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Spiral photonic crystal fiber structure for supporting orbital angular momentum modes
Optik - International Journal for Light and Electron Optics 169 (2018) 361–367
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Spiral photonic crystal fiber structure for supporting orbital angular momentum modes Nandam Ashok, Woojin Shin
T
⁎
Advanced Photonics Research Institute, Gwangju Institute of Science and Technology, Buk-gu, Gwangju, 61005, Republic of Korea
A R T IC LE I N F O
ABS TRA CT
Keywords: Fiber design Photonic crystal fibers Optical Vortices
We propose a spiral photonic crystal fiber structure for supporting orbital angular momentum modes. The structure has 12 arms, arranged in a spiral shape in the cladding region, and a big air hole at the center of the structure. It can support 14 well-separated OAM modes with a mode index difference of 10−4. Our numerical results show a nonlinear coefficient of 2.78 W−1 Km−1 for the HE21 mode at 1550-nm wavelength. The fiber shows a dispersion difference over a 600 nm bandwidth for HE21 mode is 12.1 ps/km-nm, and the structure shows a low confinement loss. We have also calculated the effect of ellipticity on the mode’s effective index. Numerical results suggest that, this structure can be utilized in high-capacity communication systems for multiplexing fiber-based OAM.
1. Introduction Space division multiplexing (SDM) has been the subject of significant recent attention because of its potential application in scaling network capacity [1–3]. Few-mode fiber communication system technologies use the multiple-input multiple-output (MIMO) signal processing method to recover the information [3] from multiplexed systems. However, MIMO is not the only available system applicable in higher-order mode fiber systems. Orbital angular momentum (OAM) beams with a helical phase are advantageous for higher order mode fiber communication systems [4,5]. As the modes of OAM are orthogonal to each other, no coupling exists between the modes. The OAM beam is expressed by exp(ilφ ), where φ represents the azimuthal angle and l is the topological charge number [5,6]. Due to these properties, OAM modes in specialty fiber show a variety of dimensions for multiplexing. Since an OAM beam comprises ring-shaped intensity distributions with a particular angular momentum, the beams can be used for several applications including super resolution imaging [7], optical communication [8–10], entanglement of photons [11], and particle trapping [12]. Generally, OAM beams are generated using spatial phase plates [13], computer generated holograms [14], ring resonators [15], and birefringent elements [16,17]. Compared to the free space OAM communication, propagation and generation of OAM beams in a fiber have considerable significance. In free space, OAM beams are enlarged with increasing propagation distance, but in the case of fiber, the beams can propagate over large distances with low crosstalk. Recently, a considerable amount of work has been carried out in the field of fiber-based OAM, in which, ring core fibers, air core fibers, and fiber couplers were reported to support the propagation of the OAM modes [18–22]. However, on the one hand, when crosstalk is reduced by the fibers, the number of OAM modes is restricted, and this number is vitally important in high-capacity communication systems. Therefore, it is necessary to investigate fiber structures having a large number of OAM modes. Over the years, photonic crystal fibers (PCFs) have been of great significance in communication [23,24]. Yue et al. showed that the transmission of OAM modes was supported by a PCF structure with As2S3. The PCF design reported the two OAM modes [25]. Zhang et al. numerically demonstrated the generation of OAM modes in a circular PCF ⁎
Corresponding author. E-mail address: [email protected] (W. Shin).
https://doi.org/10.1016/j.ijleo.2018.05.055 Received 5 March 2018; Accepted 15 May 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 169 (2018) 361–367
A. Nandam, W. Shin
Fig. 1. Cross-section of the proposed PCF.
