1. Introduction

Electromagnetically induced transparency (EIT) can reduce the linear absorption of a beam when it passes through a resonant medium [1–3], and EIT also plays an important role in generating multi-wave mixing (MWM) processes [4, 5]. EIT enhanced four-wave mixing (FWM) in rubidium vapor [3, 6, 7], the efficient six-wave mixing (SWM) in a five-level close-cycled atomic system [4], and coexisting of FWM and SWM in two ladder-type EIT windows [7] have been investigated experimentally. It is very interesting that EIT can change into electromagnetically induced absorption (EIA) [8–13] when the coupling field changes from traveling wave to standing wave, which will induce electromagnetically induced grating (EIG) [14–17] that can reflect the probe field [9, 18].

The reflection of the probe field is a result of photonic band gap (PBG) structure of the EIG [19–21]. In previous literatures, the discussed PBG only appears at the EIT places [8, 22–27], and the width of the EIT-type PBG is less than 1 MHz. As there is strong absorption in EIA region, so it is very hard to observe the EIA-type PBG. However, the EIA region is so wide that if there is PBG, it will very be much wider than EIT-type PBG, which would enhance the MWM signals through the Bragg reflection. From this point of view, we plan to construct a configuration to observe the EIA-type PBG, which is not been reported ever before.

In this paper, we experimentally and theoretically study the transmission as well as the reflection of the probe beam, and the MWM signals in the atomic ensemble. In the emission direction of MWM signals, there coexist two signals: the coherent emission (CE) signal in the MWM process and the electromagnetically induced Bragg reflection (EIBR) of the probe beam due to EIG. As the PBG structure of the EIG can be affected by the frequency detunings, powers, and angles related with the pump beams, we theoretically investigate the variation of the EIT-type and EIA-type PBGs by considering the above mentioned influence factors. In experiment, we detect the probe as well as the MWM signals, analyze the intensity variations of the signals by using the PBG theory and demonstrate the existence of EIA-type PBG.

The organization of the paper is that we briefly introduced the basic theory in Sec. 2, display the experimental results and the explanations in detail in Sec. 3, and conclude the paper in Sec.4.

2. Basic Theory

We consider an multi-level atomic system as shown in Fig. 1(a), where the transition |0〉 → |1〉 is driven by one probe field E₁ (with frequency ω₁, wavevector k₁, Rabi frequency G₁, and power P₁), the transition |1〉 ↔ |2〉 is driven by two pump fields E₂ (ω₂, k₂, G₂, P₂)&E′₂ (ω₂, k′₂, G′₂, P′₂), the transition |1〉 ↔ |4〉 is driven by E₄ (ω₄, k₄, G₄, P₄), and the transition |1〉 ↔ |3〉 is driven by E₃ (ω₃, k₃, G₃, P₃)&E′₃ (ω₃, k′₃, G′₃, P′₃). The Rabi frequency is defined as G_i = μ_ijE_i/ħ, in which μ_ij is the electric dipole moment between levels |i〉 and |j〉. The spatial geometry of these beams is displayed in Fig. 1(b), in which E₂&E′₂ with a small angle between them (∼ 0.3°) propagate oppositely to E₁, E₃&E′₃ copropagate with but respectively deviate a small angle from E₂&E′₂, and E₄ copropagate with E₂ with a small angle between them. With E₃&E′₃ off, we can observe two FWM signals E_F1 (ω₁, k_F1) and E_F2 (ω₂, k_F2), satisfying the phase matching conditions k_F1 =k₁+k₂−k′₂ and k_F2 =k₁+k₄−k₄, respectively. In addition, when E₃&E′₃ are open, there will be two SWM signals E_S1 and E_S2 as well as another FWM signal E_F3, which satisfy the phase-matching conditions k_S1 =k₁+k₃−k′₃+k₄−k₄, k_S2 =k₁+k₃−k′₃+k₂−k′₂, and k_F3 =k₁+k₃−k′₃, respectively. In experiment, the signals are detected by photomultiplier tubes that are displayed along the output directions of the signals.

 

Fig. 1 (a) Diagram of relevant energy levels. (b) Spatial beam geometry. (c) Theoretical PBG according to Eqs. (1)–(3) when only linear susceptibility is considered. (c1) Theoretical three-dimensional PBG versus Δ₁ and Δ₂. The bottom panel is the top-view of the structure. (c2) PBG versus Δ₂ corresponding to the dashed line in (c1), and the inset is the amplified view of the EIT-type PBG marked in the red circle.

