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Abstract
Correlation property of light limits the performance in related applications such as the visibility of ghost imaging or intensity interferometry. To exceed these performance limits, we here manipulate the degree of second- and higher-order coherence of light by changing controllable variables in four-wave mixing (FWM) process. The measured degree of second- and third-order coherence of the output light beams considerably exceed those of the incident pseudothermal light. Namely superbunching effects, g(2)(0) value up to 7.47 and g(3)(0) value up to 58.34, are observed experimentally. In addition, strong second- and third-order cross-correlation exist between the output light beams. Further insights into the dependence of superbunching light on the temperature of Rb vapor, the laser detuning and the optical power of all the incident light beams show that it can serve as a light source with a tunable superbunching degree.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Hanbury Brown and Twiss demonstrated that the angular size of stars can be measured by correlating the intensity fluctuations at two different locations in 1956 [1,2]. Since then such correlation measurement has proven to be a powerful tool in many areas of physics and has been applied in phase sensitivity interferometry [3], subwavelength interference [4], ghost interference [5,6] and ghost imaging [7–9], etc. The performance of light in these applications has been determined by the correlation property of the employed light [10,11]. For instance, it has been demonstrated that a light source with larger value of the second- and higher-order degree of coherence is helpful to improve the visibility of ghost imaging [12,13].
There were many different methods to realize two- and multi-photon superbunching with linear or nonlinear methods. In linear optical systems, two-photon superbunching effects were demonstrated via adding more two-photon paths [14,15], or employing more than one ground glass [16,17], or employing electro- and acousto-optic modulators [18,19], or spatial light modulators [20,21]. The studies on higher-order correlation have made great developments theoretically and experimentally in recent year [22,23]. However, the disadvantages for realizing superbunching effects in linear systems are finite degree of Nth-order coherence and increased complexity of experiments with large value of N.
In nonlinear optical systems, two-photon superbunching effects were observed in two-photon superradiance [24,25], quantum states and squeezed states [26,27], nonlinear medium [28], and quantum dots and coupled atoms in cavity [29]. Bromberg et al. [30] showed that the possibility of controlling the bunching properties of thermal light through nonlinear optical effect. There were also some studies on superbunching effects from nonlinear processes and rogue waves [31–35]. The main nonlinear processes included optical harmonics, supercontinuum (SC), electromagnetically induced transparency (EIT) and four-wave mixing (FWM).
In practical application, correlation property of light limits the performance in related applications. For instance, the visibility limit of thermal light ghost imaging is limited to 1/3 due to the degree of second-order coherence equaling 2. To exceed these performance limits, the aim of the paper is to manipulate degree of second- and higher-order coherence, the performance limits of related applications such as intensity interferometry and ghost imaging could be exceeded.
In this paper, the FWM process is employed to manipulate degree of second- and higher-order coherence of light by changing controllable variables. The proposed scheme may flexibly manipulate in two main ways: one is to employ pseudothermal light as all input beams in FWM process; the other is to control various influencing variables such as temperature, laser detuning, power of light source and incident beams in the nonlinear interaction. As a result, the second- and third-order auto- and cross-correlation functions of the output beams are measured. Further investigations about the dependence of the degree of second- and third-order coherence on influencing factors in nonlinear interaction are presented. The experimental scheme could serve as a light source with a tunable superbunching degree that be applied to intensity interference phenomena studies and potentially improve the performance of related applications such as the visibility of ghost imaging.