[4]. In the present work, our aim is to design an optical fiber that can support several OAM modes with specialty fiber. In this paper, we propose a novel spiral PCF structure with spirally arranged air holes in the cladding, which can support 14 OAM modes. Numerical simulations suggest that the use of suitable PCF parameters will permit the accommodation of many OAM modes. Our study illustrates a low dispersion variation of 10.35 ps/km-nm for the TE01 mode, and 12.1 ps/km-nm for the HE21 mode with a 600 nm bandwidth. 2. Proposed spiral design The schematic of Fig. 1 illustrates our proposed spiral PCF structure with the given design parameters to support the propagation of OAM modes. The present PCF structure consists of 12 spiral arms, where each spiral arm contains 6 air holes, in addition a big air hole arranged at the center. The optical fiber is made of silica. In each arm, the first three air holes have a diameter d1, and the next three air holes have a diameter d2. The distances between the center to the first air hole is r0 and the distance between center to second air hole is r1=(0.48×Λ)+r0 respectively, where Λ is the distance between two adjacent holes. The angular displacement is θ1 = 180/Na, where Na is the number of arms in the PCF [26,27]. The distance between the center to the nth hole is r = (0.48×Λ) +rn-1, and the nth angular displacement is θn=(n×180)/Na. Lumerical eigen-mode solver has been used to analyze the mode fields [28], and modes’ effective indices are calculated from propagation constants [29]. The structure supports TE01, TM01, and 14 OAM modes, in which the azimuthally polarized and radially polarized modes do not carry the OAM. The OAM modes in the fiber can be described as being combinations of even and odd modes of HE and EH [4]. odd OAM±±l, m = HEleven + 1, m ± i HEl + 1, m ⎫ odd ⎬ OAM∓±l, m = EHleven − 1, m ± i EHl − 1, m ⎭
(1)
± ± ± (HE21), OAM (HE31, EH11), OAM (HE41, EH21), and The 14 OAM modes supported by the structure are OAM ± 1, 1 ± 2, 1 ± 3, 1 ± OAM (HE51, EH31). We achieved a large effective index difference for the HE group modes (and EH group modes) as a result of ± 4, 1 which the PCF achieves stable OAM modes. Adjusting the design parameters, the structure allows the accommodation of a large number of OAM modes with low dispersion variation. A design of this sort would have value in high-capacity communication systems where mode division multiplexing is required. 3. Numerical results and discussions The proposed structure has been investigated numerically for the following fiber parameter values. The arms have air holes with two diameters: diameter ‘d1’ (small air holes) is 2.2 μm and diameter ‘d2’ (big air holes) is 3.4 μm. The diameter of the center air hole (2a) is 3.4 μm. Commercial software was used to calculate the effective indices of the following supported modes: TE01, TM01, HE11, HE21, HE31, EH11, HE41, EH21, HE51, and EH31. The structure contributes four OAM order modes with different topological charges l, and the l = 0 mode, or HE11 mode, has not been considered in the 14 OAM modes. Fig. 2 shows the plot of the intensity distribution for the OAM modes at a wavelength of 1550 nm. The figure shows that the OAM modes are largely restricted to the annular region. Next, we calculated the vector plots of the OAM modes, and the results are plotted in Fig. 3, which shows the vector notations of the OAM modes. As seen in the figure, similar values of m have different vector notations, which means that the same value of m results in different polarization patterns. The modal field distribution for the OAM modes along with their radius is illustrated in Fig. 4. As the mode number increases, the width of the annular intensity distribution slightly decreases. This figure also shows that the mode distribution has a good confinement in the high index region, and it can be concluded that the proposed design will overcome the coupling of multiple OAM modes into the annular index fiber system. In Fig. 5 we have plotted the propagation of HE21 even mode in 362
Optik - International Journal for Light and Electron Optics 169 (2018) 361–367
A. Nandam, W. Shin
Fig. 2. Intensity distributions of TE01, TM01, HE11, HE21, HE31, EH11, EH21 and EH31 modes.
Fig. 3. Vector notations of TE01, TM01, HE21 even, HE21 odd, HE31 even, and HE31 odd modes.
Fig. 4. Normalized Intensity distribution of modes.
a spiral PCF. It is clearly shows that guiding of the HE21 even mode, by choosing the appropriate fiber parameters. Here, we have shown the 2 m length propagation of HE21 mode. Fig. 6 shows the supported effective index of the modes as a function of the wavelength between 1.3 μm and 2 μm. The figure demonstrates that, as the wavelength increases, the effective indices of the modes decreases. To support the OAM modes in the fiber structure, the effective index difference between the modes should be 10−4, failing which, mode coupling will occur. We therefore examined the difference in the effective index for the same OAM order, l = 2,3,4, and the results are illustrated in Fig. 7. The topological number l = 2 denotes the effective index difference of HE31 and EH11 modes. As illustrated in the figure, the effective index difference for l = 2, 3 and 4 modes are on the order of 10−4 [30], but the index difference 363
Optik - International Journal for Light and Electron Optics 169 (2018) 361–367
A. Nandam, W. Shin
Fig. 5. Propagation of HE21 even mode along the fiber length.
