Interplay between six wave mixing photonic band gap signal and second-order nonlinear signal in electromagnetically induced grating

For the first time, we experimentally and theoretically research about the second-order nonlinear signal (SNS) including electromagnetically induced absorbing (EIA) and electromagnetically induced gain (EIG), six wave mixing band gap signal (SWM BGS) resulting from photonic band gap structure in an inverted Y-type four level system with the electromagnetically induced grating. The interplay between the SNS and SWM BGS is illustrated clearly for the first time. When we change the frequency detuning to make the SWM BGS and SNS overlap, the SWM BGS is suppressed and the intensity of SNS is strongest near the resonance point. We can control the intensity of the SWM BGS and EIG caused by the classic effect through changing the power of coupling field. And the changes on the EIA generated by the quantum effect are obtained by changing the power of dressing field. Since the SWM BGS is the enhancement of the four wave mixing band gap signal (FWM BGS), when we set FWM BGS as the input and SNS as the modulation role to control the amplification amplitude for the FWM BGS in our scheme, the adjustable optical amplifier can be obtained. ©2015 Optical Society of America OCIS codes: (190.4380) Nonlinear optics, four-wave mixing; (190.4180) Multiphoton processes; (300.2570) Four-wave mixing; (270.1670) Coherent optical effects. References and links 1. M. Kash, V. Sautenkov, A. Zibrov, L. Hollberg, G. Welch, M. Lukin, Y. Rostovtsev, E. Fry, and M. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82(26), 5229–5232 (1999). 2. S. Wielandy and A. Gaeta, “Investigation of electromagnetically induced transparency in the strong probe regime,” Phys. Rev. A 58(3), 2500–2505 (1998). 3. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997). 4. Y. Q. 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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][2][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][5][6][7][8][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][12][13][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.ˆ

Experimental scheme
, 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 E 1 is the only weak laser beam while other laser beams are strong.As illustrated on Fig. 1(a), the weak probe beam E 1 propagates through the 85 Rb vapors in the same direction of E´3 with a small angle between them.The dressing field E 2 propagates in the opposite direction of E 3 with a small angle between them.Due to the small angle between the probe beam E 1 and the beam E´3, the geometry not only satisfies the phase-matching (k S = k 1 + k 2 −k 2 + k 3 -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 = k 1 + k 2 −k 2 + k 3 -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 R 0 and fourth order fluorescence R 1 , R 2 are generated due to the spontaneous emission from 1 and 2 , respectively.Fluorescence signals are captured by another photodiode.
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., 1 Re( ) n χ = + . 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 . Thus we also obtain the double dressed periodic energy levels as shown in Fig. 1(d)-1(e).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 2 Δ in Fig. 2. we block the beam 1 E in Fig. 2(a3), the peak disappeared because the term 1 0 G = in (1)   10 ρ .

Results and discussions
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 3 E′ 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 2 E propagates in the opposite direction of 1 E 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.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 2 Δ , the peak must be the enhancement of four wave mixing band gap signal (FWM BGS) satisfying the phase-matching k F = k 1 + k 3 -k′ 3 which is actually SWM BGS [22].The series expansion of due to the increasing optical pumping effect of the 1 E .The corresponding fluorescence signals with the different laser beam blocked are also presented in Fig. 2(c E′ 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 3 E on the peak according to the nest-dressing term in the (4)   22D ρ .When we only block 1 E in Fig. 2(c3), the peak disappears according to the term 1 0 G = in the (4)   22D ρ .Physically, the source of fluorescence radiation is the excited transition of particles from the ground state, which disappeared with 1 E 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  In following we concentrate on the signal intensity dependence on the power of laser beam by scanning 2 Δ in Fig. 3. First, when the power of the beam 2 E (P 2 ) changes from small to large, we arrange the experimental curves from bottom to top in Fig. 3 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 P 2 increasing and then the peak will become higher since EIG and SWM BGS become larger with continuing to increase P 2 .For the left column signal, there is only a dip (EIA) locates at ρ .The peak is the fourth order fluorescence related to (4)   22D ρ .In contrast, with P 2 changes from small to large values, the dip becomes deeper and the peak becomes higher due to the increasing nest-dressing effect of E 2 according to the nest-dressing term 3(a2)-3(c2).Such theoretical calculations confirm our experimental analysis stated above.
Next we concentrate on the signal intensity dependence on the power of the probe E 1 (P 1 ) in Fig. 3(d)-3(f).When the P 1 changes from small to large, we arrange the experimental curves from bottom to top.In Fig. 3(d1), the peak caused by (1)   10 ρ locates at the position of the SNS caused by (2)  ρ .When we change P 1 from small to large values, the dip is deeper because of the dressing effect of the term  Δ is far away from the value of 1 Δ , the peak becomes higher because the dressing effect of 3 E 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 3 Δ .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 3 Δ .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 3 Δ .
One peak located at 2 ρ′ overlaps with the peak caused by (4)   22D ρ .The peak is higher when we change 3 Δ to about 1 Δ and the peak becomes smaller when 3 Δ is far away from the value of 1 Δ due to the nest-dressing effect by considering the term

