Superfluorescence is a kind of superradiance from an extended samples where the polarization is evolved spontaneously or triggered by external seed. More and more evidences indicated that population inversion is established between B (ν” = 0) and X (ν = 0) in the 800 nm pump laser fields [21,22], even though the mechanism establishing the population inversion is still far from being understood. With the help of some theoretical simulation, it is suggested that population redistribution after ionization plays the significant role for establishing the population inversion in two independent works [15,16]. The falling edge of the 800 nm pump laser fields induced the population transfer from state X to state A, which facilitates the establishment of the population inversion between B and X. Very recently, the involvement of state A to the optical gain around 391 nm is proved in the 800 nm pump laser field [23–25].

In this article, we systematically study the effects of state A to the 391 nm optical gain based on the pump-probe-seed scheme. Nitrogen molecules are ionized by strong 800 nm pump laser. Superfluorescence emission around 391 nm corresponding to the transition of B (ν” = 0)-X (ν = 0) is generated when a delayed seed laser is injected into the ionized nitrogen molecules. We observe subfemtosecond-resolved periodical modulation for the superfluorescence intensity when another weak 800 nm probe laser is injected and scanned. The period is determined to be ~2.63 fs, corresponding to the transition frequency between A (ν' = 2) and X (ν = 0). In addition to the periodical modulation, the average superfluorescence intensity depends on the time when the probe laser is injected. When the weak 800 nm probe laser is injected between the pump laser and the seed laser, the average superfluorescence intensity is higher than that without the probe laser. When the weak 800 nm probe laser lags behind the seed laser, the average superfluorescence intensity is lower than that without the probe laser. With the aid of theoretical derivation, these observations can be well explained by the laser-induced population transfer and polarization variation between the relevant electronic states of ionized nitrogen molecules.

2. Experimental setups

The 800 nm femtosecond laser pulses were launched from a Ti: Sapphire amplification system (1 kHz, 800 nm, 40 fs, 3.0 mJ). The output laser beam was first split into two parts with a beam splitter. The weak one passed through a 0.1 mm β barium borate (BBO) crystal to generate second harmonic pulses served as seed laser. In this experiment, the seed pulse is centered at 395 nm in order to control the intensity of 391 nm signal. The strong one with 2.8 mJ energy was further split into two beams and then combined with pulse energy of 1.7 mJ and 0.1 mJ respectively. These two beams were mounted on high precision delay lines with a resolution of 10 nm, i.e., a temporal resolution of 33 attoseconds. The polarizations of the pump, probe and seed pulses are all linear polarized and aligned parallel with each other. Finally these three beams were collinearly combined with a dichromatic mirror and focused into a gas chamber of nitrogen with an f = 200 mm lens. The pressure of nitrogen in the gas chamber was set as 7 mbar during the experiment. After the interaction between strong laser pulses and nitrogen gas, the output signal was focused into a fiber spectrometer. A concave mirror was placed inside the gas chamber to focus the fluorescence of side direction into the fiber spectrometer. Figure 1(a) shows the schematic diagram of our experimental setup. For the sake of clarity, we name the time delay between the seed and the pump as τ_ps and the delay between the weak 800 nm probe and the pump as τ_pw. The positive value of these two parameters means that the pump precedes and vice versa.

Fig. 1 (a) Experimental setup for cross modulation of the nitrogen ions superfluorescence. The pump pulse and the weak 800 nm probe were combined together by a beam splitter (BS). These two beams and the 391 nm seed pulses were further combined by a dichromatic mirror (DM1) and then focused into the gas chamber by an achromatic lens of f = 200 mm. The forward radiation was spectrally filtered and further measured with a spectrometer. The relative delay between the pump and the weak probe τ_pw and the delay between the pump and the seed τ_ps can be separately controlled with two mechanic delay lines. (b) Energy-level diagram of nitrogen ions where corresponding transitions of cross-modulation are indicated.

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The energy-level diagram of the N₂⁺ is shown in Fig. 1(b). The 800 nm pump ionized N₂ molecules and the generated N₂⁺ populated ground state X, first excited state A and second excited state B [15,16]. The weak 800 nm probe pulse has a spectral range of from 770 nm to 830 nm. The transitional wavelengths of P-branch and R-branch of A (ν' = 2)-X (ν = 0) transition are about 787.5 nm and 782.6 nm. Because of energy resonance, the weak 800 nm probe will couple states X (ν = 0) and A (ν' = 2) and the seed around 391 nm will couple states B (ν” = 0) and X (ν = 0).