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The two couple of pump beams E₂&E′₂ and E₃&E′₃ can interfere with each other to form two standing waves, i.e., ${\left|G_2\right|}^2=G_2^2+{G_2^{\prime }}^2+2G_2{G}_2^{\prime }\text{cos}(2k_{2}x)$ and ${\left|G_3\right|}^2=G_3^2+{G_3^{\prime }}^2+2G_3{G}_3^{\prime }\text{cos}(2k_{3}x)$ to offer periodical dressing effects. The periodic standing wave also induces EIG, which will lead to EIBR of E₁ because of the inertial PBG structure. So, the detected MWM signal includes EIBR and CE. According to the dressed perturbation chain method [28], the total susceptibility of the system can be written as

As to the PBG of the EIG, we can obtain by adoptting the plane-wave expansion method [24,25,27]. Expending χ as Fourier series and considering two-mode approximation, we obtain the expression of the Bragg wavevector as follows

Also, the absorption of the probe photon will generate fluorescence (FS) signal. In this system, without any filter, the detected FS signal is composed of three components: the single-photon FS in the decay |1〉 → |0〉 (R1), two-photon FS in the decays |2〉 → |1〉 (R2) and |4〉 → |2〉 (R3). So, the energy conservation for the system is

3. Experimental Results and Analysis

3.1. Frequency detuning modulated PBG

We first execute our experiment in the system composed by the states 5S_1/2(F = 3) (|0〉), 5P_3/2 (|1〉), 5D_5/2 (|2〉), 5D_3/2 (|4〉) and 5S_1/2(F = 2) (|3〉) in ⁸⁵Rb [29], and the temperature of the atomic cell is about 220°C, which corresponds to an atomic density of 10¹² cm⁻³. From four external cavity diode lasers (ECDL), the wavelengths of E₁, E₂&E′₂, E₃&E′₃ and E₄ are 780 nm, 775.98 nm, 795 nm and 776.16 nm, respectively. The widths of laser beams are all about 100 μm. Because ECDL is a cw laser that the high-order nonlinear susceptibilities will not be considered because of not high enough beam powers.

In Fig. 2(a), we show the experimental results versus Δ₂ with different Δ₁. With Δ₂ changing from Δ₁ + Δ₂ = 0, near resonant, to far away from resonant along the positive direction successively, the probe transmission (PT) shows a single peak, decreasing left-peak and increasing right-dip, and a single dip correspondingly (Fig. 2(a1)). Because a single EIT, reducing left-EIT and rising right-EIA, and single EIA occur at the above places successively. In addition, the concomitant PBGs are tiny, left-narrow and right-wide, and large along the positive Δ₁ direction, respectively, which would reflect and decrease the probe to make the dip in the PT more obvious. The case for Δ₁ changing along the negative direction can be understood similarly as above.

 

Fig. 2 (a1)–(a3) Measured PT, FWM, and FS signal versus Δ₂ for different Δ₁, respectively. (b1)–(b3) Similar to (a1)–(a3) but versus Δ₂ for different Δ₄. (c1) and (c2) Measured PT and FWM signal with a larger power of E₁ and dressing effects from E₂ and E₄. (d1)–(d3) Theoretical PBGs versus Δ₁&Δ₂ and Δ₂&Δ₄ correspond to (a)–(c), respectively.

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For the FWM signal E_F1, it shows two enhancement peaks and one suppression dip when Δ₁ is scanned as shown by the background profile in Fig. 2(a2), which can be also theoretically demonstrated by ${\rho }_{10}^{(3)}$. When Δ₂ is scanned with Δ₁ being fixed, E_F1 will theoretically show enhancement peak, half-enhancement peak and half-suppression dip, suppression dip, half-suppression dip and half-enhancement peak, and last an enhancement peak when Δ₁ is changing from a large negative value to a large positive value. However, all the curves exhibited in Fig. 2(a2) show two peaks, which seems contradict to the theoretical prediction mentioned above. The reason is because of the BR of PBG as shown in Fig. 2(d1) that is according to Eq. (1) from the plane-wave expansion method. And the wider the PBG, the more BR from probe will be added into E_F1, thus the BR can fill the suppression dip to lead to a small peak. For the case Δ₁ = 0, the dip is too deep for BR to fill up that there is still a small dip left over in the profile.

