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Epsilon-near-zero photonic crystal fibers for a large mode separation of orbital angular momentum modes
Optik - International Journal for Light and Electron Optics 204 (2020) 164209
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Original research article
Epsilon-near-zero photonic crystal fibers for a large mode separation of orbital angular momentum modes
T
Myunghwan Kim, Soeun Kim* Integrated Optics Laboratory, Advanced Photonics Research Institute, GIST, Gwangju 61005, South Korea
A R T IC LE I N F O
ABS TRA CT
Keywords: Photonic crystal fiber Epsilon-near-zero Orbital angular momentum Mode separation
An orbital angular momentum (OAM) mode is a good candidate as an orthogonal basis mode set for mode division multiplexing (MDM) owing to its unique phase properties. Photonic crystal fibers support large numbers of OAM modes with notable performances. However, the effective index difference between high order OAM modes is not large enough. In this work, we propose two types of epsilon-near-zero (ENZ) materials-embedded circular photonic crystal fibers for high mode separation. Compared with previously developed photonic crystal fibers, the designed fibers exhibit a high refractive index contrast owing to the inserted ENZ materials. The designed fibers support up to 42 OAM modes. Most of these OAM modes provide a large effective index difference (Δneff > 10−3) while maintaining a radial sing mode condition over a wide wavelength range from 1.3 μm to 1.8 μm. The proposed fibers could potentially be exploited in MDM for stable transmission of OAM modes and other OAM-based applications.
1. Introduction As the demand for data usage is increasing, achieving high data transmission capacity is becoming an important topic in optical communication systems. Space division multiplexing (SDM) has attracted considerable interest as a promising technique to increase data capacity. Mode division multiplexing (MDM), which is a special case of SDM, has been widely investigated for high-speed data transmission [1]. Recently, MDM in few mode fibers (FMFs) based on linearly polarized (LP) modes has been introduced. FMFs multiply the transmission capacity in proportion to the total number of modes, as opposed to single mode fibers [2,3]. However, LP modes suffered from mode coupling owing to a low effective index difference between them. Thus, large and complex multiple-input multiple-output (MIMO) processing is required to mitigate the mode crosstalk [4,5]. Orbital angular momentum (OAM) modes, carrying exp(ilφ) (where l is the topological charge number, and φ is the azimuthal angle) have been widely investigated as an orthogonal modal basis of MDM owing to the weak coupling between OAM modes [6–8]. Recently, terabit-scale MDM based on OAM fibers has been experimentally demonstrated [9]. However, this can support only a few OAM modes owing to low refractive index contrast between the core and the cladding. To solve these issues, many types photonic crystal fibers (PCFs) have been reported, such as helically twisted PCFs [10], Kagome lattice PCFs [11], hexagonal lattice PCFs [12], circular PCFs [13–15], microstructure ring fiber [16], hollow-core PCFs [17], and PCFs with As2S3 background material [12,14]. Experimental demonstration has also been achieved in a helically twisted PCF [18]. The high index contrast between air holes and a background material enables PCFs to support many OAM modes, and their flexible structure design provides unique mode properties. Although previously reported PCFs support a great deal of OAM modes with high
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Corresponding author. E-mail address: [email protected] (S. Kim).
https://doi.org/10.1016/j.ijleo.2020.164209 Received 19 September 2019; Received in revised form 24 December 2019; Accepted 7 January 2020 0030-4026/ © 2020 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 204 (2020) 164209
M. Kim and S. Kim
Fig. 1. Schematic of the proposed circular ENZ PCF. (a) Type I ENZ PCF, (b) Type II ENZ PCF; a is the period, r0 is the radius of the center hole, and dn is the radius of the n-th layer of the air holes in the cladding region. The refractive index profile of (c) Type I ENZ PCF and (d) Type II ENZ PCF.
mode performance, the effective index difference between OAM modes is not large enough to support stable transmission for higher order OAM modes because it tends to reduce as the order of mode increases. Therefore, the effective indices of the eigenmodes should be more separated to avoid coupling from OAM mode into LP mode. This calls for the use of a high refractive index material such as As2S3 as a back ground material in order to increase index contrast. However, this kind of material is not suitable for MDM owing to its high nonlinearity. In this paper, we present epsilon-near-zero (ENZ) materials-inserted circular PCFs for large effective index separation. The ENZ material provides almost zero magnitude of permittivity. This unique property has been widely utilized to enhance confinement of guided mode and mode performance [19–21]. Very recently, PCF with ENZ circular hole in the cladding has been presented. This fiber shows very high birefringence [22]. Here, we suggest two types of ENZ-based PCFs. Both feature a hollowcore circular PCF structure. We substitute the ENZ material for the center air hole or for the first layer of the air holes in the cladding region, respectively. The high index contrast between the ENZ material and the background material of the fiber supports a stable transmission of OAM modes with good mode separation. We consider both ideal and realistic ENZ materials. The designed fibers provide up to 42 OAM modes and a large effective index difference (Δneff > 10−3) between OAM modes of the same order over a wide range of wavelengths, from 1.3 μm to 1.8 μm. This index is ten times higher than many recent designs [13,15,16]. We believe that the proposed design can be employed to improve the mode stability of OAM modes and for various OAM applications.