22D
ρ .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.Next, we analyze the changes on the reflection signal, the probe transmission signal and fluorescence when we set the different 1 Δ values in Fig. 4(d1)-4(f1).Figure 4(d1) shows the influence of detuning 1 Δ on the probe transmission signal.The dip caused by the (1)   13 ρ locates at changing.When we change the value of 1 Δ from small to large, the intensity of peak is higher when the value of 1 Δ is far away from 3 Δ and peak becomes lower when 1 Δ is near 3 Δ .
Especially, the intensity of the peak is smallest on the condition of 1

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.

Appendix
According to the Liouville pathway [23]  , the first-order density matrix elements (1)   10 ρ , (1)   13 ρ can be given as follows where 1 2 3 13 According to the pathway Through the pathway According to the relation 0 E N ε χ μρ = , in which N, 0 ε are the atom density and dielectric constant respectively, so the formulations of the linear and nonlinear susceptibilities can be obtained as follows: In order to estimate the probe transmission signal and the reflection signal, we start from the nonlinear coupled wave equations [24,25], #242740 Where E 1 (x) and E r (x) represent the probe transmission signal and reflection signal, respectively.For the fluorescence signals, with the only probe beam E 1 turned on, the single-photon fluorescence R 0 generates.We can describe the expression of with the dressing effect of E 3 , the expression of density-matrix element (4)   22D ρ can be obtained as the amplitude square of which is proportional to the intensity of R 2 .

G
)-1(e) by introducing periodic standing wave field.In Fig.1(d), the level 1 will be split into two dressed states 30 G ± .The two dressed states 30 is periodic along x-axis.Thus we can obtain the periodic energy levels as shown in Fig.1(e).When E 2 is turn on, due to the second level dressing effect of E 2 , 30 G + is further split into two dressed states 30 2

Figure 2 ( 2 Δ 10 ρ 3 E′ ) and 2 E which leads to the suppression effect of 3 E′
a) represents the changes on the probe transmission signal when different laser beams are blocked.When one block 3 E′ and 3 E in Fig. 2(a1), there is a peak on the probe transmission signal locating at the position of 1 = −Δ .And the peak stands for the enhancement of probe transmission signal caused by the term (see Appendix).Next the beam 3 E′ is blocked in Fig. 2(a2), the peak is lower compared with the Fig. 2(a1) because of the strong cascade-dressing interaction between 3 E ( on the peak by considering the cascade-

Figure 2 2 ) 20 ρ 2 Δ 20 ρ 1 Δ 20 ρ
(b) has two columns of signals and we observe the left of column signals firstly.When we block beams 3E and 3 E′ in Fig.2(b1), the reflection signal only includes the SNS resulting from ((see Appendix) which has one dip and one peak.The peak stands for the EIG and appears at the position of 1 .Also the position of the EIG and EIA moves with changing .When one only block the beam 3 E′ 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 3 E′ .Now we only block the laser beam 1 E in Fig.2(b3), the EIG and EIA disappeared because the term 1 0 G = in(2)   .In Fig.2(b4), we open all the laser beams.

and 2 E 20 ρ 23 ρ 3 Δ 23 ρ . When we block 3 E′ , 3 E 23 ρ . When we open 3 E′ and 3 E with blocking 1 E 3 Δ 23 ρ
enhancement of FWM BGS and SWM BGS, in which the last term according to the condition of phase matching.And the peak is the sum of the SNS caused by(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(2)   .Also the right column signal locates about at 2 = −Δ whose position changes with 3 Δ varying by considering the dressing term (Fig.2(b1)) or only block 3 E′ (Fig.2(b2)), the SNS disappears which results from the term 3 0 G′ = in the(2)   in Fig.2(b3), one dip appears at 2 .In Fig.2(b4), we open all the laser beams, the dip is deeper compared with Fig.2(b3)

