1. Introduction

Atomic coherence which can lead to a great many of significant physical phenomena is the result of interaction between light and materials. Recently, increasingly experimental researches are reported on the atomic coherence [1–3 ]. It is well-known that the six-wave mixing (SWM) is a nonlinear optical effect generated by lights with different frequencies and different quantum properties. In the nonlinear wave-mixing process, atomic coherence of the electromagnetically induced transparency (EIT) plays a critical role has been demonstrated experimentally and theoretically in recent researches [4–9 ]. Furthermore, the increasingly effect has been paid on the switch between EIT and electromagnetically induced absorption (EIA) [3,10 ]. In the EIT environment, SWM signal can transmit through the atomic medium and the fluorescence can be generated due to spontaneous emission [11–14 ]. And the EIT-based nonlinear schemes can be driven by traveling wave beams as well as by a standing wave (SW) which formed by the two same frequency counter propagating coupling fields [15,16 ]. Thus in an atomic system driven by two counter propagating coupling fields can obtain the large nonlinearity. It is reported that the electromagnetically induced grating [17,18 ] resulted from the interaction of the SW with the atomic coherent medium can possess photonic band gap (PBG) structure in a variety of interesting research results [19,20 ].

In this paper, the optical response of hot rubidium (85Rb) atoms driven by a SW is investigated in an inverted Y-type four level system and six wave mixing band gap signal (SWM BGS) and probe transmission signal are obtained firstly in the experiment. Especially, the two photon process including EIA and electromagnetically induced gain (EIG) are researched experimentally and theoretically for the first time. And we define the two photon process as the second-order nonlinear signals (SNS) in the paper. When we change the frequency detunings and the powers of the laser fields, the interplay between the SWM BGS and SNS as well as its application are also discussed for the first time in this paper. Furthermore, we also demonstrate the relation of the probe transmission signal and fluorescence in our research.

2. Experimental scheme

The experiment was completed in a rubidium atomic vapor cell of ⁸⁵Rb. The energy levels 5S _1/2(F = 3) (|0>), 5S_1/2(F = 2) (|3>), 5P_3/2 (|1>) and 5D_5/2 (|2>) compose a inverted-Y energy system as shown in Fig. 1(b) . Probe laser beam E₁ (wavelength of 780.245 nm, frequency ω₁ and wave vector k₁) connects the transition 5S_1/2 (F = 3) (|0>) to 5P_3/2 (|1>). The laser beam E₂ (wavelength of 775.978 nm, ω₂, k₂) drive an upper transition 5P_3/2 (|1〉) to 5D_5/2 (|2〉). A pair of coupling laser beams E₃ (wavelength of 780.238 nm, ω₃, k₃) and E´₃ (wavelength of 780.238 nm, ω´₃, k´₃) connect the transition 5S_1/2(F = 2) (|3〉) to 5P_3/2 (|1〉. The coupling field E₃ and E´₃ which propagate through ⁸⁵Rb vapor in the opposite direction generate a standing wave $E_c=\widehat{y}[E_{3}cos({\omega }_{3}t-k_{3}x)+{E^{\prime }}_{3}cos({{\omega }^{\prime }}_{3}t+{k^{\prime }}_{3}x)]$, which results into an EIG. Furthermore EIG will lead to a PBG structure as shown in Fig. 1(c). In addition, the intensity of probe beam E₁ is the only weak laser beam while other laser beams are strong. As illustrated on Fig. 1(a), the weak probe beam E₁ propagates through the ⁸⁵Rb vapors in the same direction of E´₃ with a small angle between them. The dressing field E₂ propagates in the opposite direction of E₃ with a small angle between them. Due to the small angle between the probe beam E₁ and the beam E´₃, the geometry not only satisfies the phase-matching (k_S = k₁ + k₂−k₂ + k₃-k′₃) also provides a convenient spatial separation of the applied laser and generated signal beams. Thus we can detect the generated beams with highly directional [21]. The generated SWM BGS satisfies the phase-matching k_S = k₁ + k₂−k₂ + k₃-k′₃. Besides SWM BGS, there also exists SNS in the reflection signal channel as shown in Fig. 1(a). The reflection signal and the probe transmission signal are detected by a photodiode and avalanche photodiode detectors respectively. In addition, three fluorescence signals caused by spontaneous decay are measured. The second order fluorescence R₀ and fourth order fluorescence R₁, R₂ are generated due to the spontaneous emission from $|\text{1}〉$ and $|2〉$, respectively. Fluorescence signals are captured by another photodiode.