2. Structure of proposed ENZ fibers and numerical results 2.1. PCFs with an ideal ENZ material Fig. 1(a) and (b) show the cross sectional diagrams of the tow proposed ENZ-based circular PCFs: (a) center ENZ circular PCF (Type I ENZ PCF), and (b) first air hole layer ENZ circular PCF (Type II ENZ PCF), where a is the period, r0 is the radius of the center hole, and dn is the radius of the n-th layer of the air holes in the cladding region. We also plot the refractive index profiles of Type I and Type II ENZ PCFs in Fig. 1(c) and (d), respectively: x-direction refractive index profile at the center (y = 0). The two proposed types of fibers are composed of a hollow core circular PCF structure; however, the center air hole (Type I) and the first layer of air holes in the cladding (Type II) are replaced with the ENZ material. We assume in most of cases the following values of parameters: a = 2 μm, dn = 0.8a, and r0 = 5.3 μm. These values are the same as in the circular PCF for supporting OAM modes in ref. [13] for the sake of a fair performance comparison. All calculations were conducted using the finite element method (FEM) with a perfect matched layer (PML) at the boundary to determine the mode properties We chose silica as the background material owing to its low nonlinear coefficient and low loss. The refractive index of silica was defined by the Sellmeier equation [23]. We consider an ideal ENZ material with refractive index nENZ = 0.1. Note that an ideal ENZ material is studied in this section whereas a realistic ENZ material in the next section. As in conventional photonic crystal fibers, OAM 2
Optik - International Journal for Light and Electron Optics 204 (2020) 164209
M. Kim and S. Kim
Fig. 2. Effective indices of the eigenmodes in Type I ENZ PCF for (a) r0 = 5 μm, (b) r0 = 5.3 μm, (c) r0 = 5.6 μm, and (d) r0 = 5.9 μm as a function of wavelength.
modes can exist in the proposed circular ENZ PCFs as combinations of even and odd symmetry of HE and EH modes, mathematically described as: odd OAMl,m = HEeven l+1,m ± jHEl+1,m
(1)
odd OAM±l, m = EHleven − 1, m ± jEHl − 1, m
(2)
where l is the topological number, m is the order of the mode for the radial direction, and the sign +(-) denotes the right(left) phase rotation. Fig. 2(a) to (d) show the effective indices of the supported eigenmodes in the proposed Type I ENZ PCF for r0 = 5 μm, 5.3 μm, 5.6 μm, and 5.9 μm, respectively, as a function of wavelength. The proposed fibers can support up to the HE12,1 and EH10,1 modes. Thus, this fiber can transport 42 OAM modes (l ≤11) in total. The proposed fibers satisfy radial single mode condition except for r0 = 5 μm. Note that the radially higher modes for r0 = 5 μm are not shown in Fig. 2(a). We can observe that the effective indices reduce as r0 increases. The narrowing effective core region decreases the effective indices of the eigenmodes and results in mode cutoff of the higher order modes for long wavelengths. We can see that the modes from the EH8,1 mode to the EH10,1 mode are cutoff Fig. 3(a) to (d) show the effective index difference between the HEl-1 mode and EHl+1 mode, which have the same order of OAM mode, in the proposed Type I ENZ PCF for r0 = 5 μm, 5.3 μm, 5.6 μm, 5.9 μm, respectively, as a function of wavelength from 1.3 μm to 1.8 μm. Note that the effective index difference for the OAM1 mode is calculated from the effective index difference between the HE2,1 and TE0,1 modes. To exploit OAM modes in MDM, the effective index difference should be larger than 10−4, thereby avoiding mode coupling. As shown in Fig. 3(a) to (d), the effective index difference increases as r0 increases. Although the reduced effective core area decreases the number of eigenmodes, it results in the increase of Δneff. The effective index differences of all OAM modes are larger than 10−4 over a wide band of wavelength except for OAM9 at r0 = 5 μm. Moreover, Δneff for r0 = 5.9 μm is above 10-3 for all of the OAM modes. These values are ten times higher than those of many recent designs [13,16]. We also investigated the OAM mode performance of the proposed Type II ENZ PCF. Fig. 4(a) to (c) show the effective indices of the supported eigenmodes in Type II ENZ PCF for dn = 0.7a, 0.8a, and 0.9a, respectively. The parameters a and r0 are fixed at 2 μm and 5.3 μm, respectively. Similar to Type I ENZ PCF, Type II ENZ PCF supports the eigenmodes up to the HE12,1 mode and the EH10,1 mode. The effective indices decrease as dn increases owing to enhancement of the ENZ effect. Compared with Type I ENZ PCF, Type II ENZ PCF supports all eigenmodes without mode cutoff and maintains the radial single mode condition over a wide range of wavelengths. Fig. 5(a) to (c) show the effective index difference for OAM modes of the same order in the proposed Type II ENZ PCF for dn = 0.7a, 0.8a, and 0.9a, respectively. As dn increases, Δneff increases owing to enhancement of the ENZ effect. The high refractive index contrast between the ENZ material and silica enables large effective index separation between HEl-1 and EHl+1 modes. The refractive index differences for all cases are larger than 10−4. Therefore, the proposed ENZ PCFs can stably transmit a great deal of the OAM modes without incurring mode coupling into LP modes. In particular, the effective index differences of all OAM modes for dn = 0.9a are larger than 10-3 from 1.3 μm to 1.8 μm. Compared with Type I ENZ PCF in the same conditions (i.e. a = 2 μm, dn = 0.8a, and r0 = 5.3 μm), Type II ENZ PCF shows larger effective index difference. We summarize the results for Δneff at λ = 1.55 μm for Type I, Type II ENZ PCFs, Ref. [13], [16], and [17] in Table 1. 3
Optik - International Journal for Light and Electron Optics 204 (2020) 164209
M. Kim and S. Kim
Fig. 3. Effective index difference between the same OAM mode orders in the Type I ENZ PCF (a) r0 = 5 μm, (b) r0 = 5.3 μm, (c) r0 = 5.6 μm, and (d) r0 = 5.9 μm as a function of wavelength.
