Observation of the four wave mixing photonic band gap signal in electromagnetically induced grating

Zakir Ullah; Zhiguo Wang; Mengqin Gao; Dan Zhang; Yiqi Zhang; Hong Gao; Yanpeng Zhang

doi:10.1364/OE.22.029544

1. Introduction

Four-wave mixing (FWM) is a nonlinear optical effect that generates light with different frequencies and different quantum properties. Experimental and theoretical studies show that atomic coherence of the electromagnetically induced transparency (EIT) plays a critical role in the nonlinear wave-mixing process [1–6]. Under EIT conditions FWM signals can be allowed to transmit through the atomic medium and also the fluorescence induced by spontaneous emission can be generated [7–10]. Due to the two counter propagating coupling fields, the EIT-based nonlinear schemes can be driven by traveling wave beams as well as by a standing wave (SW). The large nonlinearity was obtained in an atomic system driven by two counter propagating coupling fields which form a SW if the two counter propagating coupling fields have the same frequency [11, 12]. Interaction of the SW with the atomic coherent medium results into an electromagnetically induced grating (EIG) [13, 14], which possesses photonic band gap (PBG) structure as shown in Fig. 1(c). Such EIG has a potential use in all optical switching [15], manipulation of light propagation to create a tunable photonic band gap [16, 17].This research can also be used to make optical diodes. The idea proposed here for optical diode implementation is a photonic crystal generated from a periodic modulation of the optical properties of a medium, which leads to the formation of a band gap. Light is transmitted through the crystal if its frequency is outside the gap, but it is not transmitted if the frequency is inside the gap. Due to the Doppler Effect, the counter propagating light are blue shifted and the co-propagating light is red shifted in the reference frame of the moving photonic crystal [18]. If only one of the shifted frequencies is inside the gap optical diode is formed. Optical non reciprocity of cold atom Bragg mirrors in motion may also be used to make optical diode [19].

Fig. 1 (a) Four-level energy system. (b) Schematic of an EIG formed by two coupling beams E₃ and E′₃. Together with the dressing field E₂ and probe field E₁, a dressed FWM BGS E_F will be generated according to the phase-matching condition K_F = K₁−K₃ + K′₃. (c) The setup of our experiment.

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In this paper, we investigate the optical response of hot rubidium (⁸⁵Rb) atoms driven by a stationary SW coupling field and probe field, from which the PBG structure and four wave mixing band gap signal (FWM BGS) and probe transmission signal (PTS) can be obtained. For the first time we observe optically controllable PBG structure based on an EIT medium driven by a SW field. By optically controllable PBG structure not only we experimentally observe PTS, FWM BGS and Fluorescence signal (FLS), but also observe the enhancement and suppression of these signals under the double dressing effect in an inverted Y-type four level system. Furthermore by measuring the above signals we observed that they satisfy the law of conservation of energy, this information can be used to enhance any one of the signals at the cost of suppressing the others. By finding a strong relation among the signals we are able to find the optimal condition of any one of these signal which can be of interest for particular applications. One example of this is the optimized enhancement of FWM BGS may be used to make optical amplifiers. We also demonstrate that the characteristics of PTS, FWM BGS and FLS can be controlled by frequencies, powers, and relative phase of dressing fields; this information may be useful for the enhancement and modulation of the signals which highlights the signification of this research work. Optical response of medium is also examined by resorting to a set of nonlinear coupled wave equations, which is a powerful tool [20–23] for describing the nonlinear interaction of light fields in such a dressed medium and are here used to test the validity of the experiment result.