Fig. 6. Effective indices of modes with wavelength.
Fig. 7. Effective indices difference for l = 2, 3, and 4.
for l = 3 modes decreases with increasing wavelength. At a wavelength of 1550 nm, the index difference for l = 2, 3, 4 are respectively, 5.68 × 10−4, 1.5 × 10−4 and 4.64 × 10−4. Moreover, the PCF shows an effective index difference of 10-4 over a bandwidth of 500 nm between 1.3 and 1.8 μm. The measured index differences fit well with the reported values [31,32]. The neff difference for l = 2 and 4 modes increase with wavelength; hence, the designed PCF could perhaps lead to a reduction in the coupling mode, and provide a good difference between the OAM modes. As a result, since each mode propagates separately, there is a reduction in crosstalk between the modes. A structure producing these results would be valuable in a mode division multiplexing communication system. An another important parameter in fiber is its dispersion parameter. To calculate the dispersion parameter for OAM modes, the dispersion of silica was considered. The equation used for the numerical investigation of the dispersion parameter is [33–35]:
D=−
1 ⎛ 2 d 2neff ⎞ ⎜λ 0 ⎟ λ0 c ⎝ dλ 02 ⎠
(2)
Fig. 8 shows the dispersion curve for the TE01, TM01, HE11, and other OAM modes. From the figure, we can see that as the 364
Optik - International Journal for Light and Electron Optics 169 (2018) 361–367
A. Nandam, W. Shin
Fig. 8. Dispersion of modes with wavelength.
wavelength increases, there is an increase in the dispersion parameter. The dispersion for HE11 and TE01 is almost flat, while the other OAM modes show less flat dispersion. The EH31 and HE51 modes show a high dispersion in the total bandwidth region. The proposed PCF in this work shows a dispersion variation of 4.4 ps/km-nm for the HE11 mode, 10.35 ps/km-nm for the TE01 mode, 12.1 ps/kmnm for the HE21 mode, 36.62 ps/km-nm for the HE31 mode, 73.25 ps/km-nm for the HE41 mode, 115.94 ps/km-nm for the HE51 mode, 35.63 ps/km-nm for the EH11 mode, 60.5 ps/km-nm for the EH21 mode, and 83.4 ps/km-nm for the EH31 mode over a band width of 600 nm (from 1350 nm to 1950 nm). In the case of HE11, TE01, and HE21 mode the variation in dispersion is better than the reported values [4,25]. Next, we calculated the confinement losses for OAM modes. The PCF shows a confinement loss of 5.08 × 10−3 dB/m for the HE11 mode, 5.0954 × 10−3 dB/m for the TE01 mode, 5.0951 × 10−3 dB/m for the HE21 mode, 5.11 × 10−3 dB/m for the HE31 mode, 5.13 × 10−3 dB/m for the EH11 mode, 5.31 × 10−3 dB/m for the HE41 mode, and 5.72 × 10−3 dB/m for the EH21 mode at a wavelength of 1550 nm. Next, we investigated the nonlinear coefficient of the modes. The nonlinear coefficient of the modes can be calculated from the following expression:
γ=
2πn2 λAeff
(3)
where n2 is the material nonlinear refractive index and Aeff is the mode’s effective mode area and it is calculated numerically [36]. For this calculation, the reported value of nonlinear index of silica was used: n2 = 2.3 × 10−20 m2 W−1 [37,4]. Fig. 9 depicts the plot of the nonlinear coefficients of modes against the wavelength. The figure shows that the nonlinear coefficients of the HE11, HE21, HE31, HE41, and HE51 modes decreases as the wavelength increases. HE21 mode shows a nonlinear coefficient of 2.78 W−1 Km−1 at 1550 nm wavelength, and the higher order mode HE51 has a coefficient of 2.71 W−1 Km−1. The figure demonstrates that the lower order modes have a high nonlinear coefficient, while higher order modes have a low nonlinear coefficient. Next, we have investigated the effect of ellipticity on modes. Any deformation in the shape of air hole will change the effective index of OAM modes and hence changes the dispersion properties. Therefore, we have studied the effect of deformation of central air hole on modes. Ellipticity “ε ” of the center air-hole was defined as “ε=b1/b2” where, in the design of the structure, ε = 1, it means b1=b2 (b1 = 3.4 μm). Here, we measured the effective indices of OAM modes with different ellipticity values. Fig. 10 shows a plot of effective indices of the TE01, TM01, HE11, HE21, HE31, EH11, HE41, EH21, HE51, and EH31 modes with different ellipticity values, i.e., 1.01, 1.02, 1.03, 1.04, and 1.05. The variation in the effective index of OAM modes following an ellipticity alteration proved to be
Fig. 9. Nonlinear coefficient as a function of wavelength. 365
Optik - International Journal for Light and Electron Optics 169 (2018) 361–367
A. Nandam, W. Shin
Fig. 10. Effective indices of OAM modes with ellipticity.
