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Abstract
We present a modified fiber model based on the nested hollow core anti-resonant fiber that enables the stable transmission of the orbital-angular-momentum mode HE21. By replacing a pair of nested anti-resonant tubes in the horizontal axis with resonant tubes, the coupling between core mode and cladding mode has been increased. Therefore, the relative strength of fundamental mode HE11 and the first higher mode HE21 has been modified. The numerical simulation results indicate that the loss ratio of the lowest loss HE11 to HE21 can be optimized to more than 187, while the HE21 still maintains a low confinement loss as 0.0027 dB/m. Our research has brought about a solution of low loss hollow core mode propagation in optical fiber. Those properties will make this fiber an ideal medium for blue-detuned atomic guidance.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The development of atomic manipulation and guidance technology has paved the way for the application of cold atoms in precision measurement devices such as atom interferometer [1], quantum storage [2] and atomic clock [3]. The confinement effect of laser to atoms is based on the dipole force of the gradient distribution of light field. According to the sign of laser frequency detuned to the atomic resonance frequency, the guidance mechanism can be divided into red-detuned guidance and blue-detuned guidance [4]. Using Gaussian mode laser, the red-detuned guidance has been successfully demonstrated in both free space [5] and hollow-core fiber [6,7]. As for blue-detuned guidance, it has been theoretically proved that longer life time can be achieved, since the required donut-shaped laser beam can avoid high spontaneous scattering induced by strong laser intensity [8]. In free space, hollow core beam [9] has been generated by using spatial light modulator (SLM) [10] or vortex plate [11]. The beam is usually tightly focused to generate deep trapping potential for atoms. However, the effective working region is limited by Rayleigh range due to the diffraction nature of the laser beams, which severely restricts its application on guiding atoms over long distances. The emergence of hollow-core photonic crystal fibers (HC-PCFs) solves this problem since it can transmit a low-loss longitudinally uniform light modes over a long distance. A wide variety of fiber structures have been reported to excite donut-shaped mode, such as using holographic technique to select higher-order modes in HC-PCF [12], decreasing loss of TE01 in hollow-core photonic bandgap fiber (PBF) by modifying cladding walls thickness [13] or exciting higher modes by hollow metallic waveguide [14]. However, due to the multi-mode nature of these fibers the hollow-core modes are difficult to distinguish from the fundamental mode (FM).
Among the hollow core optical fibers, hollow-core anti-resonant fiber (ARF) is [15,16] featured with extremely low-loss and wideband transmission properties, thus drawing lots of attentions since its emergence [17,18]. Moreover, the core diameter of ARF is usually larger than other types of hollow core fiber [19], which provide competitive advantages in the field of particles loading and guiding. At present, the research of ARF is mainly focused on the transmission characters of the FM [20], which is considered by inherent thinking as the lowest loss mode. But rare research has been reported on the improvement of the first-order higher-order core modes (HCMs) transmission. Fetah et al. [21] proposed a structure that by enlarging the gap of lattice tubes near the x axis, the lowest loss mode are the LP11 and LP11a since the power leakage of fundamental mode has been increased. A hollow core beam excitation was also reported by our group based on a hollow core ARF, however, the high confinement loss make it easy to be overwhelmed by other modes [22].
In this paper, we proposed a modified structure based on a nested hollow-core ARF [23] to support stable transmission of HCMs. By replacing a pair of nested anti-resonant tubes in the horizontal direction with resonant tubes, the coupling between core mode and cladding mode has been improved. Therefore, the relative transmission strength of FM and HCMs has been modified. Similar structures have been reported to improve the polarization maintaining properties of fundamental mode [24], while in this paper we focus on the hollow-core mode transmission benefit by this fiber design. To examine the impact of resonant layers, we tested characteristics of fibers under different parameters by numerical simulation. The primary goal is to obtain the stable single mode transmission properties of HCMs by enlarging the suppression ratio of FM. The results have shown that the loss ratio of FM HE11 to HE21 can be optimized to more than 187, while the HE21 still maintains a low confinement loss as 0.0027 dB/m. Such characters would make this fiber a strong candidate as the medium of blue-detuned donut-shape beam in the application of atom or nano-particles trapping and manipulation.