2. Theory
In this proposed manipulation scheme, the optical field can be described in terms of the notation and geometry of a two-level atom dynamical behavior [36]
where ${E_i}$ denotes the complex amplitude of arbitrary wave involved in FWM process, then the polarization ${P_i}\left ( \omega \right )$ corresponding to third-order nonlinear polarization process is given byWe introduce amplitudes $A_i$ of the waves propagating in the $+z$ direction by means of the equation
$k_i$ denotes the wave vector of the corresponding optical field. In the slowly varying amplitude approximation, amplitude ${A_i}$ of any field ${E_i}$ involved in nonlinear interaction must obey the set of coupled equations Here the wave vector mismatch is $\Delta k = 0$ due to satisfying the phase matching condition ${k_i} + {k_j} = {k_m} + {k_n}$. The nonlinear absorption coefficient $\alpha$ and coupling coefficient $\kappa$ are The amplitude ${A_i}$ of any field at the exit plane of the nonlinear medium is obtained by integral operation, then the intensity ${I_i}$ of any beam involved is given by The second- and third-order auto-correlation and cross-correlation functions of beams involved in FWM process can be calculated by the measured intensity. The normalized second-order temporal coherence function for intensities with time delay $\tau$ is given by [11]The normalized third-order temporal coherence function for intensities ${g^{(3)}}({t_1},{t_2},{t_3})$ is given by
It is concluded that the second- and third-order auto-correlation and cross-correlation functions of beams are determined by following controllable variables: temperature $T$, laser detuning $\Delta$, and amplitudes of other three fields ${E_j}$, ${E_m}$, ${E_n}$ involved in FWM process. Therefore we will manipulate degree of second- and third-order coherence of ${I_i}$ by changing controllable variables in FWM process.
3. Experimental results and discussion
The experimental setup is shown in Fig. 1(a). A beam from a single-mode tunable diode laser with center wavelength 780.24 nm (TOPTICA DL100 pro) went through an electro-optical modulator (EOM), whose intensity can be manipulated into pseudothermal light by regulating two linear polarizers (P$_1$ and P$_2$) and inputting corresponding signal. Then, the pseudothermal light obtained by EOM was divided into two beams by a set of half-wave plate (H$_1$) and polarization beam splitter (PBS$_1$). The reflected beam with S-polarization component was calibrated into P-polarization component via the second set of half-wave plate (H$_2$) and polarization beam splitter (PBS$_2$), named as forward pump beam. It propagated inside a rubidium atomic vapor. Meantime, the transmitted beam was further divided into two parts by a set of half-wave plate (H$_3$) and polarization beam splitter (PBS$_3$). The reflected part with S-polarization served as the backward pump beam and counter-propagated with the forward pump beam for Doppler-free [38]. The transmitted part through polarization beam splitter (PBS$_4$) was chosen as the probe beam with P-polarization. Note that, they satisfied the phase-matching condition of a degenerated-FWM process, as shown in insert of Fig. 1(a). Specifically, the H$_1$ and H$_3$ were used to adjust the appropriate intensity ratio of light beams to achieve the maximum FWM signal in this experiment. The two convex lenses (L$_1$ and L$_2$) were used to focus three incident beams at a point in Rb cell.
Fig. 1. (a) Schematics of the experimental setup and the phase-conjugate geometry for FWM process. (b) The corresponding energy diagram and transitions. (c) The measured FWM signal spectra.
The Rb cell was manipulated by a temperature control heater with digital display, which can regulate and maintain the temperature of Rb vapor in cell. It had a length of 75mm and was encased by $\mu$ metal magnetic shield to isolate from environmental magnetic fields. The atomic medium used here was ${}^{87}Rb$ vapor, the corresponding energy diagram and transitions involved in the FWM process are shown in Fig. 1(b). The frequencies of incident beams were tuned to resonant with the D2 line $(5{S_{1/2}}(F = 2) \leftrightarrow 5{P_{3/2}}(F' = 3))$ of ${}^{87}Rb$.
High-speed photoelectric detectors (Thorlabs DET-36A, D$_1$, D$_2$ and D$_3$) were employed to detect the intensity of backward pump, probe, and forward pump beam through nonlinear interaction, respectively. FWM signal was generated from the opposite direction of probe beam with S-polarization. In this scheme, the pseudothermal light were chosen as all incident beams, the FWM signal of pseudothermal light which differed from that pumped laser was firstly measured by D$_4$, as shown in Fig. 1(c).
Now we show how to manipulate degree of second- and third-order coherence of beams by changing controllable variables in FWM process experimentally. Firstly, the same pseudothermal light were selected as incident pump beams and probe beam concurrently. The forward and backward pump beam power were $253\mu W$ and $291\mu W$, the probe power was $305\mu W$. The temperature of Rb cell was set to $90^\circ C$. From the auto-correlation point of view, the measured second-order coherence functions of beams involved in FWM process are shown in Fig. 2. The error bar is standard deviation of ${g^{(2)}}(0)$ in groups of data. Initially, as show in Figs. 2(b)–2(d), ${g^{(2)}}(0)=1.80$. The reason why not 2.0 was imperfect thermal like correlations. The measured degree of second-order coherence of probe beam, forward and backward pump beam were improved from 1.80 to 6.21, 6.01 and 6.77 via FWM process, respectively. The degree of second-order coherence of generated FWM signal also reached 7.47 in Fig. 2(a). Therefore, two-photon superbunching effect can be observed for any beam involved in FWM process.