It is worth mentioning that EIA-type PBG is much wider than EIT-type PBG as shown in Fig. 2(d1). Because the refractive index contrast is much larger than the absorption, but at the dark state the former is relatively small though the latter is significantly suppressed. Considering that FS signal together with PT and FWM satisfies Eq. (2), the FS shown in Fig. 2(a3) is easy to understand.

Doubly dressing effect should be considered if we open E₄, and K for χ⁽¹⁾ will be modified as K = d₁ + |G₂|²/d₂ + |G₄|²/d₄ with d₄ = Γ₄₀ + i(Δ₁ + Δ₄). Thus, the Fourier coefficients in Eqs. (3a) and (3b) should be replaced by the doubly dressed ones. In Fig. 2(b), PT, FWM and FS signals versus Δ₂ with different Δ₄ and Δ₁ = 0 are exhibited. The background of PT shows an EIT peak because of the satisfaction of the two photon resonance (TPR) condition Δ₁ + Δ₄ = 0, and the PTs versus Δ₂ also show EIT peaks are also because the TPR condition Δ₁ + Δ₂ = 0 is satisfied. Just from the doubly dressed χ⁽¹⁾ in which Δ₂ and Δ₄ play a similar role, one can also predict that the background PT versus Δ₄ as well as the PTs versus Δ₂ should show similar profiles. The height of the PT versus Δ₂ is the smallest because K is the biggest (both the two TPR conditions are satisfied around Δ₄ = 0). The other condition is that EIBR from the PBG will also reduce the PT intensity. As shown in Fig. 2(d2), the PBG becomes wider with bigger Δ₄ that would transfer more energy from PT into FWM to fill up the dips and induce peaks as shown in Fig. 2(a2). Because the PBG is the smallest around Δ₄ = 0, the EIBR is also not big, thus the corresponding FWM is also the smallest.

If E₁ is large sufficiently to give dressing effect, we study the tri-dressing effects as shown in Fig. 2(c). Without the dressing effect from E₁, the PT will show a EIT peak because of the satisfaction of Δ₁ + Δ₂ = 0 in χ⁽¹⁾ around Δ₁ = 0. However, with Δ₁ increasing from 0, the PT first shows a dip in a peak, then a peak and a dip, and last only a dip. In order to understand the phenomenon, we introduce |G₁|²/[Γ₀₀ + |G₂|²/(Γ₂₁ + iΔ₂)] into K besides the other two dressing fields. By modifying the Fourier coefficients c₀ and c₁ with the tri-dressing ones, we display the corresponding PBG in Fig. 2(d3). In comparison with the EIT-type PBG in Fig. 2(d1), the PBG here is EIA-type because the dressing effect from E₁ divides EIT into two parts and shifts them. Thus, there are two EITs and one EIA if Δ₂ is scanned around Δ₁ = 0. Considering that the EIA-type PBG is much broader than the EIT-type, the EIBR will transfer part of the PT intensity into FWM more here than that in Fig. 2(a). So, a peak together with a dip will appear in the PT around Δ₁ = 0. If Δ₁ is large enough, only EIA can be satisfied which will swallow the peak and dig a dip in the PT.

For the FWM, it will show a peak because of the enhancement condition around Δ₁ = 0. When Δ₁ increases, the suppression and enhancement can coexist, that the peak decreases. When Δ₁ increases over a critical value, only enhancement condition can be satisfied, and considering the enhancement is much weaker than that around Δ₁ = 0, the peak here is the smallest. We note here that the FWM in Fig. 2(c2) is always enhanced by the EIBR. But, the PBG width (Eq. (5)) decreases with Δ₁ increasing as shown in Fig. 2(d3), that the EIBR from PT to FWM also reduces. Even though there is a region in which EIT-type and EIA-type PBGs coexist that can lead to a quite large EIBR, the suppression in this region still reduces the peak.

3.2. Angle and power modulated PBG

If we introduce an additional phase factor e^iΔΦ to the dressing fields, K in χ⁽¹⁾ for PT is modified as K₁ = d₁ + |G₂|²e^iΔΦ/d₂ (for single dressed) or K₂ = d₁ + |G₂|²e^iΔΦ/d₂ + |G₄|²/d₄ (for doubly dressed). Here, ΔΦ connects with the angle between the two incident beams, and it could be manipulated by the orientations of induced dipole moments [9]. Correspondingly, the doubly dressed E_F1 which is proportional to (1/K₂)² and FS which is proportional to 1/[d₂ + |G₂|²e^iΔΦ/(d₁ + |G₄|²/d₄)] can be also controlled by ΔΦ. In addition, by adjusting ΔΦ, bright and dark states could switch into each other, so that the PBGs associated with them would also be affected. In this part, we only discuss the region around Δ₁ = 0, in which the PT shows pure EIT peak. In each row of Fig. 3(a), there are three couples of signals that are recorded with ΔΦ = π/2 (left) and ΔΦ = 5π/6 (right), respectively. The configuration of Fig. 3(b) as well as that of Fig. 3(c) is same as that of Fig. 3(a).