2. Fiber modeling
For the purpose of guiding 87Rb atoms by blue-detuned dipole potential trap [25,26], the design wavelength and hollow-core diameter is chosen at 780 nm and 30 μm, respectively. The proposed circular asymmetry fiber is based on the nested hollow-core ARF mentioned above. Figure 1 illustrates an example of cross-section model and indicates key structural parameters. Six equally-spaced capillaries constitute the cladding structure of fiber. The diameter of the hollow core parameter, D, is defined as the distance of two outer capillaries symmetrical to the center. Besides, each outer capillary is been added with a nested smaller tube, attached to the cladding at the same radial position. The addition of the nested anti-resonant tubes can effectively reduce the transmission loss of light field in hollow core, as demonstrated in Ref. [23]. However, in this study the nested anti-resonant (NAr) tubes in the horizontal axis are replaced by a pair of nested resonant (NR) tubes, shown as the blue rings in Fig. 1, to suppress the transmission performance of FM. And the corresponding influence will be presented later.
Fig. 1. Schematic of circular asymmetry anti-resonant fiber, where black and blue areas represent nested anti-resonant and resonant tubes, respectively. D, the diameter of hollow core; d1 and t1, the diameter and thickness of outer capillaries; danti and tanti, the diameter and thickness of inner anti-resonant tubes; dres and tres, the diameter and thickness of inner resonant tubes
It is well known that in “ARROW” model [27], light wavelength of $\lambda = 2t\sqrt {{n_1}^2 - {n_0}^2} /m$ will be in resonance with the core-cladding interface and leak into the cladding material, where t is the capillary wall thickness, m is a positive integer, n1=1.445 and n0=1.0 are the refractive index of silica and air, respectively. In order to suppress the fundamental mode transmission in the hollow core area, the thickness tres is set to be 0.4 μm, which is a resonant value for the light guidance at λ0 = 780 nm (m=1). Whereas the parameters of other four additional rings are chosen to be far from resonance to acquire low confinement loss of hollow core modes.
To investigate the transmission behavior of both FM and HCMs and furthermore enlarge the FM extinction ratio, a finite-element method was applied to calculate the confinement and bend losses with COMSOL Multiphysics 5.0. The maximum mesh size for the hollow core and cladding silica region is chosen below λ0/6 to ensure the accuracy and reliability of the numerical calculation. And perfectly-matched layer (PML) is set outside the fiber domain, acting as an almost ideal absorber to prevent leaked light from being reflected again and affect the simulation accuracy.
3. Numerical simulation
To understand the coupling mechanism between FM and cladding modes and moreover investigate the optimal parameters to obtain the relative pure hollow-core mode, we perform numerical simulation of confinement loss as a function of the nested tubes parameters. To simplify the numerical simulation, the core diameter, D, and the major capillary parameters, d1 and t1, are fixed unless otherwise specified. The core diameter D, is 30 $\mu $m, which is large than designed wavelength [23]and also in a commonly used core size range to capture cold atoms [28]. The diameter of major tube d1, is 24 $\mu $m, defined as a relatively large value of a circle that can be inscribed inside the core. The wall thickness, t1, is 0.3 $\mu $m, which is an anti-resonant value according to “ARROW” model. In order to describe the inhibition effect of FM, a key parameter FER (fundamental mode extinction ratio) is introduced, which is contrary to the well-known HOMER (high order mode extinction ratio) [29], defined as the ratio between the lowest loss of fundamental mode and the loss in the aimed hollow core mode.
