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We propose and numerically investigate a photonic crystal fiber (PCF) based on As2S3 for supporting the orbital angular momentum (OAM) modes up to 26. The designed PCF is composed of four well-ordered air hole rings in the cladding and an air hole at the center. The OAM modes can be well separated due to the large effective index difference of above 10−4 between the eigenmodes and maintain single-mode condition radially. In addition, the dispersions of the modes increase slowly with wavelengths, while the confinement loss keeps as low as 10−9 dB/m. The proposed PCF increases the supported OAM modes which could have some potential applications in short-distance, high-capacity transmission.
© 2016 Optical Society of America
1. Introduction
Orbital angular momentum (OAM) beams described by a spatial phase form of exp(iƖθ) (Ɩ is the topological charge number) have been widely investigated in recent years due to their unique features of helical wavefront, ring-shaped intensity distribution, specific orbital angular momentum [1], which could be applied in fields of particle trapping [2,3], optical communication [4–9], photon entanglement [10,11], and imaging [12]. At present, OAM modes can be efficiently generated by using spatial-light modulators [13], spiral phase plate [14,15], hologram [16] or optical vortex beam emitters [17]. However, OAM beams transmitting in free-space will gradually enlarge with the increase of the propagation distance, which is not conducive to their applications. Therefore, to solve this problem, generating or propagating OAM beams using optical fiber attracted great interests. Generally, ring core fibers/ring fibers [18–21], graded-index fiber [22] and fiber coupler [23] were proposed to efficiently generate or transmit OAM modes. Nevertheless, in practical applications such as high capacity communication systems, the number of OAM modes is a key parameter of fiber for transmitting. Therefore, it deserves further investigation of how to design the fibers to obtain the supported OAM modes as many as possible.
On the other hand, compared with conventional optical fibers, photonic crystal fiber (PCF) is a special one, which is more convenient to obtain ideal optical performances through ingeniously engineering its structure. As we know, the PCF is structurally characterized by an array of air holes parallel to the propagation axis along the entire fiber length. Thus, it provides possibilities of changing the refractive index contrast between the core and the cladding, varying the size and the arrangement of the air holes [24,25]. As a result, the PCF could display some distinct optical properties, such as endless single-mode guiding, controllable nonlinearity, dispersion and confinement loss [26], which have many particular applications. Now that one can flexibly design the structure of PCF, it is possible to transmit OAM modes in PCF [27]. However, the number of the supported OAM modes is few. Then a question naturally arises, whether we could take full advantage of the structure characteristics of the PCF to transmit more OAM modes, which is worth studying and anticipating.
In this work, we address this issue. Herein, we propose and numerically simulate a PCF based on As2S3 for transmitting OAM modes up to 26 while maintaining the single-mode condition radially. The eigenmodes index difference can be maintained above 10−4 which ensures a good separation to avoid the near-degeneracy of the constituent vector modes. Moreover, the confinement loss of each mode is less than 10−9 dB/m from 1.1 μm to 2 μm. It is also found that the variation of the eigenmodes dispersions becomes more and more slow with the increase of the wavelength. Meanwhile the nonlinear coefficient is two orders of magnitude larger than that of the conventional single-mode fiber. Benefiting from these characteristics, the proposed PCF might have some potential applications in fields of short distance, high capacity communication, supercontinuum generation, and all-optical switch.
2. Structure of the proposed PCF
Figure 1 shows the cross-section structure of the proposed PCF for transmitting OAM modes. It is composed of 4 well-ordered air hole rings in the cladding and an air hole at the center, where d0 is the diameter of the central air hole, d1 to d4 are the diameters of the air holes in the four rings, and ʌ1 to ʌ3 are the air hole spacings, namely, the center-to-center distance between the adjacent holes in different rings. As2S3 is selected as the background material due to its high refractive index (n = 2.4373 at 1.55 μm) compared with silica and other materials doped in conventional fibers, which could easily separate different eigenmodes from each other and reduce mode coupling to support OAM modes in the PCF. The full-vector finite element method (FEM) and perfect matched layers (PMLs) are used in the simulation to obtain the electromagnetic field distributions and the effective refractive indices of the eigenmodes in the proposed PCF. All the air hole rings are used for reducing the light leakage from the designed fiber. Through appropriately controlling the size and the spacing of the air holes, it is feasible to support OAM modes as many as possible. In our design, the parameters of the PCF are selected as follows: d0 = 6.6 μm, d1 = 1.8 μm, d2 = d3 = d4 = 1.2 μm, 0 = 6 μm, 1 = 1.8 μm, 2 = 1.5 μm, and 3 = 1.5 μm.