2. Basic theory and experimental scheme

2.1 Experimental setup

The experiment was implemented in a cell with rubidium (⁸⁵Rb) atomic vapors at temperature of 53 °C with four energy levels consisting of 5S_1/2(F = 3)(|0〉), 5S_1/2(F = 2)(|3〉), 5P_3/2(|1〉) and ${5D}_{5/2} (| 2 〉)$ as shown in Fig. 1(a).The laser beams are aligned spatially as shown in Fig. 1(c). Probe laser beam E₁ (frequency ω₁ and wave vector k₁) probes the transition |0〉 to |1〉. The two coupling laser beams E₃ (ω₃, k₃) and E´₃ (ω₃, k´₃) with wavelength about 780.238 nm and a vertical polarization are split from an external cavity diode lasers (ECDL) connect the transition |3〉 to |1〉. The dressing laser beam E₂ (ω₂, k₂) with wavelength of 775.978 nm and vertical polarization is from another ECDL drive an upper transition |1〉 to |2〉. The powers of E₁, E₃ and E₃' are 2.1 mW, 13.2 mW and 8.4 mW, respectively.P₂ is set to 21mW in our detuning and phase experiments. The coupling field E₃₁ = ŷ[E₃cos(ω₃t−k₃x) + E′₃cos(ω′₃t + k′₃x)], composed by E₃ and E´₃, which propagates through ⁸⁵Rb vapor in the opposite direction, generate a standing wave, which results into an EIG. Furthermore EIG will lead to a PBG structure as shown in Fig. 1(b). Rabi frequency of the coupling field is G₃₁ = μ₃E₃₁/ħ, that we have $| G_{31} |^{2} = μ^{2} / h^{2} [E_{3}^{2} + {E^{'}}_{3}^{2} + 2 E_{3} {E^{'}}_{3} \cos 2 k_{3} x]$ .The probe field E₁ propagates in the same direction of E´₃ through the ⁸⁵Rb vapors with approximately 0.3° angle between them. The dressing field E₂ propagates in the opposite direction of E´₃ with approximately 0.3° angle between them as shown in Fig. 1(c). Generated FWM BGS and PTS are detected by a photodiode (PD1) and avalanche photodiode (PD2) detectors respectively. In addition, two fluorescence signals caused by spontaneous decay are measured. The second order FLS R₀ and fourth order FLS R₁ signals are generated due to the spontaneous emission from $| 1 〉$ to $| 0 〉$ and $| 2 〉$ to $| 1 〉$ , respectively. The fluorescence signals are captured by another photodiode(PD3).

2.2 Basic theory

According to the energy system and Liouville pathways, the first-order and third-order density matrix elements are given as follows

ρ_{10}^{(1)} = i G_{1} / (d_{1} + | G_{31} |^{2} / d_{3} + | G_{2} |^{2} / d_{2})

ρ_{10}^{(3)} = - i G_{1} G_{3} {G^{'}}_{3} / [{(d_{1} + | G_{31} |^{2} / d_{3} + | G_{2} |^{2} / d_{2})}^{2} d_{3}]

where

d_{1} = i Δ_{1} + Γ_{10},

d_{2} = i (Δ_{1} + Δ_{2}) + Γ_{20}

,

d_{3} = i (Δ_{1} - Δ_{3}) + Γ_{30}

and |G₃₁|² = |G₃|² + |G₃′|² + 2G₃G₃′cos(2k₃x).

G_{i} = μ_{i} E_{i} / ℏ

is the Rabi frequency with transition dipole moment

μ_{i}

,

Δ_{i} = Ω_{i} - ω_{i}

(

Ω_{i}

is the resonance frequency of the transition driven by

E_{i}

) and

Γ_{i j}

is transverse relaxation rate between |i〉 and |j〉. An additional phase factor

e^{i Δ φ}

can be introduced into the dressing term

| G_{2} |^{2} / d_{2}

by means of altering the incident angle of E₂ [24] in which way the relative phase

Δ φ

related with the orientations of induced dipole moments

μ_{1}

and

μ_{2}

could be manipulated and Eqs. (1)-(2) can be modified as follows:

{ρ^{'}}_{10}^{(1)} = i G_{1} / (d_{1} + | G_{31} |^{2} / d_{3} + | G_{2} |^{2} e^{i Δ φ} / d_{2})

{ρ^{'}}_{10}^{(3)} = \frac{- i G_{1} G_{3} {G^{'}}_{3}}{{(d_{1} + | G_{31} |^{2} / d {}_{3}+ | G_{2} |^{2} e^{i Δ φ} / d_{2})}^{2} (d_{3})}

According to the relation ε₀χE = Nμρ, in which μ is the transition electric dipole moment and N is the atoms density, we have formulas for the first and third order susceptibility as follows.

