1. Introduction

Bragg reflectors have periodic refractive index (RI), and are widely applied in various disciplines based on its high reflectivity in the Bragg band, such as reflective coating, distributed feedback lasers (DFBs) [1], and optical cavities [2]. Generally, conventional Bragg reflectors are based on nanoscale periodic structures. Many investigations suggested that atomic vapor is a promising candidate for RI engineering. However, most of researches only showed refractive index (RI) can be uniformly enhanced and absorption can be eliminated simultaneously, while Bragg reflector requires periodical RI distribution [3–9].

Recently, it is suggested that absorption-free Bragg reflector can be achieved by illuminating a standing-wave of a laser field in a homogeneous atomic medium with a configuration of a Ξ-system having successive inverted and non-inverted transitions with slightly different transition frequencies [10]. Compared to the conventional Bragg reflectors, this scheme could be more flexible because the amplitude and period of the RI are adjustable. In such schemes, the intensity of light increases rapidly with the frequency difference. When the difference is too large compared to the radiative decay rate, the intensity of coupling light might be extremely high, as proposed in [10] using Er³⁺ doped crystal.

In this paper, to realize an absorption-free Bragg reflector by practical power, we propose a new scheme using Zeeman levels to build an equivalent of the Ξ-system in [10] which consists two transitions with nearly the same transition frequencies. The proposed level configuration can be easily found in helium or alkaline earth atoms. The radioactive decay rates in these atoms are far greater than those of crystals, and greater than the frequency difference of the two transitions of the equivalent Ξ-system. Thus the power requirement for coupling light can be greatly reduced.

2. Model and analysis

2.1. Level configuration

We use the transitions ³S₁ → ³P_0,2, as shown in Fig. 1, to construct the equivalent Ξ-system. A coupling light (probe light) couples all the transitions |i〉 → |j〉 in ³S₁ → ³P₂ which satisfy Δm = 1 (Δm = 0), with Rabi frequencies Ω_c,ji and detuning Δ_c = ω_ji − ω_c (Δ_p = ω_ji − ω_p), where ω_ji is the resonant frequency of the transition |i〉 → |j〉. A control light couples transition |1〉 → |8〉, with Rabi frequency Ω_ct and detuning Δ_ct = ω₈₁ − ω_ct, where ω_ct refers to the frequency of the control light. We assume all decoherence rates between one sublevel of ³S₁ and one sublevel of ³P_0,2 to be γ₁, all decoherence rates among sublevels of 3S₁ to be γ₂, and all decoherence rates among sublevels of ³P_0,2 to be γ₃. All the detunings are far greater than the decoherence rates, and the frequencies of probe and coupling lights satisfy ω_p − ω_c ≈ 0.

 

Fig. 1 Level structure of transition ³S₁ → ³P_0,2. A coupling light couples all the transitions in ³S₁ → ³P₂ which satisfy Δm = 1, with detuning of Δ_c. A control light couples |³S₁, m = 0〉 → |³P₀, m = 0〉, with detuning of Δ_ct. A probe light couples all transitions of ³S₁ → ³P₂ which satisfy Δm = 0 with detuning Δ_p.

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2.2. Approximation

Assuming that all the population is pumped to |1〉, |2〉 and |3〉 by some incoherent pumping process, which is sufficiently strong to guarantee that the distribution of populations is not affected by the coupling or control light [10]. The susceptibility of the probe light χ can be regarded as the sum of contributions from two Λ-system and a simple two-level system. The first Λ-system is formed by |1〉 |2〉, and |4〉, and the second is formed by |1〉, |3〉, and |6〉, respectively. The simple two-level system includes |2〉 and |5〉. Both of the Λ-system can be regarded as an equivalent Ξ-system in dressed state system, as shown in Fig. 2(a) and Fig. 2(b).

 

Fig. 2 Both of the Λ-systems can be regarded as a Ξ-system in the dressed picture. (a) The equivalent Ξ-system of the first Λ-system in the dressed state picture. |4′〉 (or |2′〉) is shifted by ${\mathrm{\Omega }}_{c,42}^2/{\mathrm{\Delta }}_c$ (or $-{\mathrm{\Delta }}_c-{\mathrm{\Omega }}_{c,42}^2/{\mathrm{\Delta }}_c$) compared to |4〉 in the bare state picture. |1′〉 is shifted by $-{\mathrm{\Omega }}_{\mathit{ct}}^2/{\mathrm{\Delta }}_{\mathit{ct}}-{\mathrm{\Omega }}_{c,61}^2/{\mathrm{\Delta }}_c$ by the coupling |1〉 → |8〉 and |1〉 → |6〉. (b) The equivalent Ξ-system of the second Λ-system in the dressed state picture. |6′〉 (or |3′〉) is shifted by ${\mathrm{\Omega }}_{c,61}^2/{\mathrm{\Delta }}_c$ (or $-{\mathrm{\Delta }}_c-{\mathrm{\Omega }}_{c,61}^2/{\mathrm{\Delta }}_c-{\mathrm{\Omega }}_{c,\mathit{ct}}^2/{\mathrm{\Delta }}_{\mathit{ct}}$) compared to |6〉 in the bare state picture. |3′〉 is shifted by $-{\mathrm{\Omega }}_{c,73}^2/{\mathrm{\Delta }}_c$ by the coupling |3〉 → |7〉 [11].