3. Experimental results

Previous reports have shown that in the case of 800-nm pump laser, a seed is required to generate the 391-nm superfluorescent emission whether the seed is a self-seed or an external seed [6–10,26]. In the former case, the self-seed is served by the pump-laser-induced second harmonic or supercontinuum white light. In our experiment, the nitrogen gas pressure was kept below 7 mbar to avoid the self-seed effect. It is known that the population in the upper electronic state can be measured by fluorescence spectrum. Here we compared the side fluorescence intensity with and without the injection of the weak 800-nm probe laser. The side fluorescence showed no obvious variation during the scanning of the probe laser. The measurement indicates that the weak 800-nm probe laser does not cause obvious population variation under our experimental condition.

Figure 2 shows the forward spectra with and without the probe laser, in which the delay of the seed to the pump τ_ps was set at 1 ps. Figure 2(a) shows the 391 nm forward spectra without the probe laser. It can be seen that strong superfluorescence around 391 nm is observed when a delayed seed laser is injected. This observation is consistent with previous reports [10,14,22]. Figure 2(b) shows the forward spectra with the injection of the weak probe injected at different moments. When the weak 800-nm laser was injected before the pump laser, i.e., τ_pw = −1 ps, the 391 nm signal intensity was almost the same as that without the probe laser. This coincides with our side-fluorescence results, which further clarifies the effect of the weak 800-nm probe laser. However, the 391-nm superfluorescence intensity had an obvious variation when the weak 800-nm probe laser was injected after the pump laser. In addition, the variation depended on the time sequence between the probe laser and the seed laser. When the probe laser was injected before the seed laser, the superfluorescence intensity had an obvious increase relative to that without the probe laser. When the probe laser was injected after the seed laser, the superfluorescence intensity had an obvious decrease relative to that without the probe laser.

Fig. 2 (a) The forward spectrum obtained by injecting the seed at τ_ps = 1 ps without the probe laser. The spectrum of the seed is indicated in gray dashed line for comparison. (b) The forward spectrum obtained by injecting the seed and the weak 800 nm probe into the plasma. The probe was injected before the pump (blue), between the pump and the seed (red) and after the seed (gray) respectively.

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Then we set τ_ps at 0, 1, 2 and 3 ps respectively and scanned the weak 800 nm beam from τ_pw = −2 ps to τ_pw = 20 ps with a step length of 50 fs. Figure 3 shows the 391 nm signal intensity as a function of the time delay between the pump laser and the probe laser. The average intensity is higher than that without the probe laser when the weak 800 nm probe laser is injected before the seed laser. While the average intensity is lower than that without the probe laser when the weak 800 nm probe laser is injected after the seed laser. It should be mentioned that the signal intensity displays a fast decrease and slowly restoration with τ_pw > τ_ps. The restoration time here is about several tens picoseconds, which is comparable to temporal duration of 391 nm superfluorescence [14,22]. The temporal duration of 391-nm superfluorescence is about 20 ps according our previous work [27]. When the probe pulse is injected in a retarded time of 20 ps, the emission of 391-nm superfluorescence finishes and the injection of the weak probe will no longer influence the intensity of 391-nm signal.

Fig. 3 The 391-nm signal intensity obtained by scanning the weak 800 nm probe from τ_pw = −2 ps to τ_pw = −20 ps. The delay between the pump and the seed τ_ps was set at 0, 1, 2, 3 ps in (a)-(d). In (b)-(d), the figures are separated by two vertical dashed lines, which marked the moments τ_pw = 0 and τ_pw = τ_ps.

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To further investigate the effect of the weak probe laser, the 391-nm signal intensity was measured by finely scanning τ_pw with a step length of 0.1 fs. The experimental condition is the same as Fig. 2(b) with τ_ps = 1 ps. These scans were performed in two separate time domains, corresponding to 0 <τ_pw < τ_ps and τ_pw > τ_ps. The typical results are shown in Fig. 4. The 391-nm signal intensity exhibits subfemtosecond-resolved periodical modulations whether 0 <τ_pw < τ_ps or τ_pw > τ_ps. The period is determined to be 2.63 fs by fitting the experimental data with a sine function. The period corresponds to the transition frequency between state A (ν' = 2) and state X (ν = 0). It should be pointed out that this transition frequency is nearly resonant with the frequency of the weak 800-nm probe laser field.