χ_{}^{(1)} = \frac{i N μ^{2}}{h ε_{0}} \frac{1}{d_{1} + | G_{31} |^{2} / d_{3} + | G_{2} |^{2} / d_{2}}

χ_{}^{(3)} = - \frac{i N μ^{2}}{h ε_{0}} \frac{1}{{(d_{1} + | G_{31} |^{2} / d_{3} + | G_{2} |^{2} / d_{2})}^{2} d_{3}}

In order to estimate the reflection efficiency, we start from the nonlinear coupled wave equations,

\partial E_{1} (x) / \partial x = - α E_{1} (x) + k e^{- i Δ k_{x} x} E_{F} (x)

- \partial E_{F} (x) / \partial x = - α E_{F} (x) + k e^{i Δ k_{x} x} E_{1} (x)

Where E₁(x) and E_F(x) represents the probe and FWM BGS fields, respectively, α = (ω₁/c)Imχ´⁽¹⁾ /2 is the attenuation of the field due to the absorption of the medium and k = i(ω₁/c)χ´⁽³⁾/2 is the gain due to the nonlinear susceptibility. χ´⁽¹⁾ and χ´⁽³⁾ are the zero order coefficients from Fourier expansion of χ⁽¹⁾ and χ⁽³⁾, respectively.

Δ k_{x} = {2 (ω_{1} \cos θ - ω_{3}) + Re [{χ^{'}}^{(1)}] ω_{1} \cos θ} / c

is the phase mismatch magnitude, in which

θ

is the angle between probe

E_{1}

and

{E^{'}}_{3}

. If length of the sample in x direction is d_x, then by solving above equations, the reflected FWM BGS (R) and PTS (T) are given as

R = {| \frac{1}{k} \frac{e^{- λ_{2}^{+} d_{x}} - e^{- λ_{2}^{-} d_{x}}}{e^{- λ_{2}^{+} d_{x}} {(λ_{1}^{+} + α)}^{- 1} - e^{- λ_{2}^{-} d_{x}} {(λ_{1}^{-} + α)}^{- 1}} |}^{2}

T = {| \frac{e^{(λ_{1}^{+} + λ_{1}^{-}) d_{x}} (λ_{1}^{-} - λ_{1}^{+})}{(λ_{1}^{-} + α) e^{λ_{1}^{-} d_{x}} - (λ_{1}^{+} + α) e^{λ_{1}^{+} d_{x}}} |}^{2}

Where

λ_{1}^{\pm} = - i Δ k_{x} / 2 \pm {[{(α - i Δ k_{x} / 2)}^{2} - k^{2}]}^{1 / 2}

and

λ_{2}^{\pm} = λ_{1}^{\pm} + i Δ k_{x}

.Generally

ρ_{11}^{(2)}

related to second order FLS R₀ and

ρ_{22}^{(4)}

related to fourth order FLS R₁ can be obtained by solving the density-matrix equations. The fluorescence signal R₀ is described by

ρ_{00}^{(0)} \overset{E_{1}}{\to} ρ_{10}^{(1)} \overset{(E_{1}) *}{\to} ρ_{11}^{(2)}

solving the coupled density-matrix equations, the expression of the singly dressed density-matrix element

ρ_{11 S D}^{(2)}

(where SD stand for singly dressed) can be obtain as

ρ_{11 S D}^{(2)} = - {| G_{1} |}^{2} / [(d_{1} + {| G_{31} |}^{2} / d_{3}) Γ_{11}]

the amplitude squared of which is proportional to the intensity of R₀. When the beams E₂ is turned on, the fluorescence process R₀ doubly dressed, described by the Liouville pathway

ρ_{00}^{(0)} \overset{E_{1}}{\to} ρ_{\pm 0}^{(1)} \overset{(E_{1}) *}{\to} ρ_{\pm \pm}^{(2)} .

