Transverse effect of superradiation due to nonlinear effect in Rb atomic medium

Chen Niu; Chen Niu; Xuemei Cheng; Xuemei Cheng; Xuemei Cheng; Xingjia Tang; Rui Zhou; Haowei Chen; Haowei Chen; Jintao Bai

doi:10.1364/OE.27.033090

1. Introduction

Superradiation (SR) is a kind of coherent radiation that occurs in the same direction as the incident laser under the circumstance that the atoms are coherently excited and the atomic dipole moments are in phase [1]. The premise of SR generation is the so-called “coherent preparation” of samples, which is very similar to the classical macroscopic dipole radiation.

SR is one of the mainly used “no mirror” lasing methods which is able to realize light amplification in a single pass. Light sources based on SR have been developed and have unique features. SR source is an efficient way to generate X-rays, which avoids the disadvantages of stimulated X-ray lasers, such as small probability of stimulated radiation, lack of suitable cavity materials, short lifetime of laser energy levels [2]. SR sources also provide photons with extremely high photon degeneracy and extremely low quantum noise thanks to the anti-bunching and compression effects [3]. Besides, SR can be used as a quantum-beat spectroscopy method with high resolution by detecting Doppler beats in SR coming from physically different atoms [4].

Since the pioneering work done by Dicke in 1954 [1], SR has received intense and continuous research interests. Up to now, three types of theories have been developed to treat SR: Eberly’s quantum theory [5,6], MacGillivray’s semi-classical theory [7], and Bomifacio’s mean-field theory [8–10]. These theories have been successfully used to characterize many aspects of SR, such as the SR pulse width and delay time on the total number of excited atoms [11], ringing effect [12].

Compared with the above mentioned temporal effects of SR, studies on the spatial effect (transverse effect) are limited. It is always observed that SR appears ring-patterns in the far field, but the mechanism of such a phenomenon is still not clear. According to the semi-classical theory, the transverse effect occurs only at the tightly focusing (small Fresnel number) case [13], which is conflict with the experiments that ring patterns appear at the loosely focusing cases as well. According to the idea of Ben-Aryeh, the transverse effect observed from the laser-pump two-level atoms is conical emission (CE) [14]. CE is characterized by a ring pattern around the original beam when an intense light beam propagates through a nonlinear medium. It is now well accepted that CE is originated from the Kerr nonlinear effects of self-phase modulation (SPM) and cross-phase modulation (XPM) [15]. And therefore, the CE is commonly achieved when the incident laser is set red- or blue-detuned with respect to the resonant transition between atomic energy levels (to achieve large Kerr nonlinearity). However, SR is always generated when the incident laser is resonant to the atomic transitions.

To understand the mechanism of the transverse effect of SR, we loosely focus the laser in Rb atomic vapor and analyze the far-field diffraction patterns of SR around D₂ lines of ⁸⁵Rb. We also compare the cases that the Gaussian beam and the non-diffraction Bessel beam are taken as the excitation respectively. Semi-classical theory together with spatial SPM are employed to illustrate the experimental results. Different from previous work on CE, we found that it is the Kerr effect of SR that leads to the ring-patterns, not the incident beam. This work provides useful information in understanding the physics behind the transverse effect of SR, which would be of significance in the applications of SR, such as enhancement on the beam quality and efficiency of SR source.

2. Experimental and theoretical methods

Figure 1 shows the scheme of our experimental setup. The Gaussian beam output from the CW Ti: sapphire laser (Spectra-Physics, Matisse TR, tunable from 770 to 990 nm, ω_TEM00=0.7 mm) is focused into a 10 cm long Rb atomic vapor cell through a lens with the focusing length of 300 mm. Two Glan-Taylar polarizers (Union Optic, UPGT5012) were set before and after the Rb cell, acting as the polarizer and analyzer respectively. A CMOS camera (Thorlabs, DCC1240M) was used to receive the beam image of the transmission after the analyzer. During experiments, we scan the laser wavelength over the D₂ lines transitions of ⁸⁵Rb atoms, which are:

{\textrm{i}:}^{85}Rb{\kern 1pt} \;|{5{S_{1/2}}, F = 3 > \to |{5{S_{3/2}} > \textrm{at} \;780.0288}}\; \textrm{nm}

{\textrm{ii}:}^{85}Rb\;|{5{S_{1/2}}, F = 2 > \to |{5{S_{3/2}} > \textrm{at} \;780.0024}}\; \textrm{nm}

Fig. 1. (a) Scheme of the experimental setup. (b) The energy level diagram of ⁸⁵Rb D₂ lines. (c) The real part of the third-order nonlinear susceptibility χ⁽³⁾ versus ΔT₂.