very small. The designed fiber can be fabricated easily using the well-known conventional fiber fabrication techniques. The results show that by choosing the suitable PCF parameters such as d1 = 2.2 μm, d2 = 3.4 μm, and 2a = 3.4 μm we can achieve the 14 OAM modes with low dispersion values. PCF with low dispersion close to zero dispersion values are suitable for supercontinuum generation applications. A PCF meeting these criteria would be of value in the fiber-based OAM multiplexing in communication systems involving a high-capacity fiber. 4. Conclusions We have proposed a silica-based spiral PCF to support the OAM modes with low dispersion variation of OAM modes. The effective indices of the OAM modes were calculated with the aid of a lumerical eigenmode solver. The proposed PCF supports the 14 OAM modes, and the structure shows a low confinement loss of 5.0951 × 10−3 dB/m for the HE21 mode at 1550-nm wavelength. The effective index difference of 10−4 over a bandwidth of 500 nm was demonstrated, accompanied by a good separation of the OAM modes. We achieved a very low dispersion variation of 4.4 ps/km-nm for the HE11 mode, 10.35 ps/km-nm for the TE01 mode, 12.1 ps/ km-nm for HE21 mode over a band width of 600 nm. The effect of geometric deformation has also been demonstrated, and the effective indices for the HE11, HE21, HE31, HE41, HE51, EH11, EH21, and EH31 modes showed only a small deviation resulting from the deformation of the central air hole. The proposed structure should therefore be useful for fiber-based OAM multiplexing in highcapacity fiber communication and supercontinuum generation. Acknowledgments This work was supported by International Collaborative Research and Development Programme, funded by the Ministry of Trade, Industry and Energy (MOTIE) Korea. References [1] D.J. Richardson, J.M. Fini, L.E. Nelson, Space-division multiplexing in optical fibres, Nat. 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[20] A. Gaur, V. Rastogi, Design and analysis of annulus core few mode EDFA for modal gain equalization, IEEE Photon. Technol. Lett. 28 (2016) 1057–1060. [21] B. Ung, P. Vaity, L. Wang, Y. Messaddeq, L.A. Rusch, S. LaRochelle, Few-mode fiber with inverse-parabolic graded-index profile for transmission of OAMcarrying modes, Opt. Express 22 (2014) 18044–18055. [22] Y. Yan, J. Wang, L. Zhang, I.M. Jeng-Yuan Yang, N. Fazal, B. Ahmed, A.E. Shamee, K. Willner, S. Birnbaum, Dolinar, Fiber coupler for generating orbital angular momentum modes, Opt. Lett. 36 (2011) 4269–4271. [23] P.St.J. Russell, Photonic-crystal fibers, J. Lightwave Technol. 