Therefore the expression of

ρ_{11 S D}^{(2)}

can be modified as

ρ_{11 D D}^{(2)} = - {| G_{1} |}^{2} / Γ_{11} (d_{1} + {| G_{31} |}^{2} / d_{3} + {| G_{2} |}^{2} / d_{2})

(where DD stand for doubly dressed) .For signal R₁, via Liouville pathway

ρ_{00}^{(0)} \overset{E_{1}}{\to} ρ_{10}^{(1)} \overset{E_{2}}{\to} ρ_{20}^{(2)} \overset{(E_{1}) *}{\to} ρ_{21}^{(3)} \overset{(E_{2}) *}{\to} ρ_{22}^{(4)}

, Same way we can obtain the density-matrix element

ρ_{22}^{(4)}

as follows

ρ_{22}^{(4)} = {| G_{1} |}^{2} {| G_{2} |}^{2} / (Γ_{22} d_{1} d_{2} d_{4})

where

d_{4} = Γ_{21} + i Δ_{2}

, the amplitude squared of

ρ_{22}^{(4)}

is proportional to the intensity of R₁. By considering the dressing effect of E₂, the dressed fluorescence process R₁ is given as

ρ_{22 D D}^{(4)} = {| G_{1} |}^{2} {| G_{2} |}^{2} / [Γ_{22} d_{1} d_{4} (d_{2} + {| G_{2} |}^{2} / d_{1}]

By changing angle of E₂, an additional phase factor

e^{i Δ φ}

can be introduced into the dressing term

{| G_{2} |}^{2} / d_{2}

, it is obvious that the dressing effect can be modulated by manipulating the relative phase

Δ φ

, and so the switch between bright and dark states could be achieved. By means of altering the incident angle of E₂, Eqs. (12), (14) can be modified as follows.

{ρ^{'}}_{11 D D}^{(2)} = - {| G_{1} |}^{2} / Γ_{11} (d_{1} + {| G_{31} |}^{2} / d_{3} + {| G_{2} |}^{2} e^{i Δ φ} / d_{2})

{ρ^{'}}_{22 D D}^{(4)} = {| G_{1} |}^{2} {| G_{2} |}^{2} / [Γ_{22} d_{1} d_{4} (d_{2} + {| G_{2} |}^{2} e^{i Δ φ} / d_{1}]

The system obeys law of conservation of energy, according to which

R + T + I_{R_{0}} = I_{i n}

where I_in is intensity of the incident probe, I_R0 is intensity of fluorescence signal R₀, R is reflected four wave mixing band gap signal and T is the transmission of probe signal.

The condition of generating PBG structure is that the medium should have a periodic refractive index. According the relation of the refractive index with the susceptibility, i.e., $n = \sqrt{1 + Re (χ)}$ , in order to get the periodic refractive index, the susceptibility should also be periodic. Further we should generate the periodic energy level structure for getting the periodic susceptibility. Hence, by introducing periodic standing wave field E₃₁, we can obtain the periodic energy levels as shown in Fig. 2. In Figs. 2(a1)-2(a3), the level $| 1 〉$ will be split into two dressed states $| G_{31} \pm 〉$ depending on ∆₃ and |G₃₁|². The two dressed states $| G_{31} \pm 〉$ have the Eigenvalues $λ_{_{| G_{31} \pm 〉}} = - Δ {}_{3}/ 2 \pm \sqrt{Δ_{3}^{2} / 4 + | G {}_{31} | {}^{2}}$ . Since |G₃₁|² is periodic along x-axis, so $λ_{_{| G_{31} \pm 〉}}$ values are also periodic along x. Thus we can obtain the periodic energy levels as shown in Fig. 2(b1)-2(b3). When the probe reaches two-photon resonance Δ₁-Δ₃ = 0, absorption will be suppressed, i.e. the PTS becomes strong. At the same time, the FWM BGS will be suppressed correspondingly. Thus, we define Δ₁-Δ₃ = 0 as the suppression condition.

Fig. 2 (a1)-(a3) the single dressed energy level schematic diagrams and (b1)-(b3) the calculated single dressed periodic energy levels with changing Δ₃. (c1)-(c5) the double dressed energy level schematic diagrams and (d1)-(d5) the calculated double dressed periodic energy levels with changing Δ₂.

