ZHAO Lian-jie, WANG Chang-feng, YAN Xiao-jun, ZHANG Guo-wan, ZHANG An-ning(Beijing Institute of Aerospace Control Device, China Aerospace Science and Technology Corporation,Beijing 100039, China)

Atomic waveguides using blue-detuned HE11guided mode in hollow optical fiber

ZHAO Lian-jie, WANG Chang-feng, YAN Xiao-jun, ZHANG Guo-wan, ZHANG An-ning
(Beijing Institute of Aerospace Control Device, China Aerospace Science and Technology Corporation,Beijing 100039, China)

Based on the vector model of Maxwell’s equations, we exactly calculated the electromagnetic field distributions of HE11mode in a hollow optical fiber, and mainly discussed the characteristics of their components. We found that the intensity distribution of HE11mode was a doughnut-like pattern in the cross section, and the phases of the two electric-field radial components Erand Eφwere uniform, but the phases of the two magnetic-field radial components Hrand Hφwere reversed. We also exactly calculated the absolute intensity and the corresponding optical potential distributions in a typical hollow optical fiber for85Rb atom guiding, and found that, by using a incidence light (1 W) and detuning (1 GHz) blue-detuned HE11guided mode with a 0.7 μm-radius hollow optical fiber, the maximum optical potential in the hollow region is 14.9 mK, which is far greater than the temperature (120 μK) of85Rb atoms in a standard magneto-optical trap(MOT). Thus the HE11mode is absolutely adequate to guide85Rb atoms in this typical hollow optical fiber.Atomic guiding in a hollow optical fiber can be widely used in the field of atomic interference.

atomic waveguide; evanescent wave; vector model; weakly guiding approximation; hollow optical fiber

Atomic waveguides using evanescent wave in a hollow optical fiber (HOF) have been studied extensively in these years[1], which is a feasible method to realize precise manipulation of atomic motion. This atomic guiding method can be widely used to study atomic optical devices[2], especially in the field of the atom gyroscope[3], and also can be used as an effective method to solve the problem of small split angle for atomic split.With the strong dipole interaction generated between the atoms and a blue-detuned evanescent wave, atoms are repelled from the high intensity field to the darkest regions of the field. This scheme of atomic waveguide can efficiently avoid the heating effects for the guiding atoms caused by laser fluctuations and spontaneous scattering of photons. Therefore, it can be used in atomic guiding, atomic funneling, atomic trapping, and atomic interferometers etc.

The theoretical approach of evanescent wave for atomic waveguide in an HOF has been calculated by Marksteiner et al 0 and Ito et al[4]from the Maxwell equations using a vector model0. Under the weakly waveguide approximation, Ito et al also calculated the field distribution of the LP01mode in an HOF based on a scalar model[5]. In fact, the Maxwell equations are simplified to their scalar form when using the weakly waveguide approximation, which leads to the LPmlmodes, and the scalar form LPmlmodes in an HOF can be regarded as a superposition of the TE0m, TM0m,EHn-1,mand HEn+1,mvector modes[6]. Especially, the vector form HE11mode corresponds to LP01mode in the scalar model. Afterward, Jhe’s group calculated the near- and far-field distribution of an LP01mode in an HOF under the weakly waveguide approximation[7], and found that the LP01mode dramatically changed from a doughnutlike pattern in an HOF to a Gaussian profile when it propagated to the free space. However, Yin et al deemed it was not suitable to calculate the field distribution in an HOF by using the weakly waveguide approximation when the relative refractive-index difference Δn was far greater than 1%[6]. By the way, they also calculated the near- and far-field distributions of the HE11mode in an HOF based on a vector model, and resulted that the output beam will always keep doughnut-like pattern in the far field distribution. So we should keep cautious when using the weakly waveguide approximation in the future works.

Above all, it should be noted that the calculations of the field distribution of the HE11mode in an HOF are all about of the electric field distribution analysis only, and there is no magnetic field distribution0. Beside, Yin et al concluded that the magnetic field distribution is similar to the electric field distribution[6]. Is it similar completely?Are there any differences between them? In this paper,we exactly calculated the electric field and magnetic field distributions of the HE11mode in an HOF from the Maxwell equations based on a vector model, and discussed the differences between the electric and magnetic field distribution. We also exactly calculated the optical potential distributions in a typical hollow optical fiber for85Rb atom guiding, and discussed the potential application for atomic waveguides. The results proved that the HE11mode was absolutely adequate to guide85Rb atoms in this typical hollow optical fiber.

1 Electromagnetic field distribution of the HE11 mode in an HOF

In this paper, we consider a step-index cylindrical HOF, the outer cladding radius of which can be taken to be infinite. The diameter of the hollow region, the thickness of the core, and the relative refractive index differrence between the core (n1) and the cladding (n2) can be represented by 2a, d=b–a, andrespectively.