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For the case that the Bessel beam is taken as the incidence, we firstly convert the laser output into a Bessel beam using the XPM method as we reported previously [16] before focusing it into the Rb cell.

For theoretical analysis, we adopt Maxwell-Schrodinger equation to describe the SR generation, and SPM based on Kerr effect to treat the propagation of SR in the Rb atomic medium.

According to Maxwell-Schrodinger equation. The SR model is expressed as [13]:

(1)$$\frac{{\partial E}}{{\partial z}} - i\frac{1}{{4FL}}\nabla _T^2E = \frac{{4{\pi ^2}}}{\lambda } \cdot P$$

where E and P are the slowly varying complex amplitudes of the electric field and polarization respectively; $\nabla _T^2 = (1/r)(\partial /\partial r)r(\partial /\partial r)$, r is the radial coordinate; and $F = \pi {r^2}/\lambda L$ is the Fresnel number. In principle, the transverse term-the second part in Eq. (1) can be eliminated if $F \gg 1$ [17]. In our experiments, the incident spot size is 0.7 mm, the Rb atomic vapor cell length L is 100 mm, and the F is calculated to be about 79. And therefore, we can neglect the transverse part and obtain:

(2)$$\frac{{\partial E}}{{\partial z}} = \frac{{4{\pi ^2}}}{\lambda } \cdot P$$

The polarization intensity P is expressed as [13]: $P = \mu {N_0}$, in which μ is the transition dipole moment matrix element, and N₀ is the inversion density. The inversion density N₀ is closely related to the incident laser field E₀, i.e. ${N_0} \propto {E_0}$. And therefore, the shape of the radiation electric field is in constant to the incident excitation.

After generation, the SR would propagate through the medium, which would experience nonlinear optical effects. According to the Kerr effect, the refractive index of the medium is $n = {n_0} + {n_2}I$, where I is the intensity of the SR, and n₂ is the nonlinear refractive coefficient. In the atomic vapor medium, n₂ is given by [18]:

(3)$${n_2} = \frac{{12{\pi ^2}}}{{n_0^2c}}{\mathop{\rm Re}\nolimits} {\chi ^{(3)}} = \frac{{12{\pi ^2}}}{{n_0^2c}} \times \left[ { - \frac{4}{3}N{{({\rho_{bb}} - {\rho_{aa}})}^{eq}}{{|{{\mu_{ba}}} |}^4}\frac{{{T_1}T_2^2}}{{{\varepsilon_0}{\hbar^3}}}\frac{{\Delta {T_2}}}{{{{(1 + {\Delta^2}T_2^2)}^2}}}} \right]$$

where N is the atomic number density, ${({{\rho_{bb}} - {\rho_{aa}}} )^{eq}}$ is the population difference between the excited and the ground states in thermal equilibrium, T₁ is the excited state lifetime, T₂ is the dipole dephasing time, Δ is the frequency detuning, and μ_ab is the electric dipole moment. It is seen from Eq. (3), the nonlinear refractive coefficient is closely related to the light detuning Δ. The relationship between Re(χ⁽³⁾) and Δ is plotted in Fig. 1(c).

Accordingly, a nonlinear phase shift would be produced:

(4)$$\Delta \varphi = k\int_{{z_0}}^{L + {z_0}} {\Delta n({r, z} )dz = k} \int_{{z_0}}^{L + {z_0}} {{n_2}I({r, z} )} dz$$

In the case that the Gaussian beam is taken as the incident beam, the SR can be described by a Gaussian function. And the nonlinear phase shift can be further expressed as:

(5)$$\Delta \varphi (r )= k\int_{{z_0}}^{L + {z_0}} {{n_2}I({0,0} )\frac{{r_0^2}}{{r_p^2(z )}}\exp [{ - \alpha ({z - {z_0}} )} ]\exp \left[ { - \frac{{2{r^2}}}{{r_p^2(z )}}} \right]} dz$$

in which r₀ is the waist radius, I(0,0) is the central light intensity. Getting out of the sample cell, the SR carrying the nonlinear phase shift will propagate in free space. Based on the Fraunhofer approximation of the Fresnel-Kirchhoff diffraction formula [19], the far-field diffraction intensity is:

(6)$$I \propto {\left|{\int_0^\infty {\int_0^{2\pi } {E(r,{z_0} + L)\exp ( - ikr\theta \cos \varphi )\exp \left\{ { - i\left[ {\frac{{k{r^2}}}{{2R({z_0})}} + \Delta \varphi (r)} \right]} \right\}rdrd\varphi } } } \right|^2}$$

in which D is the distance from the exit surface of the medium to the receiving screen.