1 Δ 10 ρ 13 ρ 1 Δ 20 ρ
(a1)-Fig.3(c1).In Fig.#242740 3(a1), the peak shows the enhancement of the probe transmission signal, which locates 2 .When one changes P 2 from small to large values, peaks become higher due to the increasing dressing effect of 2 E with P 2 changes from small to large.Figure3(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 2 = −Δ is the sum of the SNS caused by(2)   and the SWM BGS.When we change P 2 from

23 ρ 23 ρ appears at the location of 2 3 Δ 11 ρ 1 Δ
when the value of P 2 is small.And with changing P 2 from small to large, a peak caused by the term23 d in(2)   = −Δ and becomes higher because of the classical effect of G 2 in the numerator of (2) 23ρ .For the fluorescence in Fig.3(c1), we consider the right column signal first.The dip represents that the second order fluorescence related to(2)   is suppressed by the dressing effect of E 2 , which locates at 2 analyze the left column signal.The peak shows the fourth order fluorescence caused by(4)   22 ρ′ .When we change the P 2 from small to large values, the peak becomes higher because the nest-dressing effect is increasing by considering the nestprobe transmission signal, reflection signal and fluorescence are displayed separately on Figs.

1 increasing. The dip located at 2 3 Δ 13 ρ 1 Δ 2 ) 20 ρ 20 ρ
intensity of the peak increases with P .We illustrate the reflection signal with the different P 1 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 2 = −Δ stands for the sum of SWM BGS and SNS decided by(.As P 1 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 P 1 compared with SWM BGS.The right column signal is

23 ρ 1 Δ 11 ρ 22 ρ′
. For fluorescence signal (Fig.3(f1)), the left column signal locates at the position of 2 = −Δ .The intensity of peak which is mainly dependent on the beam E 1 intensity according to (appears when the value of P 1 increases to a certain value.The dip becomes deeper with P 1 increasing for the reason that 1G has the modification effect on the dressing effect of the term and the peak gets larger with P 1 increasing, it is because that the nest-dressing effect in the term , 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.

3 Δρ′ .When we change 3 Δ 3 Δ
= −Δ is the fourth order fluorescence related to the (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 3 Δ makes the (4) 22 ρ′ (peak) smaller.Especially, when 1 = Δ , the peak related to (4) 22 position of peak moves with the value of 1 Δ

3 Δ 10 ρ 2 3Δ 23 ρ 23 ρ 1 Δ 20 ρ 1 Δ 1 Δ 1 Δρ locates at 2 1 Δ 3 Δ
. In Fig.4(e1), we analyze the changes on the SWM BGS and SNS when the value of 1 Δ is different.We consider the dip located at the position of = −Δ first.The dip is the EIA caused by(2)   .The dip becomes deeper in the case that the value of 1 Δ is near 3 Δ and the dip is shallower on the condition that 1 Δ is far away from 3 Δ when we change 1 Δ from small to big values.This is because the term makes the intensity of dip changes.Now, we analyze the signal located at 2 = −Δ .The signal is the sum of the SWM BGS and SNS caused by(2)   and the position of the signal changes with the 1 Δ varying.When 3 < Δ , the signal is a peak which is the sum of SWM BGS and SNS and the intensity of peak is smaller with the 3 Δ increasing because the SWM BGS becomes smaller.Specially, when 3 = Δ , 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 3 > Δ .Figure4(f1) illustrates the changes on the fluorescence when we set different 1 Δ and its position keeps fixed.Another peak caused by(4)   22D = −Δ and its position moves with 1 Δ changing.When we change 1 Δ from small to large values, the intensity of peak is lower on the condition of the value of 1 Δ being far away from 3 Δ and peak becomes higher when 1 Δ is near 3 Δ .Especially, the intensity of the peak is highest on the condition of 1 = Δ .It is because of the nest dressing effect of G 30 by considering the term transmission signal, reflection signal and fluorescence are displayed separately on Figs.4(d2)-4(f2).Such theoretical calculations confirm our experimental analysis stated above.

1 (
due to the nonlinear susceptibility.L χ ′ , NL χ ′are the zero order coefficients from Fourier expansion of L χ , NL χ , respectively.mismatch magnitude, in which θ is the angle between probe 1 E and 3 E′ .If the length of the sample in x direction is dx, by solving above equations, the reflection signal (R) and the probe transmission signal ( square of which is proportional to the intensity of R 0 .When we turn on the beams E 2 and E 3 , the fluorescence R , the amplitude square of which is proportional to the intensity of R 1 .Aug 2015; accepted 2 Aug 2015; published 17 Sep 2015 © 2015 OSA 21 Sep 2015 | Vol. 23, No. 19 | DOI:10.1364/OE.23.025098 | OPTICS EXPRESS 25109