Fig. 4. (a) Effective indices of the eigenmodes in the Type II ENZ PCF for (a) dn = 0.7a, (b) dn = 0.8a, and (c) dn = 0.9a as a function of wavelength.
Fig. 6(a) and (c) show the intensity of the HE3,1 modes in Type I and Type II ENZ PCFs, respectively, and Fig. 6(b) and (d) show the phase diagrams of the corresponding OAM mode in Type I and Type II ENZ PCFs, respectively. In both cases, we assumed that a = 2 μm, dn = 0.8a, and r0 = 5.3 μm. Both eigenmodes are well confined in silica area; the phase of the corresponding OAM2 mode smoothly changes by a factor of 4 π for azimuthal direction in both cases. Fig. 7(a) and (b) show confinement losses of the eigenmodes for Type I and Type II ENZ PCFs as a function of wavelength from 1.3 μm to 1.8 μm, respectively. The confinement losses increase with the wavelength owing to mode leakage through the cladding region. However, the confinement losses of the eigenmodes are negligibly small except for those in the higher order modes (≥ HE11,1 or EH8,1). These are very small in the low wavelength region. Note that the proposed PCFs with the ideal ENZ material provide lower confinement loss than air hole circular PCF with the same parameters as ours [13]. The reason is that the high refractive index 4
Optik - International Journal for Light and Electron Optics 204 (2020) 164209
M. Kim and S. Kim
Fig. 5. The effective index difference between OAM modes of the same order in Type II ENZ PCF for (a) dn = 0.7a, (b) dn = 0.8a, and (c) dn = 0.9a as a function of wavelength. Table 1 Comparison of Effective Index Difference. Structure
Δneff at λ = 1.55 μm
Type I ENZ PCF Type II ENZ PCF Circular PCF (Ref. [13]) Microstructure ring fiber (Ref. [16]) Hollow-core photonic bandgap fiber (Ref. [17])
∼ ∼ ∼ ∼ ∼
10−3 1.8 × 10−3 2 × 10−4 10−4 1.9 × 10−4
contrast by virtue of the ENZ material enhances the field confinement of eigenmodes. In addition, we observe that the ENZ effect of Type II ENZ PCF is more substantial compared with Type I PCF, which results in the low confinement loss of the Type II ENZ PCF owing to reduced mode leakage through the cladding region. The result of the dispersion characteristic calculations of the eigenmodes in both Type I and Type II ENZ PCFs as a function of wavelength from 1.5 μm to 1.6 μm are shown in Fig. 8(a) and 8(b), respectively. All the eigenmodes of Type I and Type II ENZ PCFs have a relatively flat dispersion characteristic. The total dispersion variation for all eigenmodes is less than 400 ps/nm/km, and the dispersion variation for each mode is less than 40 ps/nm/km. 2.2. PCFs with a realistic ENZ material We considered a realistic ENZ material to investigate the performance of Type I and Type II ENZ PCFs. It is well known that there are several materials that provide the ENZ property, such as KCL and AZO. We chose AZO as an ENZ material because it provides the ENZ material property at optical communication wavelengths with permittivity εAZO = 0.025 + 0.37i at λ = 1.55 μm [22,24,25]. Contrary to the ideal case, the permittivity of the AZO exhibits a loss term (i.e. Im(εAZO) ≠ 0), thereby increasing the magnitude of the permittivity of AZO. However, the ENZ property of AZO is expected to still be valid as the magnitude of permittivity of AZO is small enough. We assume a = 2 μm, dn = 0.8a, and r0 = 5.3 μm for all calculations. Fig. 9(a) and (b) show the effective index of the eigenmodes and the effective index difference between OAM modes in Type I ENZ PCF with AZO, respectively. The total number of eigenmodes is the same as that in Type I ENZ PCF with the ideal ENZ material. The effective index difference also shows similar characteristic to the ideal one. Therefore, Type I ENZ PCF with real ENZ material provides a good mode separation while maintaining the radial single mode condition. The effective index of the eigenmodes and their differences for Type II ENZ PCF with AZO are plotted in Fig. 10(a) and (b), respectively. The total number of guided modes is the same as Type II ENZ PCF with the ideal ENZ material; however, the EH10,1 mode leaks above λ = 1.75 μm owing to mode spreading as the wavelength increases. The effective index difference between OAM 5
Optik - International Journal for Light and Electron Optics 204 (2020) 164209
M. Kim and S. Kim
Fig. 6. Intensity distributions of the HE3,1 mode in (a) Type I and (c) Type II ENZ PCFs, and phase distributions of the corresponding OAM mode in (b) Type I and (d) Type II ENZ PCFs. In both cases, a = 2 μm, dn = 0.8a, and r0 = 5.3 μm.