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When E₂ is turn on, |G₃₁ + 〉 is further split into two dressed states |G₃₁ + G₂ ± 〉 due to the second level dressing effect of E₂, moreover $| G_{31} + G_{2} \pm 〉$ move with changing Δ₂ as shown in Fig. 2(c1)-2(c2).The two dressed states $| G_{31} + G_{2} \pm 〉$ have the Eigenvalues $λ_{| G {}_{31}+ G {}_{2}\pm 〉} = \frac{- Δ_{3} + \sqrt{Δ_{3}^{2} + 4 | G {}_{31} | {}^{2}}}{2} + \frac{Δ_{2}^{"} \pm \sqrt{Δ_{2}^{" 2} + 4 | G {}_{2} | {}^{2}}}{2}$ with $Δ_{2}^{"} {=Δ}_{2} -{-Δ {}_{3}+ \sqrt{Δ_{3}^{2} +4 | G {}_{31} | {}^{2}}}/2$ .The same way $| G_{31} - 〉$ is further dressed into two second level dressed states $| G_{31} - G_{2} \pm 〉$ as shown in Fig. 2(c4)-2(c5), the Eigenvalues of which are $λ_{| G {}_{31}- G {}_{2}\pm 〉} = \frac{- Δ_{3} - \sqrt{Δ_{3}^{2} + 4 | G {}_{31} | {}^{2}}}{2} + \frac{Δ_{2}^{'} \pm \sqrt{Δ_{2}^{' 2} + 4 | G {}_{2} | {}^{2}}}{2}$ , where $Δ_{2}^{'} {=Δ}_{2} -{-Δ {}_{3}- \sqrt{Δ_{3}^{2} +4 | G {}_{31} | {}^{2}}}/2$ . .In Fig. 2(c3), because of three photon resonance with $Δ {}_{1}= - Δ {}_{2}= Δ {}_{3}$ only two dressed states appear. Thus we also obtain the double dressed periodic energy levels as shown in Fig. 2 (d1)-2(d5).

3. Results and discussions

First, we observed the PTS, FWM BGS and FLS when we scan the probe frequency detuning Δ₁. When E₂ is blocked in Fig. 3(a1), there is a Doppler absorption background at the single-photon resonant condition Δ₁ = 0 due to the term d₁ = Γ₁₀ + iΔ₁ in $ρ_{10}^{(1)}$ and small PTS peak caused by the dressing effect of E₃ (E´₃) according to the term |G₃₁|²/d₃ in Eq. (1). When E₂ beam is on, due to the dressing term $| G_{2} |^{2} / d_{2}$ in $ρ_{10}^{(1)}$ of Eq. (1), the energy level $| 1 〉$ can be dressed to influence PBG structure so that the PTS increases as shown in Fig. 3(a2). The PTS reaches maximum at Δ₁ = Δ₃ = - Δ₂ due to double dressing effect of E₂ and E₃ (E´₃) according to $ρ_{10}^{(1)}$ . In Fig. 3(b1) the emission peak is FWM BGS give by R in Eq. (9) which is from the reflection of the PBG structure. With E₂ on, the FWM BGS will be suppressed due to the dressing effect of E₂ according to $| G_{2} |^{2} / d_{2}$ in $ρ_{10}^{(3)}$ as shown in Fig. 3(b2). Suppression condition is Δ₁ + Δ₂ = 0. In Fig. 3(c1), the profile curve represents the second order FLS R₀ related to $ρ_{11}^{(2)}$ . The small dip in the profile curve is induced by the dressing fields E₃ and E´₃, which appear at Δ₁ = Δ₃ according to ${| G_{31} |}^{2} / d_{3}$ in $ρ_{11 S D}^{(2)}$ . When E₂ is turn on, a narrow emission peak appears in the profile curve shown in Fig. 3(c2), which is the fourth order FLS R₁ related to $ρ_{22}^{(4)}$ .Compared to Fig. 3(c1) the dip becomes deeper because of the double dressing effect of E₂ and E₃ (E´₃) according to $ρ_{11 D D}^{(2)}$ . Compared Figs. 3(a1)-3(c1) with Figs. 3(a2)-3(c2), due to the dressing effect of E₂, the intensity of PTS increases and the intensities of FWM BGS and FLS R₀ decrease, and the whole energy is conserved according to Eq. (17).