3. Results and discussion

Firstly, we focused the Gaussian laser beam into the right center of the sample cell, set the temperature of the sample cell at 66°C (the corresponding Rb atomic number density is $5.36 \times {10^{12}}$ m⁻³ [20]), carefully increased power of the incident beam, and scanned the laser wavelength around D₂ line of ⁸⁵Rb. Emissions are detected at the resonances (and only at the resonances) once the laser power exceeds 10 mW or so. Typical images of the emission at the incident power at 40 mW are shown in Fig. 2(a), which demonstrate a ring structure. Then we set the laser wavelength at 780.0288 nm (resonant to the transition of i), and measured the spectrum of the emission, which is shown in Fig. 2(b). It is seen that the spectrum of the emission is totally different from that of the incident laser, which has a peak at the longer wavelength and has much broader range at the longer wavelength side.

Fig. 2. (a) Typical images of the emission. (b)Spectrum of the incident laser and emission.

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Many researches have demonstrated that the transverse intensity distribution can be modified due to the spatial SPM when a laser beam passes through a nonlinear medium. Furthermore, a ring-pattern with dark (bright) central spot appears when curvature of the wavefront and the nonlinear phase shift are in opposite (same) signs [21]. Previously, we observed ring-patterns with dark and bright central spots at $\Delta ={-} 0.7$ GHz and 0.7 GHz respectively [20]. Here in this experiment, the incident is fixed at $\Delta = 0$, and therefore, the ring-pattern is not the result of the propagation of the incident beam itself, which is further confirmed by the difference in polarization and spectrum. Besides, the emission is neither the ordinary fluorescence, because the emission can only be detected in the forward direction and has a power threshold at 10 mW. And hence the emission should be the SR.

Furthermore, we found the diffraction pattern is closely related to the position of the sample cell. We define $\Delta z$ as the distance from the back side of the sample cell to the focal point of the incident beam, which is negative (positive) when the back side is set before (after) the focal point. Experimentally, we set the laser wavelength at 780.0288 nm, the temperature of the sample cell at 66°C, the incident power at 50 mW and varied $\Delta z$ from −10 cm to 5 cm. The emission images under various sample position are shown in the above row of Fig. 3. It is seen that the central spot of the ring-pattern is bright at $\Delta z ={-} 10$ cm and then gradually gets dark with $\Delta z$ increasing.

Fig. 3. The experimental (above row) and simulation (bottom row) results of the far-field diffraction ring patterns with different the position of the sample cell.

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Such a phenomenon can be well explained by the SPM of the SR. From the spectrum of the SR, we know that the frequency detuning of the SR is negative with respect to the resonant line. And therefore, the nonlinear refractive coefficient for SR is negative according to Fig. 1(c), and the nonlinear phase shift is negative according to Eq . (4). As the curvature of the wavefront is negative when $\Delta z < 0$, a ring-pattern with bright spot is observed. Because the atomic medium has a defocusing effect on SR (${n_2} < 0$), the situation is just like setting a negative lens before the focal point when $\Delta z$ becomes positive. And therefore, the SR becomes divergent instead of focusing, and the curvature of the wavefront becomes positive. As a result, a ring-pattern with a dark central spot appears. The simulations based on the above analysis are shown in the bottom row of Fig. 3, which are in good agreement with the experimental results.

In order to testify that the ring-pattern is the result of SPM, we further studied the power dependence of the pattern. We set the Rb atomic cell with its back side slightly before the focal point, fixed the laser wavelength at 780.0288 nm and the temperature of the sample cell at 66°C. Then we varied the incident laser beam from 10 mW to 50 mW. The emission images under various incident powers are shown in the above row of Fig. 4. It is seen that the central spot of the ring-pattern is bright at lower incident power and then gets dark with the incident power increasing.