Fig. 4 Periodical modulations of forward 391 nm signal intensity obtained by scanning the probe with a step length of 0.1 fs when τ_ps = 1ps. Dotted lines represent experimental results and solid lines are fitted curves with sine function in a period of 2.63 fs. (a) The weak 800 nm probe precedes the seed. (b) The weak 800 nm probe lags behind the seed.

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4. Discussion

Nitrogen molecules are ionized by strong 800-nm laser pulses. The generated nitrogen ions can populate different electronic states [15,16]. The wave function can be written as:

φ = b_{0} ϕ_{B} e^{- \frac{i E_{b} t}{ℏ}} + a_{0} ϕ_{A} e^{- \frac{i E_{a} t}{ℏ}} + x_{0} ϕ_{X} e^{- \frac{i E_{x} t}{ℏ}},

where ϕ_B, ϕ_A, and ϕ_X are eigen wave functions of states B (ν” = 0), A (ν' = 2), and X (ν = 0). In the present experimental condition, only three quantum states, B (ν” = 0), A (ν' = 2) and X (ν = 0) are involved. Here, we start the theoretical deviation from the wave function of ionized nitrogen molecules by the pump laser. To simplify our discussion, we assume that the laser pulses are square waves and ignore the decay of the excited states. The weak probe pulses could be described as

E_{w} (t) = \frac{1}{2} E_{w} (t - τ_{p w}) e^{i ω_{w} (t - τ_{p w} + \frac{τ_{w}}{2})} + c . c .,

and

E_{w} (t - τ_{p w})

is defined as follows:

E_{w} (t - τ_{p w}) = {\begin{matrix} E_{w} & - \frac{τ_{w}}{2} \leq t - τ_{p w} \leq \frac{τ_{w}}{2} \\ 0 & t - τ_{p w} < - \frac{τ_{w}}{2}, t - τ_{p w} > \frac{τ_{w}}{2}, \end{matrix}

where τ_w is the pulse duration and E_w is the amplitude of the weak probe laser field. The seed is described similarly as:

E_{s} (t) = \frac{1}{2} E_{s} (t - τ_{p s}) e^{i ω_{s} (t - τ_{p s} + \frac{τ_{s}}{2})} + c . c .,

and

E_{s} (t - τ_{p s})

is defined as follows:

E_{s} (t - τ_{p s}) = {\begin{matrix} E_{s} & - \frac{τ_{s}}{2} \leq t - τ_{p s} \leq \frac{τ_{s}}{2} \\ 0 & t - τ_{p s} < - \frac{τ_{s}}{2}, t - τ_{p s} > \frac{τ_{s}}{2}, \end{matrix}

where τ_s is the pulse duration and E_s is the amplitude of the seed laser field. The arrival time of the pump is defined as zero, and the seed as well as the weak probe arrived at time τ_ps and τ_pw. When scanning the weak probe pulse, τ_pw changes and τ_ps is kept constant.

The 391-nm superfluorescence intensity is determined by the macroscopic polarization between B (ν” = 0) and X (ν = 0) [27,28]. It can be written as $P_{b x} (t) = N (μ_{b x} ρ_{b x} + c . c .),$ where N represents the number density of nitrogen ions, $μ_{b x}$ is the dipole moment and $ρ_{b x}$ determines the polarization between B (ν” = 0) and X (ν = 0). In our experimental condition, there is no superfluorescence when only the 800-nm pump laser irradiated. Therefore, no macroscopic polarization between B (ν” = 0) and X (ν = 0) is established by the pump laser, i.e., $ρ_{x b} (0) = \frac{1}{N} e^{i ω_{s} t} \sum_{k}^{} b_{0}^{k *} x_{0}^{k} = 0$ with the superscript k representing the k-th ion.

