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Characteristics of vector beams in mid-infrared waveband in an As2Se3 photonic crystal fiber with small hollow core
Optical Fiber Technology 55 (2020) 102152
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Characteristics of vector beams in mid-infrared waveband in an As2Se3 photonic crystal fiber with small hollow core
T
Weiqing Gaoa, Xiu Zhanga, Wenhui Jianga, Zhengxiong Zhanga, Panyun Gaoa, Liang Chena, Peng Wanga, Wei Zhanga, Rui Wanga, Meisong Liaob, Takenobu Suzukic, Yasutake Ohishic, ⁎ Yong Zhoua, a
Department of Optical Engineering, School of Electronic Science & Applied Physics, Hefei University of Technology, Feicui Road 420, Hefei 230601, China R&D Center of High Power Laser Components, Shanghai Institute of Optics and Fine Mechanics, Shanghai 201800, China c Research Center for Advanced Photon Technology, Toyota Technological Institute, 2–12–1 Hisakata, Tempaku, Nagoya 468–8511, Japan b
A R T I C LE I N FO
A B S T R A C T
Keywords: Vector beam Mid-infrared waveband Chalcogenide fiber Photonic crystal fiber
Characteristics of vector beams in mid-infrared waveband are analyzed numerically in an As2Se3 photonic crystal fiber (PCF) with small central hollow core (SCHH), including the mode fields, confinement loss, effective refractive index and chromatic dispersion. In the PCFs with the SCHH diameter d0 of 0–1.5 μm, the confinement losses of the four vector beams (HE11, TM01, TE01, and HE21) are smaller than 1.0 dB/m for the wavelengths up to ~5 μm. For the SCHH diameter d0 of 0.5–1.5 μm, the field enhancement of the HE11 mode occurs in the SCHHs in the central area for the wavelengths ≥5 μm. This happens because the evanescent light penetrates into the SCHH. With d0 increasing, the effective refractive index separations (δneff) between the HE11 mode and highorder modes (TM01, TE01, and HE21) decreases, while the δneff among the high-order modes increases. At the wavelength of 5 μm, the δneff of HE11-TE01, TE01-HE21, and HE21-TM01 is 0.03098, 0.03960, and 0.06867, respectively, when d0 is 1.5 μm. The second ZDWs of TE01, HE21 and TM01 mode are blueshifted with the increase of d0, but they are longer than 5 μm. The negative dispersion regions of the high-order modes (TE01, HE21, and TM01) are much wider than the HE11 modes. With the increase of d0, the HE11 mode dispersion curves have no obvious change, while the high-order modes (TE01, HE21, and TM01) dispersion curves become steep at the longer wavelength, the TM01 mode changes most obviously, and the depression appears. The transmission property for the waveband of 5–10 μm could be improved by increasing the ring number or core diameter of the fiber. The fiber losses at 10 μm are 4.81, 7.65, 5.95 and 2.47 dB/m for the four kinds of vector beams, respectively, when the diameters of the central area and SCHH are increased to 15.09 and 5.25 μm. Our simulated results will be useful for the optical applications in mid-infrared waveband, such as the generation of cylindrical vector mode and orbital angular momentum (OAM) and the particle manipulation by optical fields.
1. Introduction Mid-infrared (MIR) fiber sources, especially those within the 2–10 μm waveband, have potential applications in various fields, such as optical coherence tomography [1], spectroscopy [2], and trace gas detection [3]. Recently, some optical fiber sources have been generated in the 2–10 μm waveband by supercontinuum generation (SCG) [4], stimulated Raman scattering [5], four-wave mixing [6], and so on. The brightness of supercontinuum sources was orders of magnitude larger than even square-mile large synchrotrons [7], which have been found important applications within multi-spectral bio-imaging [8] and microscopy [9].