 

Fig. 1 (a) The setup of our experiment. (b) Inverted-Y type energy system. (c) Schematic of an electromagnetically induced grating formed by two coupling beams E₃ and E′₃. (d) The single dressed energy level schematic diagrams and the calculated single dressed period energy levels. (e) The double dressed energy level schematic diagrams and the calculated double dressed periodic energy levels.

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It is the essential to the generation of the PBG structure that the medium should have a periodic refractive index. In order to get the periodic refractive index, the susceptibility of the medium should be periodic by considering the relation of the refractive index with the susceptibility, i.e.,$n=\sqrt{1+\mathrm{Re}(\chi )}$. Thus we must get the periodic energy level structure to generate the periodic susceptibility. Hence, the periodic energy levels can be obtained in Fig. 1(d)-1(e) by introducing periodic standing wave field. In Fig. 1(d), the level $|1〉$will be split into two dressed states$|G_{30}\pm 〉$. The two dressed states$|G_{30}\pm 〉$ have the eigenvalues${\lambda }_{{}_{|G_{30}\pm 〉}}=-\Delta {}_3/2\pm \sqrt{{\Delta }_3^2/4+\left|G{}_{30}\right|{}^2}$. And the${\lambda }_{|G_{30}\pm |}$values are also periodic along x because $\left|G{}_{30}\right|{}^2$ is periodic along x-axis. Thus we can obtain the periodic energy levels as shown in Fig. 1(e). When E₂ is turn on, due to the second level dressing effect of E₂, $|G_{30}+〉$is further split into two dressed states$|G_{30}+G_2\pm 〉$. The two dressed states $|G_{30}+G_2\pm 〉$ have the eigenvalues${\lambda }_{|G_{30}+G{}_2\pm 〉}=\frac{-{\Delta }_3+\sqrt{{\Delta }_3^2+4\left|G{}_{30}\right|{}^2}}{2}+\frac{{\Delta }_2^{"}\pm \sqrt{{\Delta }_2^{"2}+4\left|G{}_2\right|{}^2}}{2}$with ${\text{Δ}}_{\text{2}}^{\text{"}}{\text{=Δ}}_{\text{2}}\text{-{-Δ}{}_{\text{3}}+\sqrt{{\text{Δ}}_{\text{3}}^{\text{2}}\text{+4}\left|G{}_{30}\right|{}^{\text{2}}}\text{}/2}$. Thus we also obtain the double dressed periodic energy levels as shown in Fig. 1(d)-1(e).