Fig. 7. Confinement loss of the eigenmodes in (a) Type I and (b) Type II ENZ PCFs as a function of wavelength. In both cases, a = 2 μm, dn = 0.8a, and r0 = 5.3 μm.
Fig. 8. Dispersion characteristic of the eigenmodes in (a) Type I and (b) Type II ENZ PCFs as a function of wavelength. In both cases, a = 2 μm, dn = 0.8a, and r0 = 5.3 μm.
modes of the same order is larger than 5 × 10−4 for all the eigenmodes, which is almost the same result as for the ideal case. The results in Figs. 9 and 10 verify that that a very large effective index difference can be achieved from the proposed PCFs with realistic ENZ material. 6
Optik - International Journal for Light and Electron Optics 204 (2020) 164209
M. Kim and S. Kim
Fig. 9. (a) Effective indices of the eigenmodes and (b) the effective index difference between the same OAM mode orders in Type I ENZ PCF with AZO as a function of wavelength.
Fig. 10. (a) Effective indices of the eigenmodes and (b) the effective index difference between the same OAM mode orders in Type II ENZ PCF with AZO as a function of wavelength.
Fig. 11(a) and (b) show the confinement loss of the eigenmodes in Type 1 and Type II ENZ PCFs with AZO as a function of wavelength, respectively. Given that the realistic ENZ material, i.e. AZO, presents material loss, this necessarily induces propagation loss. It is observed that the confinement losses are quite high owing to the high value of εi. The confinement loss of Type II ENZ PCF is slightly higher than for Type I ENZ PCF because the electric field penetrates further into the circular lossy ENZ holes in Type II PCF than into the center ENZ material in Type I ENZ PCF. While fibers with realistic ENZ material exhibit high loss, the development of ENZ materials is actively progressing. We believe that ENZ material with lower losses will be realized in the near future. The fabrication of the proposed photonic crystal fibers with the ENZ material can be accomplished through recently introduced techniques to fabricate optical fibers with complex material compositions [26,27], such as direct thermal drawing [28], pressureassisted melting technique [29], and high-pressure chemical vapor deposition [30]. For example, the pressure-assisted melting technique could be used to insert the bulk-form of the ENZ material into the appropriate holes in the host material for the PCF. This method allows selective filling of individual channels with small diameter. Recently, the PCF with an epsilon-near-zero material, KCL, holes was fabricated using this method, [31]. Because of the development of fabrication techniques and the recent much interest in ENZ materials, we believe that the proposed fiber could be realized and more advanced fabrication techniques will be available in the near future.
Fig. 11. (a) Confinement loss of the eigenmodes in (a) Type I and (b) Type II ENZ PCF with AZO as a function of wavelength. 7
Optik - International Journal for Light and Electron Optics 204 (2020) 164209
M. Kim and S. Kim
3. Conclusion We proposed two types of the ENZ circular photonic crystal fibers, where the ENZ material is inserted in the center air hole (Type I) and in the first air hole layer (Type II) of the circular PCF. The high index contrast between the ENZ material and the background material causes high effective index difference, resulting in good mode separation (Δneff > 5 ×−4) over a wide range of wavelength from 1.3 μm to 1.8 μm. This is a much higher value than for the reported PCFs for OAM transmission. The designed fibers support up to 42 OAM modes, satisfying radial single mode condition. We used AZO as the ENZ material. The fibers with AZO provide almost same the results as the ideal one: high effective index difference and large number of OAM modes. However, the confinement loss of the eigenmodes is high owing to the high material loss of AZO. We believe that the realization of the ENZ materials featuring lower loss will be achieved in the near future. Thus the proposed fibers with ENZ material may be employed for various OAM applications, particularly, MDM. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07049349) and GIST Research Institute (GRI) grant funded by the GIST in 2020 References [1] D.J. Richardson, J.M. Fini, L.E. Nelson, Space-division multiplexing in optical fibres, Nat. Photonics 7 (2013) 354–362, https://doi.org/10.1038/nphoton. 2013.94. [2] N. Bai, E. Ip, Y.-K. Huang, E. Mateo, F. Yaman, M.-J. Li, S. Bickham, S. Ten, J. Liñares, C. Montero, V. Moreno, X. Prieto, V. Tse, K. Man Chung, A.P.T. Lau, H.Y. Tam, C. Lu, Y. Luo, G.-D. Peng, G. Li, T. 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Original research article
Epsilon-near-zero photonic crystal fibers for a large mode separation of orbital angular momentum modes
T
Myunghwan Kim, Soeun Kim* Integrated Optics Laboratory, Advanced Photonics Research Institute, GIST, Gwangju 61005, South Korea
A R T IC LE I N F O
ABS TRA CT
Keywords: Photonic crystal fiber Epsilon-near-zero Orbital angular momentum Mode separation