24 (2006) 4729–4749. [24] J.M. Dudley, G. Genty, S. Coen, Supercontinuum generation in photonics crystal fiber, Rev. Mod. Phys. 78 (2006) 1135–1184. [25] Y. Yue, L. Zhang, Y. Yan, N. Ahmed, Jeng-Yuan Yang, H. Huang, Y. Ren, S. Dolinar, M. Tur, A.E. Willner, Octave-spanning supercontinuum generation of vortices in an As2S3 ring photonic crystal fiber, Opt. Lett. 37 (2012) 1889–1891. [26] M.R. Hasan, M.S. Anower, M.A. Islam, S.M.A. Razzak, Polarization-maintaining low-loss porous-core spiral photonic crystal fiber for terahertz wave guidance, Appl. Opt. 55 (2016) 4145–4152. [27] A. Agrawal, N. Kejalakshmy, B.M.A. Rahman, K.T.V. Grattan, Soft glass equiangular spiral photonic crystal fiber for supercontinuum generation, IEEE Photon. Technol. Lett. 21 (2009) 1722–1724. [28] Lumerical Solutions, Inc. http://www.lumerical.com/tcad-products/mode/. [29] N. Ashok, Woojin Shin, Effective D-shape fiber with air hole assistant design for birefringence analysis, Optik 162 (2018) 27–34. [30] S. Ramachandran, P. Kristensen, M.F. Yan, Generation and propagation of radially polarized beams in optical fibers, Opt. Lett. 34 (2009) 2525–2527. [31] Y. Yue, Y. Yan, N. Ahmed, L. Jeng-Yuan Yang, Y. Zhang, H. Ren, K.M. Huang, B.I. Birnbaum, S. Erkmen, M. Dolinar, A.E. Tur, Willner, Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber, IEEE Photon. J. 4 (2012) 535–543. [32] C. Brunet, P. Vaity, Y. Messaddeq, S. LaRochelle, L.A. Rusch, Design, fabrication and validation of an OAM fiber supporting 36 states, Opt. Express 22 (2014) 26117–26127. [33] N. Ashok, Y.L.Lee W. Shin, Chalcogenide waveguide structure for dispersion in mid-infrared wavelength, Jpn. J. Appl. Phys. 56 (2017) 032501-1-032501-5. [34] A. Nandam, M. Jung, Y.L.Lee W. Shin, Reverse ridge silicon strip waveguide and silica slot waveguide structure for the dispersion at 1550 nm, IEEE Photon. J. 8 (2016) 7102609. [35] V. Mann, N. Ashok, V. Rastogi, Coupled strip-slot waveguide design for dispersion compensation, Opt. Quant. Electron. 47 (2015) 3161–3169. [36] C.H. Jung, Hyung Su Cho, Jae Wan Han, N. Ashok, W. Shin, Chi Hwan Ouh, Leaky channel fiber design for large mode area high power application at 1 micron, Proc. SPIE 10512 (2018) 105122W-1. [37] G.P. Agrawal, Nonlinear Fiber Optics, 3rd ed, Academic, New York, 2001.
367
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Spiral photonic crystal fiber structure for supporting orbital angular momentum modes Nandam Ashok, Woojin Shin
T
⁎
Advanced Photonics Research Institute, Gwangju Institute of Science and Technology, Buk-gu, Gwangju, 61005, Republic of Korea
A R T IC LE I N F O
ABS TRA CT
Keywords: Fiber design Photonic crystal fibers Optical Vortices
We propose a spiral photonic crystal fiber structure for supporting orbital angular momentum modes. The structure has 12 arms, arranged in a spiral shape in the cladding region, and a big air hole at the center of the structure. It can support 14 well-separated OAM modes with a mode index difference of 10−4. Our numerical results show a nonlinear coefficient of 2.78 W−1 Km−1 for the HE21 mode at 1550-nm wavelength. The fiber shows a dispersion difference over a 600 nm bandwidth for HE21 mode is 12.1 ps/km-nm, and the structure shows a low confinement loss. We have also calculated the effect of ellipticity on the mode’s effective index. Numerical results suggest that, this structure can be utilized in high-capacity communication systems for multiplexing fiber-based OAM.