3. Results and discussions

First, we observe the probe transmission signal, reflection signal, and fluorescence when we block the different laser beam in case of scanning the frequency detuning${\Delta }_2$in Fig. 2 . Figure 2(a) represents the changes on the probe transmission signal when different laser beams are blocked. When one block ${E^{\prime }}_3$and $E_3$ in Fig. 2(a1), there is a peak on the probe transmission signal locating at the position of ${\Delta }_1=-{\Delta }_2$. And the peak stands for the enhancement of probe transmission signal caused by the term ${\left|G_2\right|}^2/d_{20}$in${\rho }_{10}^{(1)}$ (see Appendix). Next the beam${E^{\prime }}_3$is blocked in Fig. 2(a2), the peak is lower compared with the Fig. 2(a1) because of the strong cascade-dressing interaction between $E_3$(${E^{\prime }}_3$) and $E_2$ which leads to the suppression effect of ${E^{\prime }}_3$ on the peak by considering the cascade-dressing term${\left|G_2\right|}^2/d_{20}+{\left|G_{30}\right|}^2/d_{30}$in${\rho }_{10}^{(1)}$. When we block the beam$E_1$in Fig. 2(a3), the peak disappeared because the term $G_1=0$ in${\rho }_{10}^{(1)}$. When all laser beams are opened in Fig. 2(a4), we can find that the peak on the probe transmission signal becomes lower than peaks in Fig. 2(a1) and Fig. 2(a2). The reason is that the cascade interaction which results in the suppression effect of ${E^{\prime }}_3$ on the peak plays a vital role on the intensity of peak. Figure 2(b) confirms the existence of SWM BGS and SNS in the reflection signal channel with the different laser beam blocked. Because the beam$E_2$ propagates in the opposite direction of $E_1$ in our experiment, the Doppler-free condition is satisfied in the subsystem $|0〉-|1〉-|2〉$ and the experimental setup will be in the EIT environment. So the EIT-induced EIG will be generated in the EIT environment. Figure 2(b) has two columns of signals and we observe the left of column signals firstly. When we block beams $E_3$ and ${E^{\prime }}_3$ in Fig. 2(b1), the reflection signal only includes the SNS resulting from ${\rho }_{20}^{(2)}$ (see Appendix) which has one dip and one peak. The peak stands for the EIG and appears at the position of ${\Delta }_1=-{\Delta }_2$ according to the generating term $d_{20}$in the ${\rho }_{10}^{(3)}$(see Appendix). The dip appearing at the location of ${\Delta }_1=-{\Delta }_2+G_2$is the EIA caused by single photon term $d_{10}$dressed by the term ${\left|G_2\right|}^2/d_{20}$in the${\rho }_{20}^{(2)}$. Also the position of the EIG and EIA moves with ${\Delta }_1$ changing .When one only block the beam ${E^{\prime }}_3$in Fig. 2(b2), we can find that the peak becomes higher and dip becomes deeper compared with Fig. 2(b1) because of the optical pumping effect of the ${E^{\prime }}_3$. Now we only block the laser beam $E_1$in Fig. 2(b3), the EIG and EIA disappeared because the term $G_1=0$ in${\rho }_{20}^{(2)}$. In Fig. 2(b4), we open all the laser beams. One can find that the peak is higher than any cases illustrated in Fig. 2(b1)-2(b3) and thus a new peak is generated which does not belong to the SNS. If a peak appears in the case of scanning the dressing frequency detuning${\Delta }_2$, the peak must be the enhancement of four wave mixing band gap signal (FWM BGS) satisfying the phase-matching k_F = k₁ + k₃-k′₃ which is actually SWM BGS [22]. The series expansion of${\rho }_{10}^{(3)}$ (see Appendix),${\rho }_{10}^{(3)}=\{-i{G}_1{G}_3{G^{\prime }}_3[1-2(|G_{30}{|}^2/d_{30}+|G_2{|}^2/d_{20})/d_{10}]\}/d_{30}{d}_{10}^2$ shows us the equivalence between the enhancement of FWM BGS and SWM BGS, in which the last term $i{G}_1{G}_3{G^{\prime }}_3|G_2{|}^2/[{(d_{10})}^3{d}_{30}{d}_{20}]$is actually SWM BGS. The SWM BGS is generated by$E_1$,${E^{\prime }}_3$,$E_3$ and $E_2$according to the condition of phase matching. And the peak is the sum of the SNS caused by${\rho }_{20}^{(2)}$and SWM BGS. We can see only one peak in Fig. 2(b4) since the SWM BGS fills up the dip caused by the EIA. Next, we consider the right column signal. The right column signal is the SNS resulted from the${\rho }_{23}^{(2)}$. Also the right column signal locates about at${\Delta }_2=-{\Delta }_3$ whose position changes with ${\Delta }_3$ varying by considering the dressing term ${\left|G_2\right|}^2/d_{23}$in the${\rho }_{23}^{(2)}$. When we block${E^{\prime }}_3$,$E_3$ (Fig. 2 (b1)) or only block ${E^{\prime }}_3$ (Fig. 2(b2)), the SNS disappears which results from the term ${G^{\prime }}_3=0$in the${\rho }_{23}^{(2)}$. When we open ${E^{\prime }}_3$ and $E_3$with blocking $E_1$in Fig. 2(b3), one dip appears at ${\Delta }_2=-{\Delta }_3$ due to the term ${\left|G_2\right|}^2/d_{23}$ in the${\rho }_{23}^{(2)}$. In Fig. 2(b4), we open all the laser beams, the dip is deeper compared with Fig. 2(b3) due to the increasing optical pumping effect of the$E_1$.The corresponding fluorescence signals with the different laser beam blocked are also presented in Fig. 2(c). We analyze the left column signal first.When the beams $E_1$ and $E_2$ is turned on with ${E^{\prime }}_3$and $E_3$blocked in Fig. 2(c1), a peak described by ${\rho }_{22D}^{(4)}$ (see Appendix) appears at ${\Delta }_1=-{\Delta }_2$ whose position changes with ${\Delta }_1$ varying.Next when we open$E_1$, $E_2$ and $E_3$ with only ${E^{\prime }}_3$ blocked in Fig. 2(c2), it can be seen that the peak on the fluorescence is higher than the one in Fig. 2(c1).It is because that the nest dressing effect leads to the enhancement effect of $E_3$on the peak according to the nest-dressing term${\left|G_2\right|}^2/(d_{10}+{\left|G_{30}\right|}^2/d_{30})$ in the${\rho }_{22D}^{(4)}$.When we only block$E_1$in Fig. 2(c3), the peak disappears according to the term$G_1=0$in the${\rho }_{22D}^{(4)}$. Physically, the source of fluorescence radiation is the excited transition of particles from the ground state, which disappeared with $E_1$ blocked. In Fig. 2(c4), we open all laser beams. One can find that the peak becomes higher compared with the Fig. 2(c1) and Fig. 2(c2) because both ${E^{\prime }}_3$and $E_3$make the more enhancement effect on fluorescence. Next, we consider the right column signal. When beams $E_1$ and $E_2$ are turned on with ${E^{\prime }}_3$and $E_3$blocked (Fig. 2(c1)) or with $E_3$ opened and only ${E^{\prime }}_3$ blocked (Fig. 2(c2)), the peak caused by ${{\rho }^{\prime }}_{22}^{(4)}$disappears according to the term ${G^{\prime }}_3=0$in the${{\rho }^{\prime }}_{22}^{(4)}$. Next, we turn on beams$E_2$, $E_3$and ${E^{\prime }}_3$ with $E_1$ blocked in Fig. 2(c3), there is a peak on the fluorescence located at${\Delta }_2=-{\Delta }_3$caused by the generating term $d_{21}$in the${{\rho }^{\prime }}_{22}^{(4)}$. And the location of peak changes with the ${\Delta }_3$varying. When we open all the laser beams in Fig. 2(c4), the peak becomes higher than the one in Fig. 2(c3) because of the enhancement effect of beam $E_1$by considering the nest-dressing term ${\left|G_2\right|}^2/(d_{13}+{\left|G_1\right|}^2/d_{03})$ in the${{\rho }^{\prime }}_{22}^{(4)}$.