When a delayed seed laser is injected, we observed strong superfluorescence emission around 391 nm. After interacting with the seed, we have

\begin{array}{l} b_{s}^{k} = b_{0}^{k} \cos Θ_{s} + i x_{0}^{k} e^{i ω_{s} (τ_{p s} - \frac{τ_{s}}{2})} \sin Θ_{s} \\ x_{s}^{k} = x_{0}^{k} \cos Θ_{s} + i b_{0}^{k} e^{- i ω_{s} (τ_{p s} - \frac{τ_{s}}{2})} \sin Θ_{s}, \end{array}

where

Θ_{s} = \frac{μ_{b x} E_{s}}{2 ℏ} τ_{s},

and

τ_{s}

is the pulse duration of the seed.

ρ_{x b} (s)

can be written as:

ρ_{x b} (s) = \frac{1}{N} e^{i ω_{s} t} \sum_{k}^{} b_{s}^{k *} x_{s}^{k} = \frac{1}{N} e^{i ω_{s} t} \sum_{k}^{} [i (b_{0}^{k *} b_{0}^{k} - x_{0}^{k *} x_{0}^{k}) e^{- i ω_{s} (τ_{p s} - \frac{τ_{s}}{2})} \cos Θ_{s} \sin Θ_{s}] .

From Eq. (5), we know that the 391-nm signal intensity depends on the population difference between state B (ν” = 0) and state X (ν = 0), as well as the seed laser profile. Evidences show that population inversion is generated between B (ν” = 0) and X (ν = 0) by the 800-nm pump laser [15,16].

When the weak 800-nm probe laser is injected between the pump laser and the seed, $ρ_{x b} (w s)$ can be written as:

ρ_{x b} (w s) = \frac{1}{N} e^{i ω_{s} t} \sum_{k}^{} b_{w s}^{k *} x_{w s}^{k} = \frac{1}{N} e^{i ω_{s} t} \sum_{k}^{} [i (b_{0}^{k *} b_{0}^{k} - x_{w}^{k *} x_{w}^{k}) e^{- i ω_{s} (τ_{p s} - \frac{τ_{s}}{2})} \cos Θ_{s} \sin Θ_{s}] .

The density matrix element

ρ_{x x} (w)

represents the population of X (ν = 0) after interacting with the weak probe laser. It can be written as:

\begin{array}{l} ρ_{x x} (w) = \frac{1}{N} \sum_{k}^{} x_{w}^{k *} x_{w}^{k} \\ = \frac{1}{N} \sum_{k}^{} [x_{0}^{k *} x_{0}^{k} + (a_{0}^{k *} a_{0}^{k} - x_{0}^{k *} x_{0}^{k}) \sin^{2} Θ_{w} + (i x_{0}^{k *} a_{0}^{k} e^{- i ω_{w} (τ_{p w} - \frac{τ_{w}}{2})} \cos Θ_{w} \sin Θ_{w} + c . c .)], \end{array}

where

Θ_{w} = \frac{μ_{a x} E_{w}}{2 ℏ} τ_{w},

μ_{a x}

is the dipole moment between A (ν' = 2) and X (ν = 0), and

τ_{w}

is the pulse duration of the probe. And the average value of

ρ_{x x} (w)

on time τ_pw can be written as:

< ρ_{x x} (w) > = \frac{1}{N} \sum_{k}^{} [x_{0}^{k *} x_{0}^{k} + (a_{0}^{k *} a_{0}^{k} - x_{0}^{k *} x_{0}^{k}) \sin^{2} Θ_{w}] .

In our experiment, we observed that the 391-nm signal intensity is modulated with the period of 2.63 fs, corresponding to the transitional frequency $\frac{ω_{w}}{2 π}$ between state A (ν' = 2) and state X (ν = 0). This observation implies that the population of X (ν = 0) $ρ_{x x} (w) = \frac{1}{N} \sum_{k}^{} x_{w}^{k *} x_{w}^{k}$ is coherently modulated by the probe laser according to Eq. (6). According to Eq. (7), $e^{- i ω_{w} (τ_{p w} - \frac{τ_{w}}{2})}$ is the modulating factor with an angular frequency of ω_w and the modulation of $ρ_{x x} (w)$ requires $ρ_{x a} (0) = \frac{1}{N} e^{i ω_{w} t} \sum_{k}^{} a_{0}^{k *} x_{0}^{k} \neq 0,$ which means that states A (ν' = 2) and X (ν = 0) are coherently populated by the pump laser. This conclusion is consistent with the predication by the theories [15,16,23,24].