 

Fig. 3 (a1)–(a3) PT, FWM and FS signals versus Δ₂ with different Δ₁ and Δ₄ = 0. (b1)–(b3) Similar to (a1)–(a3) but with different Δ₄ and Δ₁ = 0. The other parameters are P₁ =3.6 mW, P₂ =30.6 mW, P′₂ =5.4 mW and P₃ =14.0 mW. (c1)–(c2) PT and FWM signals versus Δ₂ with Δ₁ fixed. (d1)–(d2) Similar to (c1)–(c2) but with Δ₁ =100 MHz when E₁, E₂ and E′₂ are turned on. Other parameters are P₁ =4 mW and P′₂ =8 mW with P₂ =2.4, 8, 12, 16 and 20 mW from bottom to top. (e1)–(e3) Theoretical PBGs correspond to (a)–(d), respectively.

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In Fig. 3(a1), the PTs with ΔΦ = π/2 show pure peaks. When ΔΦ is changed into 5π/6, the PBG as shown in Fig. 3(e1) becomes wider and leads to more EIBR and dig a deep dip in the peak of the PT profile. The corresponding FWM shown in Fig. 3(a2) at ΔΦ = π/2 shows a pure peak with slightly AT splitting because K₁ plays a slightly bigger role than d₂ in ${\rho }_{10}^{(3)}$. When ΔΦ = 5π/6, the increased EIBR from PT significantly enhances the left peak of the AT splitting FWM.

Because the reflecting properties of the EIG depend on ΔΦ, we can investigate the different angle-property for doubly dressed case with Δ₂ scanned by changing Δ₄, as shown in Fig. 3(b). As analyzed in Fig. 2(b), the PTs with ΔΦ = π/2 shown in Fig. 3(b1) will show EIT peaks. When ΔΦ = 5π/6, the width of the PBG increases sharply as shown in Fig. 3(e2) which will lead to the PT switch from a peak to a dip because of the quite large EIBR. Considering that the EIBR will transfer into FWM, the FWM displayed in Fig. 3(b2) is enhanced when ΔΦ is changed from π/2 to 5π/6. The discussion above can also explain the FS as shown in Fig. 3(b3), the intensity of which can be considered as Iin subtracting the intensities of PT and FWM.

In order to investigate the angle-dependence, we change ΔΦ in a much wide range, from −π/6 to 7π/6. The experimental results are shown in Figs. 3(c1) and 3(c2), which are the PT and FWM, respectively. We can see that the PT first shows a small peak, next to a high peak, then a small peak, then a small dip, then a deep dip, and last a small dip from top to bottom. Taking the corresponding EIA-type PBG (the EIT-type case can be neglected in comparison with the EIA-type one) shown in Fig. 3(e1) into account, as indicated by the white dashed lines, the changing of PBG width is in a small, smaller, small, high, higher, and high manner from top to bottom. Because of the idea that the wider the PBG the more EIBR, the PT changing in Fig. 3(c1) is reasonable. As to the corresponding FWM, as shown in Fig. 3(c2), can be also explained in the way discussed above, that the wide PBG will enhance the FWM due to the EIBR which can be modulated by ΔΦ. However, in the Fig. 3(c2), although the PBG width at ΔΦ = π or 7π/6 is obvious broader than that at ΔΦ = 2π/3 (see the Fig. 3(e1)), the FWM signal is much smaller, which contradicts the analysis debated above. The reason is that Δ₂ ≈ Δ₃ can be fulfilled theoretically, however it cannot exactly obtained in experiment when we scanning Δ₂. Thus, the EIBR will be reduced greatly if Δ₂ ≈ Δ₃ cannot be satisfied very well.