 

Fig. 2 Measured (a) probe transmission signal, (b) reflection signal, and (c) fluorescence versus Δ₂ from 200 MHz to 550 MHz when different beams are blocked. (a1)- (c1) ${E^{\prime }}_3$and $E_3$ blocked, (a2)-(c2) ${E^{\prime }}_3$blocked, (a3)-(c3) $E_1$blocked, and (a4)-(c4) no beam blocked. Δ₁ = −280 MHz and Δ₃ = −450MHz.

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In following we concentrate on the signal intensity dependence on the power of laser beam by scanning ${\Delta }_2$in Fig. 3 . First, when the power of the beam $E_2$(P₂) changes from small to large, we arrange the experimental curves from bottom to top in Fig. 3(a1)-Fig. 3(c1). In Fig. 3(a1), the peak shows the enhancement of the probe transmission signal, which locates ${\Delta }_2=-{\Delta }_1$decided by the term ${\left|G_2\right|}^2/d_{20}$in${\rho }_{10}^{(1)}$. When one changes P₂ from small to large values, peaks become higher due to the increasing dressing effect of $E_2$by considering the dressing term ${\left|G_2\right|}^2/d_{20}$ in ${\rho }_{10}^{(1)}$. The dip locating at ${\Delta }_2=-{\Delta }_3$ represents the suppression of the probe transmission signal caused by the term ${\left|G_2\right|}^2/d_{23}$in${\rho }_{13}^{(1)}$ and dip is deeper also due to the stronger dressing effect of$E_2$according to the term ${\left|G_2\right|}^2/d_{23}$in${\rho }_{13}^{(1)}$ with P₂ changes from small to large. Figure 3(b1) illustrates the competition between SWM BGS and SNS in the reflection signal channel. We analyze the right column signal first. The signal located at ${\Delta }_2=-{\Delta }_1$is the sum of the SNS caused by${\rho }_{20}^{(2)}$and the SWM BGS. When we change P₂ from small to large values, the signal switches from a dip to a peak and then the peak continues to become higher. It is because the SWM BGS fills up the dip caused by the EIA with the P₂ increasing and then the peak will become higher since EIG and SWM BGS become larger with continuing to increase P₂. For the left column signal, there is only a dip (EIA) locates at ${\Delta }_2=-{\Delta }_3-G_2$caused by ${\left|G_2\right|}^2/d_{23}$in the${\rho }_{23}^{(2)}$when the value of P₂ is small. And with changing P₂ from small to large, a peak caused by the term $d_{23}$in ${\rho }_{23}^{(2)}$ appears at the location of ${\Delta }_2=-{\Delta }_3$ and becomes higher because of the classical effect of G₂ in the numerator of ${\rho }_{23}^{(2)}$. For the fluorescence in Fig. 3(c1), we consider the right column signal first. The dip represents that the second order fluorescence related to ${\rho }_{11}^{(2)}$ is suppressed by the dressing effect of E₂, which locates at ${\Delta }_2=-{\Delta }_1$according to ${\left|G_2\right|}^2/d_{20}$in ${\rho }_{11}^{(2)}$. The peak is the fourth order fluorescence related to${\rho }_{22D}^{(4)}$. In contrast, with P₂ changes from small to large values, the dip becomes deeper and the peak becomes higher due to the increasing nest-dressing effect of E₂ according to the nest-dressing term${\left|G_2\right|}^2/(d_{10}+{\left|G_{30}\right|}^2/d_{30})$. Next we analyze the left column signal. The peak shows the fourth order fluorescence caused by${{\rho }^{\prime }}_{22}^{(4)}$. When we change the P₂ from small to large values, the peak becomes higher because the nest-dressing effect is increasing by considering the nest-term ${\left|G_2\right|}^2/(d_{13}+{\left|G_1\right|}^2/d_{03})$ in the${{\rho }^{\prime }}_{22}^{(4)}$. The calculated probe transmission signal, reflection signal and fluorescence are displayed separately on Figs. 3(a2)-3(c2). Such theoretical calculations confirm our experimental analysis stated above.

 

Fig. 3 For Fig. 3(a)-3(c), measured (a1) probe transmission signal, (b1) reflection signal and (c1) fluorescence versus Δ₂, when we select five different discrete values of P₂ as black (3.5 mW), red(2.4 mW), blue(1.5 mW), green(1.2 mW) and purple(0.7mW) and Δ₃ = −110 MHz, Δ₁ = −280 MHz. (a2), (b2) and (c2) are the theoretical calculations of (a1), (b1) and (c1), respectively. For Fig. 3(d)-3(f), measured (d1) probe transmission signal, (e1) reflection signal and (f1) fluorescence versus Δ₂, when we select five different discrete values of P₁ as black (10.05 mW), red(8.62 mW), blue(6.1 mW), green(3.83 mW) and purple(1.86 mW) and Δ₃ = −410 MHz, Δ₁ = −280 MHz. (d2), (e2) and (f2) are the theoretical calculations of (d1), (d1) and (f1), respectively.