In addition to the subfemtosecond-resolved modulation, the average 391 nm signal intensity is higher than that without the probe laser as shown in Fig. 3. Based on the comparison between Eqs. (5) and (6), the increase of the signal intensity can be attribtued to the decrease of the population in X (ν = 0) by the probe laser, i.e., $ρ_{x x} (w) = \frac{1}{N} \sum_{k}^{} x_{w}^{k *} x_{w}^{k} < ρ_{x x} (0) = \frac{1}{N} \sum_{k}^{} x_{0}^{k *} x_{0}^{k} .$ Accroding to Eq. (8), it is therefore deduced that $a_{0}^{k *} a_{0}^{k} < x_{0}^{k *} x_{0}^{k}$ , i.e., the population in X (ν = 0) is higher than that in A (ν' = 2) when only the 800-nm pump laser is injected.

When the weak 800-nm probe laser is injected after the seed, $ρ_{x b} (s w)$ can be written as:

\begin{array}{l} ρ_{x b} (s w) = \frac{1}{N} e^{i ω_{s} t} \sum_{k}^{} b_{s w}^{k *} x_{s w}^{k} \\ = \frac{1}{N} e^{i ω_{s} t} \sum_{k}^{} \begin{array}{l} [i (b_{0}^{k *} b_{0}^{k} - x_{0}^{k *} x_{0}^{k}) e^{- i ω_{s} (τ_{p s} - \frac{τ_{s}}{2})} \cos Θ_{s} \sin Θ_{s} \cos Θ_{w} \\ + x_{0}^{k *} a_{0}^{k} e^{- i ω_{w} (τ_{p w} - \frac{τ_{w}}{2}) - i ω_{s} (τ_{p s} - \frac{τ_{s}}{2})} \sin Θ_{s} \sin Θ_{w}] . \end{array} \end{array}

And the average value of

ρ_{x b} (s w)

on time τ_pw can be written as:

< ρ_{x b} (s w) > = \frac{1}{N} e^{i ω_{s} t} \sum_{k}^{} [i (b_{0}^{k *} b_{0}^{k} - x_{0}^{k *} x_{0}^{k}) e^{- i ω_{s} (τ_{p s} - \frac{τ_{s}}{2})} \cos Θ_{s} \sin Θ_{s} \cos Θ_{w}] .

In comparison with Eqs. (5), (9), and (10), we can see that the weak 800 nm probe laser has two effects when it is injected after the seed laser. First, the polarization between B (ν” = 0) and X (ν = 0) is periodically modulated with the period of

\frac{2 π}{ω_{w}} .

This effect is proved by our observation that the 391-nm signal intensity is modulated with a period of 2.63 fs as shown in Fig. 4(b). Secondly, a factor of

\cos Θ_{w}

is introduced for the polarization, which means that the probe injection will decrease the polarization between B (ν” = 0) and X (ν = 0). This effect is also proved by our observation that average 391-nm signal intensity is much lower than that without the probe laser as shown in Fig. 3.

5. Conclusion

We study the superfluorescence of N₂⁺ by the injection of three laser beams: a strong 800nm beam to generate N₂⁺ ions, a weak 800 nm probe to couple state A (ν' = 2) and state X (ν = 0), and a seed to induce the polarization and superfluorescence between state B (ν” = 0) and state X (ν = 0). The delay of the seed to the pump is fixed and the probe is scanned. When the probe is scanned after the pump, the signal is modulated periodically with the period of 2.63 fs, corresponding to the transitional frequency between state X (ν = 0) and state A (ν' = 2), which indicates that the initial post-pump population in X (ν = 0) and A (ν' = 2) is coherent. When the probe is injected between the pump and the seed, the average signal is increased, which testifies that the population in these two states is not inverted. When the probe is injected after the seed, the signal is decreased, which means that the polarization triggered by the seed is destroyed by the probe. The superfluorescence from state B (ν” = 0) to state X (ν = 0) could be controlled by the transfer between state A (ν' = 2) and state X (ν = 0).

Funding

National Natural Science Foundation of China (41527807, 61590933, 11625414, 21673006); National Key R&D Program of China (2018YFA0306302).

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Subfemtosecond-resolved modulation of superfluorescence from ionized nitrogen molecules by 800-nm femtosecond laser pulses

Abstract

1. Introduction

2. Experimental setups

3. Experimental results

4. Discussion

5. Conclusion

Funding

References

Cited By

Figures (4)

Equations (10)

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