An orbital angular momentum (OAM) mode is a good candidate as an orthogonal basis mode set for mode division multiplexing (MDM) owing to its unique phase properties. Photonic crystal fibers support large numbers of OAM modes with notable performances. However, the effective index difference between high order OAM modes is not large enough. In this work, we propose two types of epsilon-near-zero (ENZ) materials-embedded circular photonic crystal fibers for high mode separation. Compared with previously developed photonic crystal fibers, the designed fibers exhibit a high refractive index contrast owing to the inserted ENZ materials. The designed fibers support up to 42 OAM modes. Most of these OAM modes provide a large effective index difference (Δneff > 10−3) while maintaining a radial sing mode condition over a wide wavelength range from 1.3 μm to 1.8 μm. The proposed fibers could potentially be exploited in MDM for stable transmission of OAM modes and other OAM-based applications.
1. Introduction As the demand for data usage is increasing, achieving high data transmission capacity is becoming an important topic in optical communication systems. Space division multiplexing (SDM) has attracted considerable interest as a promising technique to increase data capacity. Mode division multiplexing (MDM), which is a special case of SDM, has been widely investigated for high-speed data transmission [1]. Recently, MDM in few mode fibers (FMFs) based on linearly polarized (LP) modes has been introduced. FMFs multiply the transmission capacity in proportion to the total number of modes, as opposed to single mode fibers [2,3]. However, LP modes suffered from mode coupling owing to a low effective index difference between them. Thus, large and complex multiple-input multiple-output (MIMO) processing is required to mitigate the mode crosstalk [4,5]. Orbital angular momentum (OAM) modes, carrying exp(ilφ) (where l is the topological charge number, and φ is the azimuthal angle) have been widely investigated as an orthogonal modal basis of MDM owing to the weak coupling between OAM modes [6–8]. Recently, terabit-scale MDM based on OAM fibers has been experimentally demonstrated [9]. However, this can support only a few OAM modes owing to low refractive index contrast between the core and the cladding. To solve these issues, many types photonic crystal fibers (PCFs) have been reported, such as helically twisted PCFs [10], Kagome lattice PCFs [11], hexagonal lattice PCFs [12], circular PCFs [13–15], microstructure ring fiber [16], hollow-core PCFs [17], and PCFs with As2S3 background material [12,14]. Experimental demonstration has also been achieved in a helically twisted PCF [18]. The high index contrast between air holes and a background material enables PCFs to support many OAM modes, and their flexible structure design provides unique mode properties. Although previously reported PCFs support a great deal of OAM modes with high
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Corresponding author. E-mail address: [email protected] (S. Kim).
https://doi.org/10.1016/j.ijleo.2020.164209 Received 19 September 2019; Received in revised form 24 December 2019; Accepted 7 January 2020 0030-4026/ © 2020 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 204 (2020) 164209
M. Kim and S. Kim
Fig. 1. Schematic of the proposed circular ENZ PCF. (a) Type I ENZ PCF, (b) Type II ENZ PCF; a is the period, r0 is the radius of the center hole, and dn is the radius of the n-th layer of the air holes in the cladding region. The refractive index profile of (c) Type I ENZ PCF and (d) Type II ENZ PCF.
mode performance, the effective index difference between OAM modes is not large enough to support stable transmission for higher order OAM modes because it tends to reduce as the order of mode increases. Therefore, the effective indices of the eigenmodes should be more separated to avoid coupling from OAM mode into LP mode. This calls for the use of a high refractive index material such as As2S3 as a back ground material in order to increase index contrast. However, this kind of material is not suitable for MDM owing to its high nonlinearity. In this paper, we present epsilon-near-zero (ENZ) materials-inserted circular PCFs for large effective index separation. The ENZ material provides almost zero magnitude of permittivity. This unique property has been widely utilized to enhance confinement of guided mode and mode performance [19–21]. Very recently, PCF with ENZ circular hole in the cladding has been presented. This fiber shows very high birefringence [22]. Here, we suggest two types of ENZ-based PCFs. Both feature a hollowcore circular PCF structure. We substitute the ENZ material for the center air hole or for the first layer of the air holes in the cladding region, respectively. The high index contrast between the ENZ material and the background material of the fiber supports a stable transmission of OAM modes with good mode separation. We consider both ideal and realistic ENZ materials. The designed fibers provide up to 42 OAM modes and a large effective index difference (Δneff > 10−3) between OAM modes of the same order over a wide range of wavelengths, from 1.3 μm to 1.8 μm. This index is ten times higher than many recent designs [13,15,16]. We believe that the proposed design can be employed to improve the mode stability of OAM modes and for various OAM applications.