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Next we concentrate on the signal intensity dependence on the power of the probe E₁ (P₁) in Fig. 3(d)-3(f). When the P₁ changes from small to large, we arrange the experimental curves from bottom to top. In Fig. 3(d1), the peak caused by ${\rho }_{10}^{(1)}$locates at the position of ${\Delta }_2=-{\Delta }_1$decided by the term ${\left|G_2\right|}^2/d_{20}$ and the intensity of the peak increases with P₁ increasing. The dip located at ${\Delta }_2=-{\Delta }_3$ is caused by the term ${\left|G_2\right|}^2/d_{23}$in ${\rho }_{13}^{(1)}$and it becomes deeper as P₁ increases because of the increasing dressing effect of the term ${\left|G_1\right|}^2/d_{03}$in${\rho }_{13}^{(1)}$. We illustrate the reflection signal with the different P₁ in Fig. 3(e1). There exist two columns of signals on the experimental curve and we consider the left column signal first. The signal located at ${\Delta }_2=-{\Delta }_1$ stands for the sum of SWM BGS and SNS decided by${\rho }_{20}^{(2)}$. As P₁ increases, the peak becomes lower and then disappears and the dip continues to become deeper. It is because that the EIA in SNS is sensitive to the variation of P₁ according to the dressing term ${\left|G_1\right|}^2/{\Gamma }_{00}$ in ${\rho }_{20}^{(2)}$ compared with SWM BGS. The right column signal is the SNS caused by${\rho }_{23}^{(2)}$. When we change P₁ from small to large values, the dip is deeper because of the dressing effect of the term ${\left|G_1\right|}^2/d_{03}$in${\rho }_{23}^{(2)}$. For fluorescence signal (Fig. 3(f1)), the left column signal locates at the position of${\Delta }_2=-{\Delta }_1$.The intensity of peak which is mainly dependent on the beam E₁ intensity according to${\rho }_{22D}^{(4)}$ get greatly larger as P₁ increases. The dip caused by the dressing term ${\left|G_2\right|}^2/d_{20}$ in ${\rho }_{11}^{(2)}$ appears when the value of P₁ increases to a certain value. The dip becomes deeper with P₁ increasing for the reason that $G_1$ has the modification effect on the dressing effect of the term ${\left|G_2\right|}^2/d_{20}$ in${\rho }_{11}^{(2)}$. The right column signal is located at${\Delta }_2=-{\Delta }_3$according to ${{\rho }^{\prime }}_{22}^{(4)}$ and the peak gets larger with P₁ increasing, it is because that the nest-dressing effect in the term ${\left|G_2\right|}^2/(d_{13}+{\left|G_1\right|}^2/d_{03})$ of ${{\rho }^{\prime }}_{22}^{(4)}$ becomes stronger. Especially, the calculated probe transmission signal, reflection signal and fluorescence are displayed separately on Figs. 3(d2)-3(f2). Such theoretical calculations confirm our experimental analysis stated above.