Fig. 2. The measured second-order auto-correlation functions of (a) FWM signal, (b) probe beam, (c) forward and (d) backward pump beams. Dashed green lines represent the normalized degree of second-order coherence ${g^{(2)}}(0)$ of thermal light in theory, that is boundary between bunching and superbunching effect. Solid blue lines are the measured second-order coherence functions of beams without Rb cell, namely the measured second-order coherence function of pseudothermal light source experimentally. Solid red lines indicate the measured second-order coherence functions of beams via FWM process.
In addition, the cross-correlation of beams involved in FWM process were studied. The second-order auto-correlation and cross-correlation degree ${g^{(2)}}(0)$ of beams via FWM process are shown in Table 1. It can be seen from the table that the generated FWM signal has the highest degree of second-order coherence ${g^{(2)}}(0)$. It might attribute to that the FWM signal was a new signal generated by the nonlinear interaction between the other three incident beams and the medium. The degree of nonlinear interaction directly affected statistical properties of FWM signal.
Table 1. The second-order auto-correlation and cross-correlation degree ${g^{(2)}}(0)$ of beams via FWM process.
Furthermore, since the degree of nonlinear interaction has been proved to be strong in the previous section, it is necessary to study the higher-order coherence functions of beams involved in the FWM process. Theoretically, the degree of third-order coherence of thermal light [23] equals 6. In this experiment, the measured degree of third-order coherence of pseudothermal light source was 5.11 due to the imperfect thermal like correlations. The third-order auto-correlation functions of beams are shown in Fig. 3. The error bar is standard deviation of ${g^{(3)}}(0)$ in groups of data. It can be seen that the degree of third-order coherence of beams via FWM process are much higher than that of pseudothermal light, which realize three-photon superbunching effects in this experiment. Figure 4 illustrates the third-order cross-correlation functions of any three beams via FWM process. This indicated that high-order correlation of beams were strong. In the above sections, it was confirmed that the degree of two- and third-order coherence of beams involved in FWM process were manipulated by selecting the statistical distributions of the incident light experimentally. The measured degree of second- and third-order coherence of beams considerably exceeded those of the pseudothermal light.
Fig. 3. The third-order auto-correlation functions of (a) FWM signal, (b) probe beam, (c) forward and (d) backward pump beam. The 3D third-order coherence function is plotted as a function of ${t_{13}}\equiv {t_1} - {t_3}$ and ${t_{23}} \equiv {t_2} - {t_3}$.
Fig. 4. The third-order cross-correlation functions of any three beams involved in FWM process, (a) $FWM$ & $Probe$ & $Pum{p_b}$, (b) $Pum{p_f}$& $Probe$ & $Pum{p_b}$, (c) $FWM$ & $Probe$ & $Pum{p_f}$.
Next we further manipulated degree of second- and third-order coherence of beams by controlling various influencing factors, such as temperature $T$, laser detuning $\Delta$, power of light source and incident beams $P_{pro}$, $P_{pump_f}$ and $P_{pump_b}$. Firstly, the dependence of measured degree of second- and third-order coherence on temperature $T$ are shown in Figs. 5(a) and 5(c). The temperature varied from $45^\circ C$ to $115^\circ C$ by heater, and other experimental parameters were the same as before. The degree of second- and third-order coherence of all beams increased to the highest values at the beginning and then decreased sharply as temperature increased. As the degree of nonlinear interaction between incident beams and medium enhanced when temperature increased, the modulation for degree of second- and third-order coherence via FWM process became stronger. To find out the reason why the degree of second- and third-order coherence decreased sharply when temperature reached a certain value, we measured the power of beams passing through Rb cell. It was found that the power decreased gradually as temperature increased. That meant absorption effect played a filter role [38], which weakened modulation for degree of second- and third-order coherence of beams.