⁎
Most of the optical sources in MIR wavebands only considered the intensity distribution. The electric field vector of light is important for applications in the MIR waveband. Vector beams constitute a solution of the Maxwell’s wave equations [10]. In addition, vector beams with spatially variant state of polarizations are expected to lead to new effects and phenomena in optical systems [11]. They have been encountered in numerous applications on optical communications [12], super-resolved focal spots [13], plasmonic nanofocusing [14], singlemolecule spectroscopy [15], optical trapping [16] and metal machining [17]. Vector beams can also be used to form the light carrying orbital angular momentum (OAM) to extend their applied fields [18]. The most typical method to generate vector beams used spatial light
Corresponding author. E-mail address: [email protected] (Y. Zhou).
https://doi.org/10.1016/j.yofte.2020.102152 Received 2 December 2019; Received in revised form 15 January 2020; Accepted 15 January 2020 1068-5200/ © 2020 Published by Elsevier Inc.
Optical Fiber Technology 55 (2020) 102152
W. Gao, et al.
vectorial mode solving technique. Fig. 2 shows the simulated evolution of the four vector beam fields (HE11, TE01, HE21, and TM01) as a function of the wavelength in the As2Se3 PCFs when the SCHH diameter changes from 0 to 1.5 μm. The parameters Λ and d are 1.5 and 1.125 μm, respectively. We defined the region surrounded by the air-hole cladding as the core area. The core diameter is 4.31 μm, which is defined as the diameter of the circle inscribed in the hexagonally arranged area. The confinement losses of the four vector beams are shown in Fig. 3(a)-3(d), and correspond to d0 values of 0, 0.5, 1.0, and 1.5 μm, respectively. The wavelength (λ1-dB) corresponding to the confinement loss of ~1 dB/m is used to express the transmission ability in MIR waveband of the vector modes. The confinement loss is part of the total fiber loss. We did not consider material loss in this simulation. According to Fig. 2, the four vector beam fields (HE11, TE01, HE21, and TM01) can be restrained effectively in the core area from 1 to 5 μm. From ~5 μm, the energy of the four vector beam will begin to appear in the air-hole cladding as the wavelength increases, and become very obvious at 10 μm. Table 1 shows the λ1-dB and λ10-dB of the four vector beams in different d0. When the value of confinement loss is equal to ~1 dB/m, HE11 mode will transmit ~6 μm, TE01, HE21, and TM01 modes will transmit ~5 μm. When the value of confinement loss is equal to ~10 dB/m, HE11 mode will transmit ~6–8 μm, TE01, HE21, and TM01 modes will transmit ~5–6 μm. It means that the four vector beams can be transmitted stably in mid-infrared waveband. The phenomenon of field enhancement is affected by the SCHH diameter [42]. When d0 is zero, there is no field enhancement at the central area of the PCF. When d0 is 0.5 μm, the light concentrates in the SCHH, and the field in this area is enhanced, as shown in Fig. 2(b). The field enhancement is also obvious when d0 increases to 1.0 and 1.5 μm, as shown in Fig. 2(c) and (d). Field enhancement occurs at longer wavelengths because the size of the SCHH (0.5–1.5 μm) is much smaller than the transmitted wavelength in the range of 5–10 μm, and is well below the diffraction limit. The evanescent light propagates into the SCHH because the refractive index of the SCHH is smaller than the As2Se3 glass around it. This effect enhances as the wavelength increases and causes field enhancements in the SCHH. However, for shorter wavelengths, the size of the SCHH is not much lower than the diffraction limit so that the field enhancement does not occur.