2. Structure of proposed ENZ fibers and numerical results 2.1. PCFs with an ideal ENZ material Fig. 1(a) and (b) show the cross sectional diagrams of the tow proposed ENZ-based circular PCFs: (a) center ENZ circular PCF (Type I ENZ PCF), and (b) first air hole layer ENZ circular PCF (Type II ENZ PCF), where a is the period, r0 is the radius of the center hole, and dn is the radius of the n-th layer of the air holes in the cladding region. We also plot the refractive index profiles of Type I and Type II ENZ PCFs in Fig. 1(c) and (d), respectively: x-direction refractive index profile at the center (y = 0). The two proposed types of fibers are composed of a hollow core circular PCF structure; however, the center air hole (Type I) and the first layer of air holes in the cladding (Type II) are replaced with the ENZ material. We assume in most of cases the following values of parameters: a = 2 μm, dn = 0.8a, and r0 = 5.3 μm. These values are the same as in the circular PCF for supporting OAM modes in ref. [13] for the sake of a fair performance comparison. All calculations were conducted using the finite element method (FEM) with a perfect matched layer (PML) at the boundary to determine the mode properties We chose silica as the background material owing to its low nonlinear coefficient and low loss. The refractive index of silica was defined by the Sellmeier equation [23]. We consider an ideal ENZ material with refractive index nENZ = 0.1. Note that an ideal ENZ material is studied in this section whereas a realistic ENZ material in the next section. As in conventional photonic crystal fibers, OAM 2
Optik - International Journal for Light and Electron Optics 204 (2020) 164209
M. Kim and S. Kim
Fig. 2. Effective indices of the eigenmodes in Type I ENZ PCF for (a) r0 = 5 μm, (b) r0 = 5.3 μm, (c) r0 = 5.6 μm, and (d) r0 = 5.9 μm as a function of wavelength.
modes can exist in the proposed circular ENZ PCFs as combinations of even and odd symmetry of HE and EH modes, mathematically described as: odd OAMl,m = HEeven l+1,m ± jHEl+1,m
(1)
odd OAM±l, m = EHleven − 1, m ± jEHl − 1, m
(2)
where l is the topological number, m is the order of the mode for the radial direction, and the sign +(-) denotes the right(left) phase rotation. Fig. 2(a) to (d) show the effective indices of the supported eigenmodes in the proposed Type I ENZ PCF for r0 = 5 μm, 5.3 μm, 5.6 μm, and 5.9 μm, respectively, as a function of wavelength. The proposed fibers can support up to the HE12,1 and EH10,1 modes. Thus, this fiber can transport 42 OAM modes (l ≤11) in total. The proposed fibers satisfy radial single mode condition except for r0 = 5 μm. Note that the radially higher modes for r0 = 5 μm are not shown in Fig. 2(a). We can observe that the effective indices reduce as r0 increases. The narrowing effective core region decreases the effective indices of the eigenmodes and results in mode cutoff of the higher order modes for long wavelengths. We can see that the modes from the EH8,1 mode to the EH10,1 mode are cutoff Fig. 3(a) to (d) show the effective index difference between the HEl-1 mode and EHl+1 mode, which have the same order of OAM mode, in the proposed Type I ENZ PCF for r0 = 5 μm, 5.3 μm, 5.6 μm, 5.9 μm, respectively, as a function of wavelength from 1.3 μm to 1.8 μm. Note that the effective index difference for the OAM1 mode is calculated from the effective index difference between the HE2,1 and TE0,1 modes. To exploit OAM modes in MDM, the effective index difference should be larger than 10−4, thereby avoiding mode coupling. As shown in Fig. 3(a) to (d), the effective index difference increases as r0 increases. Although the reduced effective core area decreases the number of eigenmodes, it results in the increase of Δneff. The effective index differences of all OAM modes are larger than 10−4 over a wide band of wavelength except for OAM9 at r0 = 5 μm. Moreover, Δneff for r0 = 5.9 μm is above 10-3 for all of the OAM modes. These values are ten times higher than those of many recent designs [13,16]. We also investigated the OAM mode performance of the proposed Type II ENZ PCF. Fig. 4(a) to (c) show the effective indices of the supported eigenmodes in Type II ENZ PCF for dn = 0.7a, 0.8a, and 0.9a, respectively. The parameters a and r0 are fixed at 2 μm and 5.3 μm, respectively. Similar to Type I ENZ PCF, Type II ENZ PCF supports the eigenmodes up to the HE12,1 mode and the EH10,1 mode. The effective indices decrease as dn increases owing to enhancement of the ENZ effect. Compared with Type I ENZ PCF, Type II ENZ PCF supports all eigenmodes without mode cutoff and maintains the radial single mode condition over a wide range of wavelengths. Fig. 5(a) to (c) show the effective index difference for OAM modes of the same order in the proposed Type II ENZ PCF for dn = 0.7a, 0.8a, and 0.9a, respectively. As dn increases, Δneff increases owing to enhancement of the ENZ effect. The high refractive index contrast between the ENZ material and silica enables large effective index separation between HEl-1 and EHl+1 modes. The refractive index differences for all cases are larger than 10−4. Therefore, the proposed ENZ PCFs can stably transmit a great deal of the OAM modes without incurring mode coupling into LP modes. In particular, the effective index differences of all OAM modes for dn = 0.9a are larger than 10-3 from 1.3 μm to 1.8 μm. Compared with Type I ENZ PCF in the same conditions (i.e. a = 2 μm, dn = 0.8a, and r0 = 5.3 μm), Type II ENZ PCF shows larger effective index difference. We summarize the results for Δneff at λ = 1.55 μm for Type I, Type II ENZ PCFs, Ref. [13], [16], and [17] in Table 1. 3
Optik - International Journal for Light and Electron Optics 204 (2020) 164209
M. Kim and S. Kim
Fig. 3. Effective index difference between the same OAM mode orders in the Type I ENZ PCF (a) r0 = 5 μm, (b) r0 = 5.3 μm, (c) r0 = 5.6 μm, and (d) r0 = 5.9 μm as a function of wavelength.