Fig. 5. Dependence of the degree of second- and third-order coherence on (a)(c) temperature and (b)(d) laser detuning.
The dependence of the measured degree of second- and third-order coherence on laser detuning $\Delta$ are shown in Figs. 5(b) and 5(d). The laser detuning varied from resonant to 10 MHz by tuning laser, and other experimental conditions were the same as before. We found that the degree of second- and third-order coherence of FWM signal decreased gradually as laser detuning increased, while the other three beams were basically stable and then decreased at some point. Because the other three beams were still in the strong resonance range at the beginning, and then gradually away from the strong interaction area as the laser detuning increased. Since the FWM frequency spectrum was relatively narrow, the acquisition points got far away from the resonance range slowly, and the degree of nonlinear interaction weakened.
Then the influence of pseudothermal light source power $P_{ther}$ on degree of the second- and third-order coherence are shown in Figs. 6(a) and 6(c). The power varied by adding variable attenuator, and other experimental parameters were the same as before. The degree of second- and third-order coherence of each beam gradually ascended and then basically remained stable as power of pseudothermal light source increased. The reason was that the light intensity was proportional to the degree of nonlinear interaction. However, when the power reached a certain value, the degree of second- and third-order coherence remained unchanged because the nonlinear interaction was basically saturated.
Fig. 6. Dependence of the degree of second- and third-order coherence on power of (a)(c) pseudothermal light source and (b)(d) probe beam.
Finally, the relationships between the degree of second- and third-order coherence and the power of incident beams $P_{pro}$, $P_{pump_f}$, $P_{pump_b}$ are shown in Figs. 6(b), 6(d) and Fig. 7. From Figs. 6(b), 6(d) and Figs. 7(a), 7(c), the degree of second- and third-order coherence of FWM signal increased first and then became stable as power of probe beam and forward pump increased, while the other three beams were basically stable. It showed that power of probe beam and forward pump had effects on FWM signal, but had little effect on the other three beams. Similarly, from Figs. 7(b) and 7(d), the degree of second- and third-order coherence of forward pump increased first and then became stable as power of backward pump increased, while the other three beams were basically stable. It showed that power of backward pump had effects on forward pump, but had little effect on the other three beams. Thus, we can choose the most appropriate incident power combination to get the maximum degree of second- and third-order coherence.
Fig. 7. Dependence of the degree of second- and third-order coherence on power of (a)(c) forward pump and (b)(d) backward pump beam.
We also suspected that the superbunching effect was not caused by nonlinear interaction, but by the aperiodic occlusion of light intensity by atoms in Brownian movement. In order to eliminate doubts, we assumed that the incident beams only pass through Rb cell at $90^\circ C$ without nonlinear effect, the measured result was still pseudothermal light with ${g^{(2)}}(0)=1.80$. Therefore, it was demonstrated that the degree of the second- and third-order coherence of beams involved in FWM process were manipulated by nonlinear interaction.
Interesting though, compared with other nonlinear effects, there are more controllable variables in FWM process. Unlike optical harmonic process, the advantage of FWM is that the pump field can also be replaced by a field different from the probe field, or even a field with more fluctuation than the probe field to improve the auto-correlation and cross-correlation of the output beams. Even unlike parameter conversion, the degree of second- and higher-order coherence of beams can be flexibly controlled by changing various experimental parameters in nonlinear interaction.
4. Conclusion
In conclusion, the degree of second- and third-order coherence of light are manipulated by changing controllable variables in FWM process. By selecting the statistical distributions of the incident light and controlling various influencing factors in the nonlinear interaction, the measured degree of second- and third-order coherence of the output beams considerably exceed those of incident pseudothermal light, namely two- and multi-photon superbunching effect are observed experimentally. Additionally, strong second- and third-order cross-correlation exist between the output light beams. We also investigate the dependence of superbunching light on the temperature of Rb vapor, the laser detuning and the optical power of all incident light beams. This experimental scheme could serve as a light source with a tunable superbunching degree that be applied to intensity interference phenomena studies and potentially improve the performance of related applications such as the visibility of ghost imaging.
Funding
Shaanxi Key Research and Development Project (2019ZDLGY09-10); Key Innovation Team of Shaanxi Province (2018TD-024); 111 Project (B14040).
Disclosures
The authors declare no conflicts of interest.
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