modulators [19]. Chen et al. researched the generation and propagation of a partially coherent vector beam with special correlation functions theoretically and experimentally [20]. Yue et al. proposed and experimentally demonstrated an approach to generate vector vortex beams with a single metasurface by locally tailoring phase and transverse polarization distributions [21]. Recently, several other methods have been reported to generate vector beams. Matsuba et al. generated vector beams by superposing two optical vortex beams from helical undulators [22]. Kozawa et al. obtained the vector beam output from vertical cavity, surface-emitting lasers, using a birefringent lens inside the cavity to select the single transverse mode of the vector beam [23]. Li et al. generated a perfect optical vortex and elliptic perfect vector beam by modulating the dynamic and geometric phases [24]. Vector beams could also be generated in optical fibers, such as in the cases where only few-mode fibers (FMFs) are used [25] by adjusting the misalignment in the angle and transverse dimensions between fibers [26], by applying acoustic flexural waves on the fibers [27], or by using mode selective couplers [28] and long-period fiber gratings [29]. The pulsed vector beams could also be generated in fibers [30]. To extend the application to MIR wavebands, the generation of vector beams in soft-glass fibers is attractive. Compared to silica fibers, soft-glass fibers have the advantages of high nonlinearity and wide transmission windows within the MIR waveband [31]. In 2006, Littler et al. first reported the acousto-optic resonances in non-silica fibers, which could be used for the generation of vector beams in soft-glass fibers [32]. In 2012, Yue et al. demonstrated the SCG using different vortex modes in As2S3 chalcgenide fibers [33]. In 2016, Ji et al. presented numerically the propagation of radially and azimuthally polarized cylindrical vector modes in tellurite fibers [34,35]. To generate the light source covering the entire spectral range of 2–10 μm with high efficiency, it is important to design the fiber so that the optical modes inside the core area can be confined for wide spectral ranges. The fibers transmitting the light in the MIR waveband are developed with different lateral microstructures, including photonic crystal [36,37], hollow-core [38], negative curvature [39], and twisted fibers [40]. It is significant to develop vector beams in MIR wavebands to extend the applications and generate new effects in the region. The first key problem is to develop the fibers with the structures that can transmit vector beams covering MIR wavebands using single fiber. In this work, the characteristics in the MIR waveband of four vector beams (HE11, TM01, TE01, and HE21) in an As2Se3 PCF with SCHH, are analyzed versus the diameter of the SCHH, including the effective refractive index, chromatic dispersion, confinement loss, and vector beam field. As the SCHH diameter and wavelength increase, the energy distribution changes greatly. With the SCHH diameter increasing, the effective refractive index separations (δneff) between the HE11 mode and high-order modes (TM01, TE01, and HE21) decreases, while the δneff among the high-order modes increases. At the wavelength of 5 μm, the δneff between each mode can be greater than 0.03. This can reduce the probability of mode coupling efficiently and enable good differentiation among different vector modes. The confinement loss increases with the SCHH diameter increasing, but it is also can transmit in the spectral range up to approximately 5 μm with the loss lower than 1 dB/m.
3. Effective refractive index and chromatic dispersion The effective refractive index and chromatic dispersion profiles of the As2Se3 PCF are shown as in Fig. 4. When d0 is zero, the effective refractive indices of the four vector beams decrease as the wavelength increases, as shown in Fig. 4(a). The effective refractive index separation (δneff) between the HE11 and high-order modes (TE01, HE21, and TM01) is obvious, but the δneff among the high-order modes (TE01, HE21, and TM01) is smaller. The δneff between the HE21 and TM01 mode is the smallest. For the same wavelength, the HE11 mode has the highest effective refractive index, followed by the TE01 mode. When the d0 are 0.5, 1.0, and 1.5 μm, the effective refractive indices of the four vector beams also decrease with the wavelength increasing, as shown in Fig. 4(b)-(d). In Fig. 4(b) and (c), the δneff between the HE11 mode and the higher order modes is obvious, but the δneff among the higher order modes are still small. At the same wavelength, the HE11 mode has the highest refractive index, followed by the TE01 mode. In Fig. 4(d), the δneff between the HE11 mode and the higher order modes is obvious, and the δneff among the high-order modes is also obvious. In Ref. [43], a high index ring was added around the fiber core to realize the separated coupling from the fundamental mode to the constitutive vector modes of the LP11 group by an effective refractive index separation δneff is 1.8 × 10−4. Patrick et al. demonstrated that an air core applied into the high index ring fiber was helpful to support 12 OAM modes with δneff is 1 × 10−4 [44]. A FMF was characterized by an inverse-parabolic graded-refractive-index profile to realize the effective
2. Vector beam fields in the As2Se3 PCFs with different SCHH diameters The As2Se3 PCF has a SCHH and three periods of hexagonally arranged holes, as shown in Fig. 1(a). The parameters Λ, d, and d0, represent the pitch distance, air-hole diameter and SCHH diameter, respectively. Fig. 1(b) shows the material refractive index of the As2Se3 glass calculated according to Sellmeier’s equation [41]. The finite element method (FEM) was used to solve Maxwell’s equations in combination with the initial conditions and boundary conditions to achieve the distribution of the vector beam fields. The vector beams of HE11, TE01, HE21 and TM01 were simulated by the full2
Optical Fiber Technology 55 (2020) 102152
W. Gao, et al.
Fig. 1. (a) Cross-section of the As2Se3 PCF. (b) Refractive index profile of material As2Se3 glass.