Finally, we observe the probe transmission signal, reflection signal and fluorescence under different detunings by scanning ${\Delta }_2$in Fig. 4 . First, when the coupling detuning ${\Delta }_3$ changes from small to large values, we arrange the experimental curves from bottom to top in Figs. 4(a1)-4(c1). For the probe transmission signal, the dip located at${\Delta }_2=-{\Delta }_3$ moves toward peak due to the term ${\left|G_2\right|}^2/d_{23}$in ${\rho }_{13}^{(1)}$and the dip becomes smaller until invisible when we change ${\Delta }_3$to about the value of${\Delta }_1$. The peak locates at ${\Delta }_2=-{\Delta }_1$according to the term${\left|G_2\right|}^2/d_{20}$in${\rho }_{10}^{(1)}$. When one changes ${\Delta }_3$ to about${\Delta }_1$, the intensity of the peak becomes lower due to the term $|G_{30}{|}^2/d_{30}$ in${\rho }_{10}^{(1)}$. The intensity of the peak will become stronger when ${\Delta }_3$ far away${\Delta }_1$. For the reflection signal (Fig. 4(b1)), the dip located at ${\Delta }_2=-{\Delta }_3$ stands for the EIA caused by the term ${\left|G_2\right|}^2/d_{23}$in${\rho }_{23}^{(2)}$. And the position of EIA changes with the value of ${\Delta }_3$ varying. The dip will be deepest when ${\Delta }_3$ is near the value of ${\Delta }_1$ and it becomes shallower when ${\Delta }_3$ is far away from the value of ${\Delta }_1$because of the term ${\left|G_1\right|}^2/d_{03}$in${\rho }_{23}^{(2)}$. The peak located at ${\Delta }_2=-{\Delta }_1$ is the sum of SWM BGS and SNS induced by${\rho }_{20}^{(2)}$. The peak is smaller when we change ${\Delta }_3$ to about ${\Delta }_1$ because of the dressing effects of the term $|G_{30}{|}^2/d_{30}$in${\rho }_{20}^{(2)}$and the term $|G_{30}{|}^2/d_{30}$in${\rho }_{10}^{(5)}$. When ${\Delta }_3$ is far away from the value of${\Delta }_1$, the peak becomes higher because the dressing effect of $E_3$ decreases. Since the intensity of SNS is very small, the intensity of peak almost comes from SWM BGS. In Fig. 4(b1), we can modulate the intensity of SWM BGS through changing${\Delta }_3$. Thus, the schematic diagram of the adjustable optical amplifier can be illustrated in Fig. 4(g) where we set the FWM BGS as input and SWM BGS is the enhancement of the FWM BGS. Further, SNS plays the modulation role which can control the amplification amplitude for FWM BGS through changing the detuning${\Delta }_3$. Then SWM BGS modulated by SNS is the output of adjustable optical amplifier. In Fig. 4(c1), we observe the changes on the fluorescence when one changes the detuning${\Delta }_3$. One peak located at ${\Delta }_2=-{\Delta }_3$ is the fourth order fluorescence related to the${{\rho }^{\prime }}_{22}^{(4)}$. Another peak which locates at ${\Delta }_2=-{\Delta }_1$ is the fourth order fluorescence caused by${\rho }_{22D}^{(4)}$. First, we consider the peak related to ${{\rho }^{\prime }}_{22}^{(4)}$.When we change ${\Delta }_3$ from small to large values, the location of the peak is changing and the intensity of the peak is smaller because the increasing value of ${\Delta }_3$ makes the ${{\rho }^{\prime }}_{22}^{(4)}$(peak) smaller. Especially, when${\Delta }_1={\Delta }_3$, the peak related to ${{\rho }^{\prime }}_{22}^{(4)}$overlaps with the peak caused by ${\rho }_{22D}^{(4)}$. The peak is higher when we change ${\Delta }_3$ to about ${\Delta }_1$ and the peak becomes smaller when ${\Delta }_3$ is far away from the value of ${\Delta }_1$ due to the nest-dressing effect by considering the term ${\left|G_2\right|}^2/(d_{10}+{\left|G_{30}\right|}^2/d_{30})$ in${\rho }_{22D}^{(4)}$. The calculated probe transmission signal, reflection signal and fluorescence are displayed separately on Figs. 4(a2)-4(c2). Such theoretical calculations confirm our experimental analysis stated above.

 

Fig. 4 For Fig. 4(a)-4(c) measured (a1) probe transmission signal, (b1) reflection signal and (c1) fluorescence versus Δ₂, when we select five different discrete values of Δ₃ as black (−150MHz), red (−200MHz), blue (−280MHz), green (−350MHz) and purple (−450MHz) and Δ₁ = −280 MHz. (a2), (b2) and (c2) are the theoretical calculations of (a1), (b1) and (c1), respectively. For Fig. 4(d)-4(f) measured (d1) probe transmission signal, (e1) reflection signal and (f1) fluorescence versus Δ₂, when we select five different discrete values of Δ₁ as black (−80MHz), red (−180MHz), blue (−280MHz), green (−300MHz) and purple (−380MHz) and Δ₃ = −280 MHz. (d2), (e2) and (f2) are the theoretical calculations of (d1), (d1) and (f1), respectively. (g) the schematic diagram of the adjustable optical amplifier.