3. Numerical results
3.1 OAM mode properties of the proposed PCF
The supported eigenmodes in the designed PCF include HEm1, EHn1, TM01 and TE01, where m = 1~8, n = 1~6. OAM modes in this fiber can be obtained by combining the even and odd modes of HEm + 1 1 or EHn-1 1 (i.e., , where the topological charge numbers are ± m and ± n, respectively.). It is noted that one can change the direction of the circularly polarized incident light to control the sign of topological charge. In our designed fiber, we need to get enough light-area to support OAM modes as many as possible. However, too large area is not favorable for maintaining ‘single-mode’ in the radial direction [21], and the fiber which owns ‘multi-mode’ in the radial direction may have radial crosstalk [20]. Under these considerations, we choose a suitable area by controlling the size of d0 and d1 to support OAM modes and maintain ‘single-mode’ in the radial direction. Figure 2 shows the intensity distributions of two eigenmodes (HE31,HE81) and the phase diagrams of the corresponding OAM modes (OAM02,OAM07) in the designed PCF operating at 1.55 μm, which is calculated by using a commercial finite element mode solver COMSOL. As seen from Fig. 2, the eigenmodes are well confined in the area of high refractive index and the phase distribution of the corresponding OAM0,Ɩ mode has a 2Ɩπ change azimuthally. The designed PCF can support OAM modes up to 26 (combined by HEm1 or EHn1, m = 2~8, n = 1~6) over hundreds of nanometers optical bandwidth while maintain ‘single-mode’ in the radial direction, which is improved compared with those of a single-ring in the traditional ring core fibers/ring fibers [18–21] and graded-index fiber [22]. In fact, more OAM modes could be obtained through enlarging the area of high refractive index, but it may break single-mode condition radially.
Fig. 2 (a) and (c) Intensity distributions of the eigenmodes HE31and HE81, respectively; (b) and (d) phase diagrams of the corresponding OAM modes in the proposed PCF.
3.2 The effective indices of the eigenmodes
Figure 3 manifests the effective indices of the supported eigenmodes as a function of wavelength from 1.1 μm to 2 μm in the designed PCF. It is clear that the effective indices decrease monotonously. However, the effective index differences between the eigenmodes are maintained above 10−4 over hundreds of nanometers optical bandwidth to ensure the good separation between eigenmodes. Noted that the effective index differences between these eigenmodes increase with the wavelength, for example, at 1.55 μm, the effective index difference between the eigenmodes HE31 and EH11 can reach above 10−3. It means that the designed PCF could effectively reduce the mode coupling and keep each mode propagating separately. Therefore the crosstalk between the supported OAM modes can also be reduced as the effective index differences between the eigenmodes are beyond the threshold value [28]. In the simulation process, we also found that the effective index difference between the eigenmodes can be further increased by shrinking the structure of the fiber in a certain proportion. For instance, the effective index difference between HE31 and EH11 at 1.1 μm is kept 10−4, when the designed PCF is shrinked by 40%, it can be up to 10−3. However, it should be pointed out that the fabrication challenges for PCF are much more than those of the ring core fibers. Then shrinking the structure of the PCF too much will further increase the challenges. In addition, it will also increase the coupling difficulty of external light into the PCF if the size of the fiber is too small.