Fig. 2. Simulated evolution of mode fields in As2Se3 PCF. (a) d0 = 0 μm, (b) d0 = 0.5 μm, (c) d0 = 1.0 μm, and (d) d0 = 1.5 μm.
index separations by δneff is 2.1 × 10-4 [45]. Fig. 5(a)-(b) shows the effective refractive indices of the four vector beams at the wavelengths of 3 and 5 μm for hollow core diameters d0 in the range of 0–1.5 μm. According to Fig. 5(a)–(b), the neff of the four vector beams decreases as d0 increases. The δneff between the HE11 mode and the high-order modes decreases as d0 increases, but it still arrives 0.00854 at the wavelength of 3 μm and 0.03098 at the wavelength of 5 μm when d0 is 1.5 μm. The δneff among the high-order modes increases as d0 increases. As shown in Table 2, for a d0 of 1.5 μm, the δneff of HE11-TE01, TE01-HE21, and HE21-
TM01, are equal to 0.03098, 0.03960, and 0.06867, respectively. The δneff among all the vector modes is high compared to the common fibers. The increased δneff can reduce effectively the possibility of mode coupling between the vector modes, and avoids vector mode degeneration into linearly polarized (LP) modes. It can also suppress mode disorder. The As2Se3 PCF then becomes beneficial for the transmission of vector beams for wavelengths up to 5 μm. The chromatic dispersion profiles are shown by the dashed lines in Fig. 4(a)-(d). The first zero-dispersion wavelengths (ZDWs) and the 3
Optical Fiber Technology 55 (2020) 102152
W. Gao, et al.
it decreases when d0 is 1.5 μm. When d0 is 0, 0.5 μm, 1.0 μm, and 1.5 μm, the first ZDW of the TE01 modes are redshifted gradually, while the first ZDW of TM01 modes are blueshifted. The second ZDWs of TE01, HE21 and TM01 mode are blueshifted with the increase of d0, but they are longer than 5 μm. The negative dispersion regions of the high-order modes (TE01, HE21, and TM01) are much wider than the HE11 modes. Obviously, the dispersion curve of HE11 mode is flatter than that of the higher-order modes (TE01, HE21, and TM01). With the increase of d0, the HE11 mode dispersion curves have no obvious change, while the highorder modes (TE01, HE21, and TM01) dispersion curves become steep at the longer wavelength, the TM01 mode changes most obviously, and the depression appears. 4. Improvement of the transmission property To transmit the light at 10 μm waveband with low loss, we have made some attempts to improve the transmission property of the proposed fiber. The transmission loss is affected by the limitation extent of light into the central area, which is dominated chiefly by the ring number of cladding and core diameter. Firstly, the ring number was increased to four and five, respectively, and the other fiber parameters were kept the same as in Fig. 3 with the d0 of 1.5 μm. The simulated loss curves for the four kinds of vector beams are shown as in Fig. 7. For the HE11, TE01, HE21 and TM01 mode, λ1-dB are 6.6, 6.2, 6.1 and 5.6 μm in the PCF with four rings, respectively, while λ1-dB are 7.7, 7.0, 6.7 and 6.3 μm in the PCF with five rings, respectively. We can see that the loss is decreased obviously for the wavelengths longer than 5 μm compared to three rings in Fig. 3. But the losses at 10 μm are still higher than 100 dB/m. Secondly, the Λ and d were kept the same in the PCF with three rings as in Fig. 3, but the diameters of the central area and SCHH were increased to 15.09 and 5.25 μm, respectively, with the same ratio of 3.5. The simulated loss curves are shown as in Fig. 8. For the HE11, TE01, HE21 and TM01 mode, λ1-dB are 8.9, 8.6, 8.8, and 9.4 μm, respectively, which are decreased greatly compared to Fig. 7. In Fig. 8, the fiber losses at 10 μm are 4.81, 7.65, 5.95 and 2.47 dB/m for the four kinds of vector beams, respectively, which are much lower than those in Fig. 3. Then the As2Se3 PCF with the transmission property as in Fig. 8 can transmit the MIR wavelengths close to 10 μm. In brief, increasing ring number or core diameter can improve the transmission property for the waveband of 5–10 μm. The increase of ring number will increase the difficulty of fiber fabrication, especially for the soft-glass PCFs. While increasing core diameter will not bring additional requirements for fiber fabrication. With the development of fiber fabrication techniques in the future, the two ways could be combined together to improve the transmission at MIR waveband of 10 μm or longer. Then the MIR fibers with more excellent transmission characteristics for vector beams could be realized.