The powers of dressing and probe fields also can affect the PBG. In Figs. 3(d1)–3(d3), with Δ₁ fixed and P₂ being very low, a single-peak structure with a not obvious dip in the PT versus Δ₂ can be observed. With P₂ increases, the peak moves leftward and a dip appears on right of the peak. Further increasing P₂, the dip becomes wider and deeper, and the peak also becomes higher gradually. In the transmission probe as shown in Fig. 3(d1), the peak (dip) higher (lower) than baseline is EIT (EIA) induced by E₂(E′₂) through the term |G₂|²/d₂ when Δ₂ is scanned. The corresponding bright state is at Δ₁ + Δ₂|G₂|²/|G₁|² = 0 and dark state at Δ₂ = 0. With the power of E₂(E′₂) increased, the height of EIT (EIA) gets higher (lower) as shown in Fig. 3(d1), which leads to the enhancement of the FWM as shown in Fig. 3(d2) because of the increasing EIT-type as well as EIA-type PBG width which is shown in Fig. 3(e3). At last, the EIA-type PBG is much wider than the EIT-type one, that the two peaks of FWM merge into one wide peak (the top curve in Fig. 3(c2)).

3.3. Modulated EIA-type PBG in SWM

To further understand the EIA-type PBG, we investigate the SWM signal E_S1 in the system by turning all fields. Because of the multi-dressing effects, the K in χ⁽¹⁾ will be modified into d₁ +e^iΔΦ[|G₁|²/(Γ₀₀ +|G₂|²/d₂₁)+|G₂|²/d₂ +|G₄|²/(d₄₁ +|G₂|²/d₁₂₄)] with d₂₁ = Γ₂₁ +iΔ₂, d₄₁ = Γ₄₁ + i(Δ₁ + Δ₄) and d₁₂₄ = Γ₂₄ + i(Δ₁ + Δ₂ + Δ₄). We show the intensities of the probe transmission, SWM and FS signals versus Δ₂ (from negative to positive) with different ΔΦ in Figs. 4(a)–4(c), respectively. The corresponding theoretical PBG is shown in Fig. 4(d).

 

Fig. 4 PT (a), SWM (b) and FS signals (c) versus Δ₂ at Δ₁ = −40 MHz with ΔΦ = −3π/4, −5π/8, −π/2, −π/4, 0 and π/4 from the bottom to top, respectively. (d) Theoretical PBG versus Δ₂ and ΔΦ, and the dashed lines correspond to the curves in (a). Other parameters are P₁ =7 mW, P₂ =17 mW, and P′₂ =8 mW.

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Let’s explore the PBG shown in Fig. 4(d) first. Indicated by the dashed lines from bottom to top, the PBG width first decreases, and then increases; when it reaches a maximum it decreases again. Recalling the EIBR effect of PBG, it can be understood that the PT first shows a dip, then a peak-dip structure, and last a peak along the increasing of ΔΦ from bottom to top in Fig. 4(a). Specifically, at ΔΦ = −3π/4, EIA-type PBG is narrow, so the EIBR is weak and the PT shows a peak. With ΔΦ changing from −3π/4 to −π/2, the width of EIA-type PBG decreases, which leads the peak in PT to get larger. The peak-dip structure is for the reason that the PBG gets modulated as ΔΦ is altered. When ΔΦ changes from −π/2 to π/4, the PBG is much broadened although PBG is narrow at ΔΦ = π/4, which will lead to much stronger EIBR and the switch from peak into dip. As to the E_S1 signal shown in Fig. 4(b), it can be understood by the changing of EIBR analyzed above.

For the FS signal shown in Fig. 4(c), it can be determined by the PT and E_S1 signals according to Eq. (6), which is also affected by the changing of PBG. For example, with the increasing of PBG, the PT decreases, while the sum of E_S1 and FS signals increases.

Next, we consider a five-level atomic system as shown in Fig. 1(a) is composed of energy levels 3S_1/2(F = 1) (|0〉), 3P_3/2 (|1〉), 4D_3/2 (|2〉), 5S_1/2 (|4〉) and 3S_1/2(F = 2) (|3〉) of sodium [29]. The transitions |0〉 → |1〉, |1〉 → |2〉, |1〉 → |4〉, |1〉 → |3〉 are driven by fields with wavelengths 589.0 nm, 568.8 nm, 616.1 nm, and 589.0 nm, respectively [30]. The atomic density as well as the width of the beams is similar to rubidium vapour as mentioned before. Due to the pulsed laser beams are applied into the system, the third-order nonlinear susceptibility need to be considered.

In Figs. 5(a1) and 5(a2), the PTs (first row) as well as FWM (second row) versus Δ₂ with different Δ₁ around and away from the resonance region are shown, respectively. The corresponding theoretical simulations of PBG are shown in Figs. 5(b1) and 5(b2), respectively.