Fig. 3 Measured (a) PTS, (b) FWM BGS and (c) FLS versus Δ₁ from −100 MHz to 100 MHz when different beams are blocked. (a1)-(c1) E₂ blocked with Δ₃ = 10 MHz and (a2)-(c2) no beam blocked with Δ₂ = −10 MHz and Δ₃ = 10 MHz.

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Furthermore, we observe second level dressing effect on PTS, FWM BGS and FLS by scanning Δ₂, at different discrete values of Δ₁. Figure 4(a1) represents the second level dressed PTS. The profile (dashed curve) of these PTS baselines shows Doppler absorption background with a peak on it. The profile peak is the intensity of single-dressed PTS induced by E₃ and E₃', which appear at Δ₁ = Δ₃ according to $| G_{31} |^{2} / d_{3}$ of $ρ_{10}^{(1)}$ in Eq. (1). In each sub curve, the peak on the baseline stands for the enhanced PTS induced by the second level dressing effect of E₂, which meets Δ₂ = -Δ₁ according to the dressing term $| G_{2} |^{2} / d_{2}$ in Eq. (1). The total PTS reaches maximum at Δ₂ = -Δ₁ = -Δ₃. In Fig. 4(b1), profile (dashed curve) of the baselines shows the FWM BGS related to R in Eq. (9) from reflection of the PBG structure. Dip in each sub curve shows that FWM BGS is suppressed due to the dressing effect of E₂ at Δ₂ = -Δ₁ according to $| G_{2} |^{2} / d_{2}$ of $ρ_{10}^{(3)}$ in Eq. (2). The deepest dip appears at Δ₂ = -Δ₁ = -Δ₃ corresponding to the total strongest PTS. In Fig. 4(c1), the profile(dashed curve) of the baselines is the second order FLS R₀ signal suppressed by E₃ and E′₃ and it reaches minimum at Δ₁-Δ₃ = 0 due to the term |G₃₁|²/d₃ in Eq. (12). In each sub curve, the peak is the fourth order FLS R₁ related to $ρ_{22}^{(4)}$ , and it reaches the smallest one at Δ₁ = 0 because of the strongest dressing effect of E₂ according to ${| G_{2} |}^{2} / d_{1}$ of $ρ_{22 D D}^{(4)}$ in Eq. (14) as shown in Fig. 4(c3).

Fig. 4 Measured (a1) PTS, (b1) FWM BGS and (c1) FLS versus Δ₂, when we select five different discrete values of Δ₁ as black(−47 MHz), red(−23 MHz), blue(0 MHz), pink(28 MHz) and green(47 MHz) and Δ₃ = 0 MHz. (a2), (b2) and (c2) are the theoretical calculations of (a1), (b1) and (c1), respectively.

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Next, we observe the power dependences of the PTS, FWM BGS and FLS versus Δ₂. Variations in the three types of signals are shown from bottom to top with increasing power of E₂ (P₂) as shown in Figs. 5(a)-5(c). In Fig. 5(a), the baselines of the signals represent the intensity of the PTS induced by E₃ (E´₃). Peaks in the baselines at Δ₁ = Δ₃ shows the enhancement of the PTS induced by the second level dressing effect of E₂.Changing P₂ from small to large values, the peaks becomes higher due to the dressing term $| G_{2} |^{2} / d_{2}$ in $ρ_{10}^{(1)}$ of Eq. (1). The FWM BGS signal intensity is shown by the baselines in Fig. 5(b). Dip in the baseline shows suppression of reflected FWM BGS signal because of the second level dressing effect of E₂ according to the dressing term $| G_{2} |^{2} / d_{2}$ in $ρ_{10}^{(3)}$ of Eq. (2). The dip is shallow at small values of power and become deeper with increasing P₂ due to the enhanced dressing effect of E₂.In Fig. 5(c), the baselines represent the second order FLS R₀ suppressed by E₃. By changing P₂ from small to large values the dip change from shallow to a deeper one because of the enhanced dressing effect of E₂. Variation of the dips shows that R₀ are further suppressed by E₂ according to the term ${| G_{2} |}^{2} / d_{2}$ in $ρ_{11 D D}^{(2)}$ of Eq. (12). R₀ signal with the deepest dip correspond to the FWM BGS with deepest dip and PTS with higher peak. Peaks in the baseline are R₁ related to $ρ_{22 D D}^{(4)}$ , which become higher with increasing the power of the dressing field E₂. By observing these three signals we conclude, with the PTS increasing, the FWM BGS and FLS R₀ decrease to ensure the conservation of energy according to Eq. (17).