As the preliminary simulation results shown in Fig. 2, by introducing a pair of resonant layers and reasonably selecting numerical parameters, the light field of FM is coupled to the cladding material through the resonant layer while the HE21 is not. Consequently, the confinement loss of some certain hollow core mode has the potential to be reduced under fundamental mode.
Fig. 2. The contour profile of electric field of (a) FM HE11 and (b) HE21, respectively. The legends represent the normalized electric field strength.
3.1 Influence of NR tubes
To start with, the influence of NR tubes is studied and the calculation is performed when danti = 16 μm and tres = 0.3 μm. Figure 3 shows the loss of TE01, HE21 and HE11 varies with dres, respectively. It can be seen that the diameter of NR tubes has a strong impact on the propagation of transmission mode. Over a wide range of dres (4 μm < dres < 12 μm) the loss of fundamental mode remains lower than the first high-order modes group and the FER is even less than 10−3. However, when dres goes to the range between 13 μm and 14 μm the simulation curves of TE01 and HE21 show the opposite variation trend with HE11. The loss of the hollow-core modes drops sharply, while the leakage effect of FM to the cladding modes is enhanced. Consequently, FERs of two hollow-core mode TE01 and HE21 are peaked at 14.2 μm, as shown in Fig. 3(b). And the maximum FER is obtained by HE21, where the loss of HE11 and HE21 are 0.2573 dB/m and 0.0032 dB/m, respectively.
Fig. 3. The dependence of transmission performance on the diameter of resonant tubes (dres). (a) The confinement loss of the lowest-loss HE11, the first higher-mode TE01 and HE21 respectively. (b) FER of TE01 and HE21, the black horizontal line in (b) indicates the level where the loss of FM equals the other modes.
In our study, the typical radially polarized TM01 mode is severely disturbed by the asymmetry fiber structure, so it is not discussed in this paper. It also needs to be emphasized that the high fundamental mode inhibition ratio is obtained by sacrificing the constraint condition of fiber to a certain extent. In fact, the absolute losses of FM and HCMs are all increased compared with the normal nested ARF (dashed line in Fig. 3(a), where tres= t1=0.3μm).
To further understanding the effect of NR tubes on the transmission behavior of both HE11 and HE21, the effective mode indexes of relative four modes are calculated and illustrated in Fig. 4. When the diameter of NR tubes is small, the light field is likely to leak into the moon-shape region between the outer capillaries and NR tubes on the x axis, which is called tube mode in this paper. The effective mode indexes of HE21 (blue dashed line) are much close to the tube mode (magenta solid line) when 4 μm < dres < 10 μm, whereas the effective mode index of FM (red dashed line) is far away from the tube mode. As a result, within this range the confinement losses of HE21 are much higher than FM and the existence of NR tubes would not play a significant role on FER, as shown in Fig. 3.
Fig. 4. Four typical core modes and cladding modes of the nested anti-resonant fiber: (a) FM HE11 (b) HCM HE21 (c) nested tube mode (d) tube mode. (e) The corresponding effective mode indexes at different diameter of inner NR tubes (dres). The interval of simulation data is less than 0.1μm.
The mode which actually contributes to the loss of FM is the nested tube mode, represented by the black solid line in Fig. 4. The nested tube mode is a dielectric mode, which means that most power is confined in the dielectric material [30]. With dres increasing, the effective mode index difference of nested tube mode and FM has sharply reduced until dres = 14 μm. As a consequence, the nested tube mode has strongly coupled with FM at this position, as shown in the inset mode figure. Such result is consistent with the optimum diameter of NR tubes in Fig. 3, where the maximum FER is achieved when dres = 14.2 μm.