 

Fig. 5 (a1) PT (first row), FWM (second row) signals versus Δ₂ and PT images (third row) with Δ₁ =10, 15, 20, 25, and 30 MHz from left to right, respectively. (a2) Similar to (a1) but with Δ₁ far away from TPR. The x–labels just display the values of Δ₁, at which the experimental results are obtained (have noting to do with the x–scale). (b1) and (b2) Theoretical PBGs correspond to (a1) and (a2), respectively.

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When Δ₁ is set Δ₁ = 10, 15 and 20 MHz (away from Δ₁ = 0), a dip and a very tiny peak within this dip can be seen in the PT as shown in Fig. 5(a1). If only the linear susceptibility χ⁽¹⁾ is considered, only an EIA can be obtained, but if the third-order nonlinear susceptibility χ⁽³⁾ is taken into consideration, an EIT with narrower width appears in the center of EIA (the three left-up curves in Fig. 5(a1)). Here, EIA is concomitant with wide PBG as shown in the Fig. 5(b1), and the effect of the PBG concomitant with EIT can be neglected. As discussed above, the wider PBG due to EIA can result in strong FWM signal via both enhancing the CE and EIBR components. However, the EIT will lead to a dip in this FWM peak via suppressing both the two components, as shown by the three left-low curves in Fig. 5(a1). However, as shown by the two right-up curves in Fig. 5(a1), with Δ₁ increasing, the wide dip in the PT becomes shallower (Δ₁ = 25 MHz) and even disappears (Δ₁ = 30 MHz), but the peak within it becomes higher. The reason is that EIA-type PBG width decreases with Δ₁ increasing as depicted by Fig. 5(b1), which leads to the decreasing of EIBR of the probe and enhancement of the PT. The disappearance of the wide dip at Δ₁ = 30 MHz is because that the EIT due to χ⁽¹⁾ becomes stronger, while the EIA due to the combined effect of χ⁽¹⁾ and χ⁽³⁾ becomes weaker. Therefore, there is only single peak in the PT. However, though the EIBR component decreases, the CE component is not influenced because the atomic coherence effect is still significant in this region, so the peaks of FWM is not completely corresponding to the dips in the PT signal as described by the two right-low curves in Fig. 5(a1).

PT and FWM affected by the PBG can be also verified by the splitting of the probe images, which connects with the nonlinear phase shift by ϕ_NL ∝ 1/I₁ [31]. There are two reasons responsible for such phenomena. First, according to the relative position between the probe E₁ and the strong dressing E₂ (E′₂), E₁ overlaps with E₂ and E′₂ in y direction, which leads to the splitting of E₁ in y direction due to the attraction of E₂ and E′₂ beam. Second, when Δ₁ changes from 10 MHz to 30 MHz, the intensity of PT is affected by the modulated PBG due to changing of Δ₁, i.e, the PT increases with the increasing of Δ₁ due to the decreasing PBG width. Therefore, with I₁ increasing, the splitting of E₁ image weakens, as shown by the images in Fig. 5(a1).

Figure 5(a2) shows the PT (top row) and FWM signal (bottom row) versus Δ₂ but with Δ₁ far away from TPR when χ⁽¹⁾ and χ⁽³⁾ are considered simultaneously. The corresponding PBGs are shown by the dashed lines in Fig. 5(b2), in which the EIT is indicated by the slope region inbetween the dashed lines at Δ₁ = −100 MHz and Δ₁ = 50 MHz. As discussed above, the PT shown in Fig. 5(a2) can be easily understood according to the PBG sketch in Fig. 5(b2). For example, the dip in the PT at Δ₁ = −300 MHz can be explained by the wide EIA-type PBG indicated by the bottom dashed line in Fig. 5(b2). Corresponding to the PT, FWM shown in Fig. 5(a2) can be also understood through the PBG width together with the discussions mentioned above.

4. Conclusion

In summary, we experimentally and theoretically investigated the enhancement of the MWM signals by the EIBR from EIT- and EIA-type PBGs in atomic vapors by considering two EIGs simultaneously. We found that the EIA-type PBG is much wider than the EIT-type one, so the former one can bring more EIBR and make the MWM signal at EIA place be enhanced significantly. Such enhancement effect can be affected by the frequency detunings, powers, and the related angles of the involved fields. The theory and experimental results agree with each other very well. The investigation provides a novel perspective on PBG formed in atomic vapors, and has potential applications in all-optical communications as well as signal processing.