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Figure 2 (a) shows the contribution to χ from the first Λ-system can be regarded as the sum of two Lorentzians due to two transitions |1′〉 → |2′〉 and |1′〉 → |4′〉, respectively, and can be written

In addition, the contribution of the simple two-level structure to the susceptibility of the probe light is

 

Fig. 3 The effective Ξ-system. The probe light couples the two transitions of nearly the same frequency.

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2.3. Validity of the approximation

The approximation above assumes that the two Λ-systems are independent, and the control light only brings Stark shift to level |1〉. Actually the two Λ-systems in Fig. 1 are coupled to each other, which can be easily seen by expanding Liouville equation ρ̇ = −i[H, ρ]/h̄ + decay terms + pump terms. The evolution of density matrix elements of these two Λ-system satisfying

Doppler broadening will greatly influence this scheme. While the atomic vapor is warm, the coupling light must be extremely collinear with the probe light to ensure they interact with the same Doppler population. In our scheme, cold atoms are needed to avoid the Doppler broadening since counter-propagating coupling light are necessary, and the propagation of the coupling light should be set to perpendicular to the probe light to ensure they have the proper polarization.

2.4. Possible experimental setup

A possible experimental scheme is proposed in Fig. 4, which can be realized in helium atoms or alkaline earth atoms. The propagation direction of the coupling light is perpendicular to the propagation of the probe light and the control light to ensure the polarization is σ₊.In order to avoid Doppler broadening, cold atoms should be used, which can be realized via magnetic optical trap (MOT) [12, 13].

 

Fig. 4 A possible experimental scheme of the Bragg reflector. The polarization of the signal and control light are set to be π and the polarization of the control light is set to be σ₊ to ensure they couple the transitions stated above.

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3. A proposal in helium atoms

In order to prove the effectiveness of the scheme, we give an example in helium atoms. In this case, |μ₆₃|² = |μ₅₂|² = 7.67 × 10⁻⁵⁹(C · m)², |μ₄₁|² = 4|μ₆₃|²/3. The resonance wavelength (in vacuum) of ³S₁ → ³P₂ and ³S₁ → ³P₀ are λ₁ = 1083.33nm and λ₂ = 1083.21nm, respectively [14]. We assume the number density of the helium atoms is N = 1 × 10¹⁵/cm³. The decay rates are set to be γ₁ = 20MHz, γ₂ = 0.01γ₁ and γ₃ = 0.1γ₁. The intensity of coupling light are set to satisfy Ω_c,42 = 2γ₁. ${\mathrm{\Omega }}_{c,61}=\sqrt{3}{\mathrm{\Omega }}_{c,42}$, ${\mathrm{\Omega }}_{c,73}=\sqrt{6}{\mathrm{\Omega }}_{c,42}$, and the detuning Δ_c = 100γ₁. Obviously ξ₁γ₁ ≪ γ₂ and ξ₂γ₁ ≪ γ₂ are satisfied. Hence, γ_2′1′ ≈ γ_3′1′ ≈ γ₂ is guaranteed. We assume the incoherent pump process ρ̇₁₁ = −Γ_pρ₁₁ + (9/5) Γ_pρ₃₃, ρ̇₂₂ = −Γ_pρ₂₂ and ρ̇₃₃ = Γ_pρ₂₂ + Γ_pρ₁₁ − (9/5)Γ_pρ₃₃ with Γ_p = γ₁, i.e., this process depletes ρ₂ and sets the ratio between ρ₁ and ρ₃ to be 9/5. Hence populations will be ρ₁ = 9/14, ρ₂ = 0, and ρ₃ = 5/14 to ensure that the gain and absorption has the same amplitude. The detuning of the probe light is set as Δ_p = 99.90γ₁. We set ${\mathrm{\Omega }}_{\mathit{ct}}=\sqrt{2}{\gamma }_1\text{cos}(kx)$ and Δ_ct = 50γ₁ to achieve the Bragg reflector without absorption, where k is the wave vector of the control light. Figure 5 compares χ calculated directly from Liouville equation and derived from Eq. (1). It is clear that these two results match well and periodical RI without absorption nor gain has been achieved. In this scheme, the intensity of the coupling light and control light is of the order of 100mW/mm², which is feasible.