3.2 Optimization of FER
In order to optimize the structural parameters and achieve the maximum FER, the influence of nested NAr tubes and relative dimension ratio of nested tubes on confinement loss are analyzed. First of all, the diameter of NAr tubes danti has been investigated independently. The basic fiber geometry is set as dres = 14.2 μm, cause a large FER value is obtained as the result mentioned above. As illustrated by the black solid line in Fig. 5, the influence of the inner NAr tubes on the transmission loss of HE11 is almost negligible, and the propagation behavior of it sustains significant suppression by the horizontal resonant layers. Similarly, in a wide range the performance of HE21 is generally unaffected by the NAr tubes, except that when danti < 8 μm the losses abruptly start to be larger than FM. By looking at the electric field distribution in Figs. 5(a) and 5(b), we can realize that when the nested tubes are much smaller relative to the outer capillary, the first high-order modes would be easily leak into the domain between them.
Fig. 5. The dependence of confinement loss and FER on the diameter of NAr tubes (danti). (a) (b) are electric field of HE21 and TE01, respectively, when danti is 7 μm.
For a more intuitive representation, the variation trend of FER is given as well, shown as blue and red dashed line respectively. The FM extinction ratios of TE01 and TM01, are more than one order of magnitude lower than HE21, because the polarization symmetry is likely to be perturbed by the isometric structure [31].
Secondly, the influence of relative dimension dres/danti on FER has been carefully studied. Figure 6 shows the loss ratio between HE11 and HE21 as a function of danti and dres/danti. Most notably, the diameter of NR tubes implements a dominant effect on FER, as illustrated by the white dashed lines, where ① dres = 14.16 μm and ② dres = 16 μm. Apart from that, the relative size dres/danti also plays an important role on the optimization of FER. The dependence of FER on danti is shown on the right side of Fig. 6, corresponding to three typical dres values. The maximum FER is 187, acquired where dres = 14.16μm and danti = 12.68μm, and the corresponding confinement loss of HE11 and HE21 are 0.5049 dB/m and 0.0027 dB/m respectively.
Fig. 6. Contour plot of FER as a function of danti and dres /danti. The variation of FER with danti is shown on the right side corresponding to the different dres values.
3.3 Wavelength performance
As mentioned in Section 2, the thickness of nested resonant tube tres = 0.4 μm is chosen specially for the wavelength of 780 nm. In order to verify the credibility of this fiber design on the FM suppression effect, the dependence of effective mode index and the fundamental mode suppression ratio on light wavelength has been analyzed and displayed in Fig. 7. In the vicinity of 780 nm, the black solid line displays a narrow valley with its minimum value close to the effective mode index of FM, which indicates that the strong mode coupling between FM and the nested tube mode occurs only in a small range close to the designed wavelength. Correspondingly, owing to the NR tube induced FM loss, the simulation of FER also shows a sharp peak at 780 nm with a limited bandwidth of 0.28 nm. And except for the designed wavelength, the fundamental mode will not be affected and remains transmitted as the lowest-loss mode. Such wavelength performance is different from the previous mentioned works, which have FM suppression ability within a large wavelength bandwidth [21]. Therefore, the designed fiber can satisfy the application requirements of different light modes at multi wavelengths in atomic guidance and detection experiments.
Fig. 7. Effective mode index of FM and nested tube mode at different light wavelength and the corresponding FER. (dres=14.16 μm, danti = 12.68 μm)
4. Conclusion
In summary, a modified isometric nested ARF has been theoretically demonstrated to support low loss and stable transmission of HE21 at the wavelength of 780 nm. By introducing a pair of nested resonant tubes in the horizontal direction, the coupling strength between different air-core modes and cladding surface has been modified. At some certain geometry parameters, the confinement loss of the first high-order mode HE21 can be reduced to much smaller than FM. The numerical simulations proves that the largest FER can reach 187, and the corresponding confinement loss of HE21 is as low as 0.0027 dB/m. The effective wavelength of the proposed fiber to support lowest loss transmission of HE21 are within a narrow bandwidth. At other wavelengths, the transmission property of fundamental mode will not be broken. Such characteristics would make the proposed fiber a flexible tool to generate multi-mode in different wavelength and furthermore be applied to blue-detuned guidance in atom or particle transportation technique.
Funding
National Natural Science Foundation of China (61935002).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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