3.3 The dispersion characteristics
Figure 4 shows the dispersion characteristics of the eigenmodes which form the OAM modes, calculated by combining the material dispersion and the waveguide dispersion. The material dispersion is taken into account by using the Sellmeier dispersion equation [29]:
where n is the refractive index of As2S3, λ is the wavelength of the incident light. The waveguide dispersion Dw and the material dispersion Dm are calculated respectively with the following formulas: where neff is the effective index of the eigenmode, c is the speed of light in vacuum. Then the total dispersion can be obtained with:As seen from Fig. 4, the dispersion of each mode increases with the wavelength from 1.1μm to 2.0 μm, but the growth rate is more and more slow. Noted that the value of the dispersion is larger compared with that of the conventional fiber due to As2S3 as the background material. In order to obtain more flat dispersion curve, the modes in this fiber need experience a more loose confinement [27], which can be realized by reducing the value of d0 or d1. However, considering the demand of maintaining ‘single-mode’ in the radial direction, reducing the size of d1 is more practical. Furthermore, it is possible to choose different material which owns appropriate refractive index to optimize the dispersion of the fiber and support the same number of OAM modes.3.4 Nonlinear coefficient and confinement loss
We also studied the nonlinear coefficients of the eigenmodes, which were calculated from the mode distributions. In the simulation, the nonlinear refractive index n2 for As2S3 is 3 × 10−18 m2/W [30]. Figure 5 shows the nonlinear coefficients of HE31 and HE81 modes. Notably, the nonlinear coefficient decreases with the wavelength due to the increased effective mode area, and the nonlinear coefficient difference between the eigenmodes increases with the wavelength. Actually, the lower order modes have a more smooth growth in effective mode area compared with the higher order modes, because that the higher order modes are easier to leakage from the air cladding and it is more difficult to limit eigenmodes at longer wavelengths. As shown in Fig. 5, the nonlinear coefficient is two orders of magnitude larger than that of conventional single-mode fiber. As an example, at 1.55 μm, HE31 and HE81 modes have large nonlinear coefficients of 356.2 /W/km and 325.7 /W/km, respectively. Due to the high nonlinear coefficient, the proposed PCF may have potential applications in fields of supercontinum generation [27], all-optical switch [31], and so on. It should be pointed out that the high nonlinear coefficient is unfavorable for optical communication, thus it needs to find some materials which own appropriate refractive index and lower nonlinear refractive index to overcome this shortcoming. Furthermore, the confinement loss is calculated from the imaginary part of the effective index of the eignmodes. Generally, higher order mode has larger confinement loss, but in this PCF, both the confinement loss of modes HE31 and HE81 are less than 10−9 dB/m in the range from 1.1 μm to 2 μm. It should be pointed out that through increasing the number of cladding rings, it is convenient to maintain the confinement loss at an acceptable level.
4. Discussions for fabrication challenges
As a special fiber, PCF possesses many unique optical characteristics due to its microstructure feature. However, its structure also determines that the PCF fabrication is more complicated and costly than traditional fibers. Generally speaking, to fabricate a PCF, it consists of making a preform with the designed microstructure pattern and then drawing the preform into a fiber form by using the fiber-drawing apparatus. Although through appropriately designing the structure of the proposed PCF leading to supporting 26 OAM modes over hundreds of nanometers optical bandwidth, its fabrication difficulty and the cost are much greater than those of ring core fibers. The potential fabrication challenges for this design mainly include:1) maintaining the air holes at a certain size and position in different rings since the air holes are close and small; 2) precisely controlling the temperature and the drawing speed during the drawing process to prevent collapsing of the air holes. Despite the fabrication of this special PCF is relatively difficult, with the development of the fabrication techniques, we believe that the proposed PCF could be realized in the future and provide more choices for transmitting OAM modes.
5. Conclusion
In conclusion, we propose and numerically simulate a PCF based on As2S3 to support OAM modes as many as 26 while the fiber is ‘single-mode’ in the radial direction. The large effective index difference of above 10−4 between the eigenmodes could keep the OAM modes well separated. In addition, the PCF owns larger dispersion and nonlinear coefficient compared with those of the conventional single-mode fiber due to As2S3 as the background material, while the confinement loss is less than 10−9 dB/m. Meanwhile, the fabrication challenges of the proposed PCF are also discussed. The designed PCF might apply in fields of short-distance, high capacity transmission, supercontinum generation and so on.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 61378036, 61307058, 11304101, 11474108), Key Program of Natural Science Foundation of Guangdong Province, China (2014A030311037), Open Fund of the State Key Laboratory of Luminescent Materials and Devices (South China University of Technology) (Grant No. 2016-skllmd-12). Z.-C. Luo acknowledges the financial support from the Guangdong Natural Science Funds for Distinguished Young Scholar (Grant No. 2014A030306019), Program for the Outstanding Innovative Young Talents of Guangdong Province (Grant No. 2014TQ01X220), and the Zhujiang New-star Plan of Science & Technology in Guangzhou City (Grant No. 2014J2200008).
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