1. Introduction Space division multiplexing (SDM) has been the subject of significant recent attention because of its potential application in scaling network capacity [1–3]. Few-mode fiber communication system technologies use the multiple-input multiple-output (MIMO) signal processing method to recover the information [3] from multiplexed systems. However, MIMO is not the only available system applicable in higher-order mode fiber systems. Orbital angular momentum (OAM) beams with a helical phase are advantageous for higher order mode fiber communication systems [4,5]. As the modes of OAM are orthogonal to each other, no coupling exists between the modes. The OAM beam is expressed by exp(ilφ ), where φ represents the azimuthal angle and l is the topological charge number [5,6]. Due to these properties, OAM modes in specialty fiber show a variety of dimensions for multiplexing. Since an OAM beam comprises ring-shaped intensity distributions with a particular angular momentum, the beams can be used for several applications including super resolution imaging [7], optical communication [8–10], entanglement of photons [11], and particle trapping [12]. Generally, OAM beams are generated using spatial phase plates [13], computer generated holograms [14], ring resonators [15], and birefringent elements [16,17]. Compared to the free space OAM communication, propagation and generation of OAM beams in a fiber have considerable significance. In free space, OAM beams are enlarged with increasing propagation distance, but in the case of fiber, the beams can propagate over large distances with low crosstalk. Recently, a considerable amount of work has been carried out in the field of fiber-based OAM, in which, ring core fibers, air core fibers, and fiber couplers were reported to support the propagation of the OAM modes [18–22]. However, on the one hand, when crosstalk is reduced by the fibers, the number of OAM modes is restricted, and this number is vitally important in high-capacity communication systems. Therefore, it is necessary to investigate fiber structures having a large number of OAM modes. Over the years, photonic crystal fibers (PCFs) have been of great significance in communication [23,24]. Yue et al. showed that the transmission of OAM modes was supported by a PCF structure with As2S3. The PCF design reported the two OAM modes [25]. Zhang et al. numerically demonstrated the generation of OAM modes in a circular PCF ⁎
Corresponding author. E-mail address: [email protected] (W. Shin).
https://doi.org/10.1016/j.ijleo.2018.05.055 Received 5 March 2018; Accepted 15 May 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 169 (2018) 361–367
A. Nandam, W. Shin
Fig. 1. Cross-section of the proposed PCF.
[4]. In the present work, our aim is to design an optical fiber that can support several OAM modes with specialty fiber. In this paper, we propose a novel spiral PCF structure with spirally arranged air holes in the cladding, which can support 14 OAM modes. Numerical simulations suggest that the use of suitable PCF parameters will permit the accommodation of many OAM modes. Our study illustrates a low dispersion variation of 10.35 ps/km-nm for the TE01 mode, and 12.1 ps/km-nm for the HE21 mode with a 600 nm bandwidth. 2. Proposed spiral design The schematic of Fig. 1 illustrates our proposed spiral PCF structure with the given design parameters to support the propagation of OAM modes. The present PCF structure consists of 12 spiral arms, where each spiral arm contains 6 air holes, in addition a big air hole arranged at the center. The optical fiber is made of silica. In each arm, the first three air holes have a diameter d1, and the next three air holes have a diameter d2. The distances between the center to the first air hole is r0 and the distance between center to second air hole is r1=(0.48×Λ)+r0 respectively, where Λ is the distance between two adjacent holes. The angular displacement is θ1 = 180/Na, where Na is the number of arms in the PCF [26,27]. The distance between the center to the nth hole is r = (0.48×Λ) +rn-1, and the nth angular displacement is θn=(n×180)/Na. Lumerical eigen-mode solver has been used to analyze the mode fields [28], and modes’ effective indices are calculated from propagation constants [29]. The structure supports TE01, TM01, and 14 OAM modes, in which the azimuthally polarized and radially polarized modes do not carry the OAM. The OAM modes in the fiber can be described as being combinations of even and odd modes of HE and EH [4]. odd OAM±±l, m = HEleven + 1, m ± i HEl + 1, m ⎫ odd ⎬ OAM∓±l, m = EHleven − 1, m ± i EHl − 1, m ⎭
(1)