Fig. 5 Measured (a) PTS, (b) FWM BGS and (c) FLS versus Δ₂ from −120 MHz to 120 MHz with Δ₃ = Δ₁ = 0, when we set the power of E₂ (P₂) from bottom to top as (1) 9.2 mW, (2) 13.0 mW, (3) 17.1 mW, (4) 21.6 mW, (5) 25.7 mW, respectively.

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Finally, we regulate the PTS, FWM BGS and FLS with the relative phase of E₂ (Δφ) by changing its incident angle. The experimental results can be obtained by scanning Δ₂ with Δ₁ = 0 as shown in Figs. 6(a)-6(c). With the relative phase Δφ changing from $- π /3$ to $2 π /3$ , the PTS in Fig. 6(a) can be switched from a dip to a peak due to the change of dressing effect of E₂ according to ${ρ^{'}}_{10}^{(1)}$ . The peaks stand for the transmission enhancement of probe signal and the dips stand for the absorption enhancement of the PTS. During this process, the deepest dip and the highest peak separately appear at Δφ = $2 π /3$ and Δφ = $- π /6$ , respectively. In Fig. 6(b) and Fig. 6(c) where Δφ is altered from $- π /3$ to $2 π /3$ , a change from shallow to deep then to shallow occurs due to the regulating of the relative phase Δφ. With the variation of the dressing effect of G₂ caused by Δφ, the deepest suppression dips in FWM BGS and R₀ appear at Δφ = $- π /6$ which corresponds to the strongest PTS peak. Compared with the signals at the reference phase Δφ = 0, the FWM BGS and R₀ can be suppressed, i.e. the suppression dip become deeper, with Δφ changed as $- π /6$ . And the FWM BGS and R₀ can be enhanced, i.e. the suppression dip become shallower, with Δφ changed as $- π /3$ and $2 π /3$ . These variations are because of the switch of the dressing effect caused by regulating the relative phase Δφ according to the term $| G_{2} |^{2} e^{i Δ φ} / d_{2}$ in Eqs. (3)-(4) and Eqs. (15)-(16). The controllable enhancement of FWM BGS may be used to make optical amplifiers.

Fig. 6 Measured (a) PTS, (b) FWM BGS and (c) FLS versus Δ₂ with Δ₁ = Δ₃ = −120 MHz, when we change the relative phase of E₂ as $2 π /3$ , $π /3$ , 0, $- π /6$ and $- π /3$ , respectively.

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4. Conclusion

In summary, the single-dressed and double-dressed PTS, FWM BGS and FLS are compared for the first time to deeply comprehend the double-dressing effect on the PBG. We experimentally and theoretically demonstrated that, PTS and FWM BGS and FLS can be manipulated by multiple parameters like, changing power, detuning and relative phase of incident beams. We also observed, the three types of signals satisfy law of conservation of energy. Such research could find its applications in optical amplifiers and quantum information processing.

Acknowledgments

This work was supported by the 973 Program (2012CB921804), NSFC (61108017, 11474228, 61308015, 11104214, 11104216, 61205112), KSTIT of Shaanxi Province (2014KCT-10), NSFC of Shaanxi Province (2014JZ020), KLP of Shaanxi Province (2013SZS04-Z02), FRFCU (2012jdhz05, xjj2012080), and CPSF (2014M560779).

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Observation of the four wave mixing photonic band gap signal in electromagnetically induced grating

Abstract

1. Introduction

2. Basic theory and experimental scheme

2.1 Experimental setup

2.2 Basic theory

3. Results and discussions

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (17)

Optics Express