 

Fig. 5 χ calculated by numerical calculation (blue) and by Eq. (1) (red). The real part (dispersion) and the imaginary part (absorption) are illustrated by solid and dashed curve, respectively. It can be seen that the absorption is very small and the dispersion is modulated periodically along the x-axis. |Im(χ)| < 0.006. x is normalized by 1/k, where k is the wavevector of the control light. Parameters are set as ${\mathrm{\Omega }}_{c,61}=\sqrt{3}{\mathrm{\Omega }}_{c,42}$, ${\mathrm{\Omega }}_{c,73}=\sqrt{6}{\mathrm{\Omega }}_{c,42}$, Δ_c = 100γ₁, ${\mathrm{\Omega }}_{\mathit{ct}}=\sqrt{2}{\gamma }_1\text{cos}(kx)$ and Δ_c = 50γ₁.

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The wavelength of the probe is in the Bragg band gap. The susceptibility of the control light can be written as χ_ct = Nρ₁|μ₈₁|²/(Δ_cε₀h̄) ≈ 0.036 (|μ₈₁| = 5.10 × 10⁻⁵⁹(C · m)²). Hence λ_ct = λ₂/ (1 + χ_ct/2) = 1058.99nm and λ_p = λ₁/ (1 +χ_bg/2) = 1064.11nm, where χ_bg ≈ 0.046 is the background susceptibility of the probe light. The difference between λ_p and Bragg wavelength λ_ct is |λ_p − λ_ct| = 5.1nm, far less than the Bragg band gap width Δλ = λ_ct Δn/(πn_bg) = 23nm in our case. With 100 periods, We can achieve reflectivity of about 1.082 and transmittance of 3.25 × 10⁻⁷. The reflectivity is larger than 1 because the average of the imaginary part is less than 0, providing a little gain.

The parameters are chosen to ensure the approximation to be valid. In Fig. 6 we study the influence of relative parameters of the coupling and control light on the validity of the approximation. Figures 6(a) and 6(b) show the comparison between ρ₆₄ and ρ₄₁. From Eq. (1) we can conclude that the achieved susceptibility increases with the deduction of Δ_c and the increase of the intensity of the coupling light. However, Fig. 6(a) and Fig. 6(b) clearly shows that ρ₆₄ will increase rapidly if we increase the intensity or decrease the detuning of coupling light, making two Λ-system coupling to each other, thus violating the assumption. Similarly, by increasing the intensity or decreasing the detuning of the control light we can increase the variation amplitude of the Bragg reflector, however, it will also cause ρ₈₄ to be too large to maintain the approximation the control light only shifts |1〉. In the simulation we choose the optimum parameters Ω_c,42 = 2γ₁, Δ_c = 50γ₁, ${\mathrm{\Omega }}_{\mathit{ct}}=\sqrt{2}\text{cos}(kx){\gamma }_1$ and Δ_ct = 50γ₁.

 

Fig. 6 The frequency of the probe light is ω_p = ω_2′1′. (a) The relation between |ρ₆₄/ρ₄₁| and Δ_c. (b)The relation between |ρ₆₄/ρ₄₁| and Ω_c,42.(c)The relation between |ρ₈₄/ρ₄₁| and Δ_ct. (d)The relation between |ρ₈₄/ρ₄₁| and Ω_ct.

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Another assumption is that the incoherent process must be strong enough to make sure the required distribution of population can be achieved, i.e., ρ₁ = 9/14, ρ₂ = 0 and ρ₃ = 5/14 must be preserved. When Γ_p = γ₁, we can numerically calculate the steady population distribution, which varies little spatially. We find ρ₁ ≈ 0.64 ≈ 9/14, ρ₃ ≈ 0.36 ≈ 5/14, and ρ₂ = 3 × 10⁻⁴. The sum of population of the excited states is 2 × 10⁻³. The deviation from the assumption is very small and can be neglected.

4. Conclusion

We propose a feasible scheme to realize an atomic absorption-free Bragg reflector by a level configuration common in helium and alkaline earth atoms. We use two-photon resonance to construct an equivalent Ξ-system with two nearly the same transition frequency, thus greatly reduce the necessary laser power. The scheme can be used to design a narrow-band reflector with high reflectivity, which is useful in developing narrow linewidth DFB lasers.