± ± ± (HE21), OAM (HE31, EH11), OAM (HE41, EH21), and The 14 OAM modes supported by the structure are OAM ± 1, 1 ± 2, 1 ± 3, 1 ± OAM (HE51, EH31). We achieved a large effective index difference for the HE group modes (and EH group modes) as a result of ± 4, 1 which the PCF achieves stable OAM modes. Adjusting the design parameters, the structure allows the accommodation of a large number of OAM modes with low dispersion variation. A design of this sort would have value in high-capacity communication systems where mode division multiplexing is required. 3. Numerical results and discussions The proposed structure has been investigated numerically for the following fiber parameter values. The arms have air holes with two diameters: diameter ‘d1’ (small air holes) is 2.2 μm and diameter ‘d2’ (big air holes) is 3.4 μm. The diameter of the center air hole (2a) is 3.4 μm. Commercial software was used to calculate the effective indices of the following supported modes: TE01, TM01, HE11, HE21, HE31, EH11, HE41, EH21, HE51, and EH31. The structure contributes four OAM order modes with different topological charges l, and the l = 0 mode, or HE11 mode, has not been considered in the 14 OAM modes. Fig. 2 shows the plot of the intensity distribution for the OAM modes at a wavelength of 1550 nm. The figure shows that the OAM modes are largely restricted to the annular region. Next, we calculated the vector plots of the OAM modes, and the results are plotted in Fig. 3, which shows the vector notations of the OAM modes. As seen in the figure, similar values of m have different vector notations, which means that the same value of m results in different polarization patterns. The modal field distribution for the OAM modes along with their radius is illustrated in Fig. 4. As the mode number increases, the width of the annular intensity distribution slightly decreases. This figure also shows that the mode distribution has a good confinement in the high index region, and it can be concluded that the proposed design will overcome the coupling of multiple OAM modes into the annular index fiber system. In Fig. 5 we have plotted the propagation of HE21 even mode in 362
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Fig. 2. Intensity distributions of TE01, TM01, HE11, HE21, HE31, EH11, EH21 and EH31 modes.
Fig. 3. Vector notations of TE01, TM01, HE21 even, HE21 odd, HE31 even, and HE31 odd modes.
Fig. 4. Normalized Intensity distribution of modes.
a spiral PCF. It is clearly shows that guiding of the HE21 even mode, by choosing the appropriate fiber parameters. Here, we have shown the 2 m length propagation of HE21 mode. Fig. 6 shows the supported effective index of the modes as a function of the wavelength between 1.3 μm and 2 μm. The figure demonstrates that, as the wavelength increases, the effective indices of the modes decreases. To support the OAM modes in the fiber structure, the effective index difference between the modes should be 10−4, failing which, mode coupling will occur. We therefore examined the difference in the effective index for the same OAM order, l = 2,3,4, and the results are illustrated in Fig. 7. The topological number l = 2 denotes the effective index difference of HE31 and EH11 modes. As illustrated in the figure, the effective index difference for l = 2, 3 and 4 modes are on the order of 10−4 [30], but the index difference 363
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Fig. 5. Propagation of HE21 even mode along the fiber length.
Fig. 6. Effective indices of modes with wavelength.
Fig. 7. Effective indices difference for l = 2, 3, and 4.
for l = 3 modes decreases with increasing wavelength. At a wavelength of 1550 nm, the index difference for l = 2, 3, 4 are respectively, 5.68 × 10−4, 1.5 × 10−4 and 4.64 × 10−4. Moreover, the PCF shows an effective index difference of 10-4 over a bandwidth of 500 nm between 1.3 and 1.8 μm. The measured index differences fit well with the reported values [31,32]. The neff difference for l = 2 and 4 modes increase with wavelength; hence, the designed PCF could perhaps lead to a reduction in the coupling mode, and provide a good difference between the OAM modes. As a result, since each mode propagates separately, there is a reduction in crosstalk between the modes. A structure producing these results would be valuable in a mode division multiplexing communication system. An another important parameter in fiber is its dispersion parameter. To calculate the dispersion parameter for OAM modes, the dispersion of silica was considered. The equation used for the numerical investigation of the dispersion parameter is [33–35]:
D=−
1 ⎛ 2 d 2neff ⎞ ⎜λ 0 ⎟ λ0 c ⎝ dλ 02 ⎠
(2)
Fig. 8 shows the dispersion curve for the TE01, TM01, HE11, and other OAM modes. From the figure, we can see that as the 364
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Fig. 8. Dispersion of modes with wavelength.