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Next, we analyze the changes on the reflection signal, the probe transmission signal and fluorescence when we set the different ${\Delta }_1$ values in Fig. 4(d1)-4(f1). Figure 4(d1) shows the influence of detuning ${\Delta }_1$on the probe transmission signal. The dip caused by the${\rho }_{13}^{(1)}$locates at ${\Delta }_2=-{\Delta }_3$according to the term${\left|G_2\right|}^2/d_{23}$. The peak caused by the${\rho }_{10}^{(1)}$locates at ${\Delta }_2=-{\Delta }_1$ denoting by the term ${\left|G_2\right|}^2/d_{20}$ and the position of peak moves with the value of ${\Delta }_1$ changing. When we change the value of ${\Delta }_1$ from small to large, the intensity of peak is higher when the value of ${\Delta }_1$ is far away from ${\Delta }_3$ and peak becomes lower when ${\Delta }_1$ is near${\Delta }_3$. Especially, the intensity of the peak is smallest on the condition of${\Delta }_1={\Delta }_3$. It is because of the dressing effect of the term $|G_{30}{|}^2/d_{30}$ in ${\rho }_{10}^{(1)}$. In Fig. 4(e1), we analyze the changes on the SWM BGS and SNS when the value of ${\Delta }_1$is different. We consider the dip located at the position of ${\Delta }_2=-{\Delta }_3$first. The dip is the EIA caused by${\rho }_{23}^{(2)}$. The dip becomes deeper in the case that the value of ${\Delta }_1$ is near ${\Delta }_3$ and the dip is shallower on the condition that ${\Delta }_1$ is far away from ${\Delta }_3$when we change ${\Delta }_1$ from small to big values. This is because the term ${\left|G_1\right|}^2/d_{03}$ in ${\rho }_{23}^{(2)}$makes the intensity of dip changes. Now, we analyze the signal located at${\Delta }_2=-{\Delta }_1$. The signal is the sum of the SWM BGS and SNS caused by ${\rho }_{20}^{(2)}$and the position of the signal changes with the ${\Delta }_1$varying. When${\Delta }_3<{\Delta }_1$, the signal is a peak which is the sum of SWM BGS and SNS and the intensity of peak is smaller with the ${\Delta }_3$increasing because the SWM BGS becomes smaller. Specially, when${\Delta }_3={\Delta }_1$, the peak and the dip overlap. And then the signal becomes a dip which is the EIA because the SWM BGS disappeared in the case of${\Delta }_3>{\Delta }_1$. Figure 4(f1) illustrates the changes on the fluorescence when we set different${\Delta }_1$. One peak caused by ${{\rho }^{\prime }}_{22}^{(4)}$locates at ${\Delta }_2=-{\Delta }_3$ and its position keeps fixed. Another peak caused by ${\rho }_{22D}^{(4)}$ locates at ${\Delta }_2=-{\Delta }_1$ and its position moves with ${\Delta }_1$ changing. When we change ${\Delta }_1$ from small to large values, the intensity of peak is lower on the condition of the value of ${\Delta }_1$ being far away from ${\Delta }_3$ and peak becomes higher when ${\Delta }_1$ is near ${\Delta }_3$. Especially, the intensity of the peak is highest on the condition of${\Delta }_1={\Delta }_3$. It is because of the nest dressing effect of G₃₀ by considering the term ${\left|G_2\right|}^2/(d_{10}+{\left|G_{30}\right|}^2/d_{30})$in${\rho }_{22D}^{(4)}$. Especially, the calculated probe transmission signal, reflection signal and fluorescence are displayed separately on Figs. 4(d2)-4(f2). Such theoretical calculations confirm our experimental analysis stated above.

4. Conclusion

In summary, the probe transmission signal, reflection signal, and fluorescence are compared for the first time in the case of scanning the dressing detuning. We experimentally and theoretically demonstrated the interplay between SNS and SWM BGS when we change the frequency detunings of the coupling and probe field as well as the powers of the probe and dressing field. We also observed the relation of the probe transmission signal and fluorescence. Such research could find its applications in adjustable optical amplifiers.