Based on the cylindrical coordinate system(r ,φ,z),the electromagnetic field distribution can be divided into two components, the transverse component (r ,φ)and the longitudinal component (z). For the guidance of atoms along the fiber, the longitudinal component of the HE11mode in an HOF can be written as follows:

Applying the continuous boundary conditions at both boundaries,r=aandr=b, we obtain the following set of linear and homogeneous equations:

Where ω is the angular frequency of the incident field,0ε,1ε, and2ε, are the vacuum, the core and the cladding dielectric constants respectively, and0μ is the vacuum permeability.

By analyzing the equation (2), we find that the analytical solutions for the equation (2) are quite difficult to obtain. In this case we select a typical HOF which structural parameters can be set to be: a=0.7μm,b=2.7μm, n2=1.45,, and the wavelength λ of the incidence field is 780nm. By assigning these fiber parameters to equation (2), we can obtain a nontrivial numerical solution forβ, and then we can get the values of u, v and w. Furthermore, we can get the eight coefficients A, B, C, D, P, M, N, Q based on the boundary conditions.

All of the above, we can get the transverse components of electromagnetic field distributions of the HE11mode: Er, Eφ, Hr, Hφagainst the radial position r in the typical HOF, which is shown in figure 1. By analyzing the figure 1, we can get the conclusions as below: 1) Following the cross section of the HOF, the electromagnetic field distributions of the HE11mode in the HOF are mainly concentrated in the core, the electromagnetic field (the evanescent wave) in the hollow region can be used to guide the atoms in the HOF when the incidence field is blue-detuned. 2) The ratio between the electric field amplitude and the magnetic field one is about 260 times. 3) The electric field components Erand Eφshare the same phases, but the magnetic field components Hrand Hφpossess the opposite phases. 4) Based on the continuous boundary conditions, the transverse componentsEφandHφare consecutive at the boundariesr=aandr=b, but the componentsErandHrare inconsecutive at these boundaries. The relationship between the neighbor media can be represented as, wheren1andare the refractive indices between the neighbor media, andEr1,Er2are the electric field components against the radial positionrrespectively. Hence the electric field amplitudes of the HE11mode exhibit abrupt jumps at the boundaries.

Fig.1 Transverse components of electromagnetic field distributions of the HE11 mode: Er, Eφ, Hr, Hφ against the radial position rin a typical hollow optical fiber. (a) Er ( r)～r; (b)Eφ(r) ～r; (c)H r (r)～r; (d)Hφ(r) ～r.

For the visualization of figure 1 it is desirable to draw the gradient vector diagrams of the transverse electromagnetic field components in the cross section of the HOF, which is shown in figure 2. The electromagnetic field components in the core possess the larger amplitudes obviously. Similarly, the electric field componentsErandEφshare the same phases, but the magnetic field componentsHrandHφpossess the opposite phases. All the above are also corresponds to the figure 1.

Fig.2 Electromagnetic gradient vector diagrams of the HE11-mode in the typical hollow optical fibers. (a) The electric field componentsErandEφ; (b) The magnetic field componentsHrandHφ.

2 Intensity distribution of HE11 mode in HOF

Based on the definition of the energy flux density,the optical intensity can be represented by the time averaged Poynting vector along thezcomponent[3].Thus the intensity distribution of the HE11mode in an HOF can be expressed as

WhereEr,Eφ,Hr,Hφare the transverse components of the electromagnetic field of the HE11mode, and “?”denotes the corresponding conjugated component.

Figure 3 shows the intensity distributionI(r) of the HE11mode against the radial positionrin the HOF.As well as the electromagnetic field distributions, the intensity distributions of the HE11mode in the HOF are mainly concentrated in the core, and the profile of which assumes a doughnut-like shape. The evanescent wave in the hollow region is really weak, but the gradient of the intensity is very strong. Thus, the system can be used for the guidance of atoms along the fiber when the incidence field is blue-detuned.

3 Potential application

For the guidance of atoms along the fiber it is desirable that the atoms dose not hit the wall. In this case there should have a force to repel or attract the atoms to the center. The field of the HE11mode in the hollow region is a doughnut-like pattern, can generate dipole forces to repel atoms to the darkest region. This scheme can be used to guide atoms when the incidence field is blue-detuned. The figure 4 is the sketch map of atomic waveguide in the HOF. From the figure 4, we can see that the gradient forces generated by the evanescent wave in the cross section of the fiber confine the atoms to the dark centre of the fiber, and if the atomic flux has a velocity along the z axis, the atoms will move along the fiber.

As well as the optical intensity distributions as shown in figure 3, we also need to calculate the absolute optical intensity distributions when the power of the incidence field is definite. The relationship between the power and intensity of the HE11mode in the HOF can be expressed as[9]:

Where P is the power of the HE11 mode, I( r) is the intensity distributions against the radial positionr.Applying the law of energy conversation, we calculate the absolute intensity distrribution I( r)of the HE11mode against the radial position r in the typical HOF with the incidence field power set to P=1W, and we draw the curve in figure 5.