Fig. 4. (a) Effective indices of the eigenmodes in the Type II ENZ PCF for (a) dn = 0.7a, (b) dn = 0.8a, and (c) dn = 0.9a as a function of wavelength.
Fig. 6(a) and (c) show the intensity of the HE3,1 modes in Type I and Type II ENZ PCFs, respectively, and Fig. 6(b) and (d) show the phase diagrams of the corresponding OAM mode in Type I and Type II ENZ PCFs, respectively. In both cases, we assumed that a = 2 μm, dn = 0.8a, and r0 = 5.3 μm. Both eigenmodes are well confined in silica area; the phase of the corresponding OAM2 mode smoothly changes by a factor of 4 π for azimuthal direction in both cases. Fig. 7(a) and (b) show confinement losses of the eigenmodes for Type I and Type II ENZ PCFs as a function of wavelength from 1.3 μm to 1.8 μm, respectively. The confinement losses increase with the wavelength owing to mode leakage through the cladding region. However, the confinement losses of the eigenmodes are negligibly small except for those in the higher order modes (≥ HE11,1 or EH8,1). These are very small in the low wavelength region. Note that the proposed PCFs with the ideal ENZ material provide lower confinement loss than air hole circular PCF with the same parameters as ours [13]. The reason is that the high refractive index 4
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Fig. 5. The effective index difference between OAM modes of the same order in Type II ENZ PCF for (a) dn = 0.7a, (b) dn = 0.8a, and (c) dn = 0.9a as a function of wavelength. Table 1 Comparison of Effective Index Difference. Structure
Δneff at λ = 1.55 μm
Type I ENZ PCF Type II ENZ PCF Circular PCF (Ref. [13]) Microstructure ring fiber (Ref. [16]) Hollow-core photonic bandgap fiber (Ref. [17])
∼ ∼ ∼ ∼ ∼
10−3 1.8 × 10−3 2 × 10−4 10−4 1.9 × 10−4
contrast by virtue of the ENZ material enhances the field confinement of eigenmodes. In addition, we observe that the ENZ effect of Type II ENZ PCF is more substantial compared with Type I PCF, which results in the low confinement loss of the Type II ENZ PCF owing to reduced mode leakage through the cladding region. The result of the dispersion characteristic calculations of the eigenmodes in both Type I and Type II ENZ PCFs as a function of wavelength from 1.5 μm to 1.6 μm are shown in Fig. 8(a) and 8(b), respectively. All the eigenmodes of Type I and Type II ENZ PCFs have a relatively flat dispersion characteristic. The total dispersion variation for all eigenmodes is less than 400 ps/nm/km, and the dispersion variation for each mode is less than 40 ps/nm/km. 2.2. PCFs with a realistic ENZ material We considered a realistic ENZ material to investigate the performance of Type I and Type II ENZ PCFs. It is well known that there are several materials that provide the ENZ property, such as KCL and AZO. We chose AZO as an ENZ material because it provides the ENZ material property at optical communication wavelengths with permittivity εAZO = 0.025 + 0.37i at λ = 1.55 μm [22,24,25]. Contrary to the ideal case, the permittivity of the AZO exhibits a loss term (i.e. Im(εAZO) ≠ 0), thereby increasing the magnitude of the permittivity of AZO. However, the ENZ property of AZO is expected to still be valid as the magnitude of permittivity of AZO is small enough. We assume a = 2 μm, dn = 0.8a, and r0 = 5.3 μm for all calculations. Fig. 9(a) and (b) show the effective index of the eigenmodes and the effective index difference between OAM modes in Type I ENZ PCF with AZO, respectively. The total number of eigenmodes is the same as that in Type I ENZ PCF with the ideal ENZ material. The effective index difference also shows similar characteristic to the ideal one. Therefore, Type I ENZ PCF with real ENZ material provides a good mode separation while maintaining the radial single mode condition. The effective index of the eigenmodes and their differences for Type II ENZ PCF with AZO are plotted in Fig. 10(a) and (b), respectively. The total number of guided modes is the same as Type II ENZ PCF with the ideal ENZ material; however, the EH10,1 mode leaks above λ = 1.75 μm owing to mode spreading as the wavelength increases. The effective index difference between OAM 5
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Fig. 6. Intensity distributions of the HE3,1 mode in (a) Type I and (c) Type II ENZ PCFs, and phase distributions of the corresponding OAM mode in (b) Type I and (d) Type II ENZ PCFs. In both cases, a = 2 μm, dn = 0.8a, and r0 = 5.3 μm.