Fig. 3. Loss curves of the As2Se3 PCF with the following hollow core diameters: (a) d0 = 0 μm, (b) d0 = 0.5 μm, (c) d0 = 1.0 μm, and (d) d0 = 1.5 μm. Unit of vertical axis of the partial enlarged view is dB/m.
Table 1 Values of λ1-dB and λ10-dB, of the four vector beams in different d0. d0 (μm)
0
0.5
modes
HE11
TE01
HE21
TM01
HE11
TE01
HE21
TM01
λ1-dB (μm) λ10-dB (μm)
6.6 7.6
5.4 6.1
5.4 6.0
5.4 6.1
6.3 7.3
5.4 6.1
5.4 6.0
5.4 5.9
d0 (μm)
1.0
modes
HE11
TE01
HE21
TM01
HE11
TE01
HE21
TM01
λ1-dB (μm) λ10-dB (μm)
5.8 6.7
5.3 6.0
5.3 5.9
5.1 5.7
5.3 6.1
5.2 5.8
5.0 5.6
4.8 5.3
1.5
5. Conclusions The characteristics of the optical field distribution, confinement loss, effective refractive index, and chromatic dispersion, were analyzed numerically for four types of vector beams for the HE11, TE01, HE21, and TM01 modes in the As2Se3 PCFs with different SCHH diameters. For SCHH diameters in 0, 0.5, 1.0, and 1.5 μm, the HE11 mode could transmit within the MIR waveband at wavelengths up to approximately 6.6, 6.3, 5.8, and 5.3 μm, respectively. For the high-order modes, the maximum wavelength that can be transmitted is approximately 5 μm. The field enhancement of the HE11 mode in the central area occurred for wavelengths ≥5 μm for SCHH diameters in 0.5, 1.0, and 1.5 μm. This was caused because the evanescent light propagated into the SCHH. The profiles of the refractive index and chromatic dispersion were analyzed for the four types of vector modes. The effective refractive index separations between the HE11 mode and the high-order modes decreased with increased SCHH diameters, while the effective
Fig. 4. Effective refractive index (solid lines) and chromatic dispersion (dotted lines) curves of the As2Se3 PCF with the following hollow core diameters: (a) d0 = 0 μm, (b) d0 = 0.5 μm, (c) d0 = 1.0 μm, and (d) d0 = 1.5 μm.
second ZDWs (ZDW2) of the modes HE11, TE01, HE21, and TM01, are listed as in Table 3. When the values of d0 are equal to 0 and 0.5 μm, the HE11 mode has only one ZDW in the range of 1–10 μm. The first ZDW (ZDW1) of the HE11 mode is larger than those for high-order modes. As shown in Fig. 6, when d0 is 0.5, 1.0, and 1.5 μm, the first ZDW of the fundamental mode are blueshifted gradually. The first ZDW of HE21 mode is redshifted when d0 becomes equal to 0, 0.5, and 1.0 μm, while 4