Fig.5 Absolute intensity distribution I( r) of the HE11 mode against the radial position r in the typical HOF with the incidence field power set to P=1W.

From the figure 5,we can see that the maximum absolute optical intensitty in the hollow region is, and the corresponding intensity gradient isSuch great intensity gradient is absolutely adequate to guide atoms for a long distance along this typical HOF.

For a two level atom,the potentialgenerated by the repulsive gradient forces canbe written as follows[3]:

Where δ is the detuning of the incidence laser frequency off-resonance of an atom, ISand Γ are the saturation intensity and the natural line width of an atom respectively.

Figg.6 Optical potential distributtion U( r) of the HE11 mode against the radial position r for 85Rb atoms in the typical HOF, when the incidence field power (P=1W) and the detunning (δ=1 GHHz) are definitely.

From the figure 6, we can see that the maximum optiical potential U(r) in thehollow region is 14.9 mK,which is far greater than the temperature(120 μK) of85Rb atoms in standard magneto-optical trap (MOT).Thus if the atom flux has a velocity along the z axis,the atom flux will move towards ahead along the fiber.

The influerce on the optical potential due to the detuning should be considered in the atomic waveguide.Figure 7(a) shows that the optical potential U(r) in the typical HOF can be affected obviously with the different detuning.At the time, we also study the relationship between the optiical potential distribution U( r) of the HE11mode and the detuning δ for85Rb atoms at the boundary r=a. Form figure 7 (b), we can see that the optiical potential can get its maximum 38.3mK when the detuning set to 9.4 GHz.

Fig.7 When the power of the incidence field is set toP=1W: (a) is the optical potential distribution U( r) of the HE11 mode against the radial position r for 85Rb atoms with different detuning; (b) is the optical potential distribution U( r) of the HE11 mode against the detuning δ for 85Rb atoms.

4 Conclusions

In conclusion, we exactly calculated the electromagnetic field distributions of the HE11mode in the HOF using the vector model of Maxwell’s equations, and discussed the differences between them. We also obtained the intensity distribution of this typical HOF,and resulted that the electromagnetic field distribution of the HE11mode in the HOF was mainly concentrated in the core, and the profile of which assumes a doughnutlike pattern. We also briefly discussed the potential application of the HE11mode in atomic waveguide, and calculated the absolute intensity and optical potential distributions for85Rb atoms in the typical HOF. If we set the incidence field powerP=1W, the detuning , the maximum optical potential in the hollowregion is 14.9 mK, which is far greater than the temperature of85Rb atoms in a standard magneto-optical trap(MOT). Obviously, such HE11mode in this typical HOF is absolutely suitable for atomic waveguide. Beside we also discussed the influences of detuning to optical potential for85Rb atoms waveguide.

The authors acknowledge Prof. Yin for useful discussion on hollow optical fibers. This work is supported by the National Nature Science Foundation of China (Grant No. 6150101, 11304007).

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空芯光纖藍(lán)失諧HE11模原子導(dǎo)引

趙連潔，王長峰，嚴(yán)小軍，張國萬，張安寧

（北京航天控制儀器研究所 中國航天科技集團(tuán)公司，北京 100039）

基于矢量模型，從麥克斯韋方程組出發(fā)精確計(jì)算了空芯光纖中HE11模的電磁場(chǎng)分布，并分析了電磁場(chǎng)各分量的傳播特性。研究表明，HE11模強(qiáng)度沿光纖橫截面呈環(huán)狀分布，電場(chǎng)分量Er和Eφ相位相同，磁場(chǎng)分量Hr和Hφ相位相反。以85Rb原子為例，當(dāng)入射激光功率為1 W，失諧量為1 GHz，空芯光纖空心區(qū)半徑為0.7 μm時(shí)，計(jì)算了空芯光纖中HE11模的絕對(duì)強(qiáng)度分布和光學(xué)勢(shì)分布，計(jì)算發(fā)現(xiàn)空心區(qū)最大光學(xué)勢(shì)為14.9 mK，遠(yuǎn)大于標(biāo)準(zhǔn)磁光阱中85Rb原子溫度(120 μK)，因此空芯光纖中的HE11?？捎糜趯?dǎo)引85Rb原子。空芯光纖原子導(dǎo)引有望應(yīng)用于原子干涉領(lǐng)域。

原子導(dǎo)引；消逝波；矢量模；弱波導(dǎo)近似；空芯光纖

U666.1

A

1005-6734(2016)05-0643-06

10.13695/j.cnki.12-1222/o3.2016.05.015

2016-05-05；

2016-07-08

國家自然科學(xué)基金（6150101, 11304007）

趙連潔（1985—），女，博士研究生，從事新型慣性器件研制。E-mail: ap_mail@yeah.net

（References）：

聯(lián) 系 人：嚴(yán)小軍（1972—），男，研究員，博士生導(dǎo)師。E-mail: yanxiaojun@139.com