Fig. 4. The experiment (above row) and simulation (bottom row) results of the far-field diffraction ring patterns with different incident power.

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The results are to be expected based on the SPM of the SR. From the analysis for cell position dependence, we know that the nonlinear refractive coefficient for SR is negative according to Fig. 1(c), and the nonlinear phase shift is negative according to Eq. (4). As the Rb atomic cell is set with its back side slightly before the focal point, curvature of the wavefront is also negative. So the ring-pattern with bright spot is observed. When the laser power increases, the power of the SR should increase as well. Because the Rb atomic medium has a defocusing effect on the SR when it propagates through the Rb cell, the focusing SR becomes divergent when its intensity increases (similar to the case that a defocusing lens is set before the focal point). Under this circumstance, curvature of the wavefront becomes positive, i.e. in the opposite sign with the nonlinear phase shift, and the center of the ring-pattern becomes dark correspondingly. The simulations based on the above analysis are shown in the bottom row of Fig. 4, which are in good agreement with the experimental results.

From the above studies we know that the modification of the SR beam profile is mainly due to its propagation in the nonlinear medium. Theoretically, a Gaussian shaped SR can be generated when the atomic sample is excited by a Gaussian incidence. According to the particular beam profile and diffraction of the Gaussian, the nonlinear phase shift occurs and produce the diffraction ring-patterns in the far-field. To further make this point clear, we replaced the Gaussian beam into a Bessel beam as the excitation. Bessel beam is a kind of well-known non-diffraction beam, which means its beam size and profile keep invariant during propagation. Previous work has indicates that the non-diffraction property can effectively help Bessel beam to enhance the nonlinear optical processes [22–24]. Here in this work, we compare the performance of Bessel beam with that of Gaussian beam. During experiments, the laser wavelength was set at 780.0288 nm, the temperature of the sample cell was kept at 66°C, and the cell was fixed with its back side at focal point ($\Delta z = 0$ cm). Typical SR images under various incident power with Gaussian beam and Bessel beam as the excitation are shown in the above and bottom row of Fig. 5 respectively. It is seen that a Bessel SR is generated with Bessel beam as the excitation. Moreover, the central spot of the Bessel SR keeps almost invariant except that more rings in the Bessel SR become visible with the incident power increasing.

Fig. 5. Comparison of the far-field diffraction patterns of SR produced by Gaussian beam (above row) and Bessel beam (bottom row) as excitation under different incident power.

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The performance of the Bessel excitation can be attributed to its non-diffraction property. That is, the wavefront curvature of the Bessel beam remains constant during propagation and almost no additional nonlinear lateral phase shift is generated for Bessel SR. And therefore, its beam profile keeps invariant. Moreover, we can see the entire Bessel SR is of higher intensity compared with Gaussian SR at higher incident power, which is also the result of the non-diffraction. So, it is better to get SR emission with higher power by using the excitation with non-diffraction beam as the excitation than the commonly used Gaussian beam.

4. Conclusion

In conclusion, we study the mechanism of the transverse effect of SR via analyzing its far-field diffraction patterns around D₂ lines of ⁸⁵Rb. Two types of beams - Gaussian and Bessel beam are employed as the excitation, respectively. The results indicate that the Gaussian SR shows a ring-pattern beam in far-field and its beam profile is closely dependent on the incident power and the position of sample cell. The results are well explained by considering its propagation in the nonlinear medium. The experimental results are in good agreement with the simulations using semi-classical theory of SR generation together with spatial SPM. By contrast, the beam profile of Bessel SR is hardly dependent on the input power thanks to its non-diffraction property. The comparison confirms that the transverse effect of the SR is mainly the result of the nonlinear effect for SR. Besides, the finding in Bessel SR suggest a way to enhance SR emission by using the non-diffraction beam as the excitation. This work provides useful information in understanding the physics behind the transverse effect of SR, which would be of significance in the applications of SR, such as enhancement on the beam quality and efficiency of SR source.

Funding

National Natural Science Foundation of China (NSFC) (61805200, 11874299); Key Laboratory of Spectral Imaging Technology of Chinese Academy of Sciences (LSIT201908W).

Disclosures

The authors declare no conflicts of interest.

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Transverse effect of superradiation due to nonlinear effect in Rb atomic medium

Abstract

1. Introduction

2. Experimental and theoretical methods

3. Results and discussion

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (5)

Equations (8)

Optics Express