wavelength increases, there is an increase in the dispersion parameter. The dispersion for HE11 and TE01 is almost flat, while the other OAM modes show less flat dispersion. The EH31 and HE51 modes show a high dispersion in the total bandwidth region. The proposed PCF in this work shows a dispersion variation of 4.4 ps/km-nm for the HE11 mode, 10.35 ps/km-nm for the TE01 mode, 12.1 ps/kmnm for the HE21 mode, 36.62 ps/km-nm for the HE31 mode, 73.25 ps/km-nm for the HE41 mode, 115.94 ps/km-nm for the HE51 mode, 35.63 ps/km-nm for the EH11 mode, 60.5 ps/km-nm for the EH21 mode, and 83.4 ps/km-nm for the EH31 mode over a band width of 600 nm (from 1350 nm to 1950 nm). In the case of HE11, TE01, and HE21 mode the variation in dispersion is better than the reported values [4,25]. Next, we calculated the confinement losses for OAM modes. The PCF shows a confinement loss of 5.08 × 10−3 dB/m for the HE11 mode, 5.0954 × 10−3 dB/m for the TE01 mode, 5.0951 × 10−3 dB/m for the HE21 mode, 5.11 × 10−3 dB/m for the HE31 mode, 5.13 × 10−3 dB/m for the EH11 mode, 5.31 × 10−3 dB/m for the HE41 mode, and 5.72 × 10−3 dB/m for the EH21 mode at a wavelength of 1550 nm. Next, we investigated the nonlinear coefficient of the modes. The nonlinear coefficient of the modes can be calculated from the following expression:
γ=
2πn2 λAeff
(3)
where n2 is the material nonlinear refractive index and Aeff is the mode’s effective mode area and it is calculated numerically [36]. For this calculation, the reported value of nonlinear index of silica was used: n2 = 2.3 × 10−20 m2 W−1 [37,4]. Fig. 9 depicts the plot of the nonlinear coefficients of modes against the wavelength. The figure shows that the nonlinear coefficients of the HE11, HE21, HE31, HE41, and HE51 modes decreases as the wavelength increases. HE21 mode shows a nonlinear coefficient of 2.78 W−1 Km−1 at 1550 nm wavelength, and the higher order mode HE51 has a coefficient of 2.71 W−1 Km−1. The figure demonstrates that the lower order modes have a high nonlinear coefficient, while higher order modes have a low nonlinear coefficient. Next, we have investigated the effect of ellipticity on modes. Any deformation in the shape of air hole will change the effective index of OAM modes and hence changes the dispersion properties. Therefore, we have studied the effect of deformation of central air hole on modes. Ellipticity “ε ” of the center air-hole was defined as “ε=b1/b2” where, in the design of the structure, ε = 1, it means b1=b2 (b1 = 3.4 μm). Here, we measured the effective indices of OAM modes with different ellipticity values. Fig. 10 shows a plot of effective indices of the TE01, TM01, HE11, HE21, HE31, EH11, HE41, EH21, HE51, and EH31 modes with different ellipticity values, i.e., 1.01, 1.02, 1.03, 1.04, and 1.05. The variation in the effective index of OAM modes following an ellipticity alteration proved to be
Fig. 9. Nonlinear coefficient as a function of wavelength. 365
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Fig. 10. Effective indices of OAM modes with ellipticity.
very small. The designed fiber can be fabricated easily using the well-known conventional fiber fabrication techniques. The results show that by choosing the suitable PCF parameters such as d1 = 2.2 μm, d2 = 3.4 μm, and 2a = 3.4 μm we can achieve the 14 OAM modes with low dispersion values. PCF with low dispersion close to zero dispersion values are suitable for supercontinuum generation applications. A PCF meeting these criteria would be of value in the fiber-based OAM multiplexing in communication systems involving a high-capacity fiber. 4. Conclusions We have proposed a silica-based spiral PCF to support the OAM modes with low dispersion variation of OAM modes. The effective indices of the OAM modes were calculated with the aid of a lumerical eigenmode solver. The proposed PCF supports the 14 OAM modes, and the structure shows a low confinement loss of 5.0951 × 10−3 dB/m for the HE21 mode at 1550-nm wavelength. 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