Fig. 7. Confinement loss of the eigenmodes in (a) Type I and (b) Type II ENZ PCFs as a function of wavelength. In both cases, a = 2 μm, dn = 0.8a, and r0 = 5.3 μm.
Fig. 8. Dispersion characteristic of the eigenmodes in (a) Type I and (b) Type II ENZ PCFs as a function of wavelength. In both cases, a = 2 μm, dn = 0.8a, and r0 = 5.3 μm.
modes of the same order is larger than 5 × 10−4 for all the eigenmodes, which is almost the same result as for the ideal case. The results in Figs. 9 and 10 verify that that a very large effective index difference can be achieved from the proposed PCFs with realistic ENZ material. 6
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Fig. 9. (a) Effective indices of the eigenmodes and (b) the effective index difference between the same OAM mode orders in Type I ENZ PCF with AZO as a function of wavelength.
Fig. 10. (a) Effective indices of the eigenmodes and (b) the effective index difference between the same OAM mode orders in Type II ENZ PCF with AZO as a function of wavelength.
Fig. 11(a) and (b) show the confinement loss of the eigenmodes in Type 1 and Type II ENZ PCFs with AZO as a function of wavelength, respectively. Given that the realistic ENZ material, i.e. AZO, presents material loss, this necessarily induces propagation loss. It is observed that the confinement losses are quite high owing to the high value of εi. The confinement loss of Type II ENZ PCF is slightly higher than for Type I ENZ PCF because the electric field penetrates further into the circular lossy ENZ holes in Type II PCF than into the center ENZ material in Type I ENZ PCF. While fibers with realistic ENZ material exhibit high loss, the development of ENZ materials is actively progressing. We believe that ENZ material with lower losses will be realized in the near future. The fabrication of the proposed photonic crystal fibers with the ENZ material can be accomplished through recently introduced techniques to fabricate optical fibers with complex material compositions [26,27], such as direct thermal drawing [28], pressureassisted melting technique [29], and high-pressure chemical vapor deposition [30]. For example, the pressure-assisted melting technique could be used to insert the bulk-form of the ENZ material into the appropriate holes in the host material for the PCF. This method allows selective filling of individual channels with small diameter. Recently, the PCF with an epsilon-near-zero material, KCL, holes was fabricated using this method, [31]. Because of the development of fabrication techniques and the recent much interest in ENZ materials, we believe that the proposed fiber could be realized and more advanced fabrication techniques will be available in the near future.
Fig. 11. (a) Confinement loss of the eigenmodes in (a) Type I and (b) Type II ENZ PCF with AZO as a function of wavelength. 7
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3. Conclusion We proposed two types of the ENZ circular photonic crystal fibers, where the ENZ material is inserted in the center air hole (Type I) and in the first air hole layer (Type II) of the circular PCF. The high index contrast between the ENZ material and the background material causes high effective index difference, resulting in good mode separation (Δneff > 5 ×−4) over a wide range of wavelength from 1.3 μm to 1.8 μm. This is a much higher value than for the reported PCFs for OAM transmission. The designed fibers support up to 42 OAM modes, satisfying radial single mode condition. We used AZO as the ENZ material. The fibers with AZO provide almost same the results as the ideal one: high effective index difference and large number of OAM modes. However, the confinement loss of the eigenmodes is high owing to the high material loss of AZO. We believe that the realization of the ENZ materials featuring lower loss will be achieved in the near future. Thus the proposed fibers with ENZ material may be employed for various OAM applications, particularly, MDM. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07049349) and GIST Research Institute (GRI) grant funded by the GIST in 2020 References [1] D.J. Richardson, J.M. Fini, L.E. Nelson, Space-division multiplexing in optical fibres, Nat. Photonics 7 (2013) 354–362, https://doi.org/10.1038/nphoton. 2013.94. [2] N. Bai, E. Ip, Y.-K. Huang, E. Mateo, F. Yaman, M.-J. Li, S. Bickham, S. Ten, J. Liñares, C. Montero, V. Moreno, X. Prieto, V. Tse, K. Man Chung, A.P.T. Lau, H.Y. Tam, C. Lu, Y. Luo, G.-D. Peng, G. Li, T. 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