Optical Fiber Technology 55 (2020) 102152
W. Gao, et al.
Fig. 5. (a) Effective refractive index (neff) of the four vector beams at 3 μm. (b) Corresponding neff of the four vector beams at 5 μm. Table 2 δneff among the four vector beams at the wavelength of 5 μm in the As2Se3 PCF. δneffd0 d0 d0 d0 d0
= = = =
0 μm 0.5 μm 1.0 μm 1.5 μm
δneff (HE11-TE01)
δneff (TE01-HE21)
δneff (HE21-TM01)
0.15487 0.11283 0.06516 0.03098
0.02065 0.02146 0.02780 0.03960
0.00495 0.01801 0.04218 0.06867
Fig. 7. Loss curves of the As2Se3 PCF with different cladding number for the four kinds of vector beams: (a) HE11, (b) TE01, (c) HE21, and (d) TM01.
Table 3 ZDWs of different optical modes in As2Se3 PCF. d0 (μm)
d0 = 0 μm d0 = 0.5 μm d0 = 1.0 μm d0 = 1.5 μm
ZDW (μm)
ZDW1 ZDW2 ZDW1 ZDW2 ZDW1 ZDW2 ZDW1 ZDW2
(μm) (μm) (μm) (μm) (μm) (μm) (μm) (μm)
MODES HE11
TE01
HE21
TM01
3.304 – 3.821 – 3.324 4.580 2.950 4.361
2.636 8.251 2.644 8.248 2.698 8.169 2.742 7.765
2.546 7.730 2.558 7.725 2.590 7.612 2.563 7.092
2.528 6.798 2.499 6.592 2.430 6.091 2.326 5.407
Fig. 8. Loss curves of the As2Se3 PCF with the diameters of the central area and SCHH of 15.09 and 5.25 μm, respectively, for the four kinds of vector beams.
TM01) dispersion curves become steep at the longer wavelength, the TM01 mode changes most obviously, and the depression appears. Increasing the ring number or core diameter of the fiber could improve the transmission property for the waveband of 5–10 μm. With the increase of core diameter to 15.09 μm, the fiber losses could be lower than 10 dB/m for the four kinds of vector beams. The MIR fibers with more excellent transmission characteristics for vector beams could be realized with the development of fiber fabrication techniques. Our simulated results would be helpful and instructive for experimental research in the future.
CRediT authorship contribution statement Weiqing Gao: Conceptualization, Funding acquisition, Project administration. Xiu Zhang: Data curation, Writing - original draft. Wenhui Jiang: Data curation. Zhengxiong Zhang: Data curation. Panyun Gao: Resources. Liang Chen: Resources. Peng Wang: Resources. Wei Zhang: Investigation. Rui Wang: Investigation. Meisong Liao: Conceptualization. Takenobu Suzuki: Writing - review & editing. Yasutake Ohishi: Writing - review & editing. Yong Zhou: Conceptualization, Software, Funding acquisition.
Fig. 6. ZDW1 and ZDW2 curve of the As2Se3 PCF with the following modes: (a) HE11, (b) TE01, (c) HE21, and (d) TM01.
refractive index separations among the high-order modes increased at increased SCHH diameters. At the wavelength of 5 μm, the δneff values of HE11-TE01, TE01-HE21, and HE21-TM01, were 0.03098, 0.03960, and 0.06867, respectively, when d0 was 1.5 μm. The dispersion curve of HE11 mode is flatter than that of the higher-order modes (TE01, HE21, and TM01). With the increase of d0, the HE11 mode dispersion curves have no obvious change, while the high-order modes (TE01, HE21, and
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.
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Acknowledgments
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The work was supported by National Key R&D Program of China (No. 2018YFB0504500). It was also supported by the National Natural Science Foundation of China (NSFC) (Nos. 61875052 and 11374084), the Anhui Provincial Natural Science Foundation (No. 1908085QF273) and the Fundamental Research Funds for the Central Universities (Nos. PA2019GDQT0007 and JZ2019HGTA0037). References [1] N.M. Israelsen, C.R. Petersen, A. Barh, D. Jain, G. Hannesschlager, P. TidemandLichtenberg, C. Pedersen, A. Podoleanu, O. Bang, Real-time high-resolution midinfrared optical coherence tomography, Light-Sci. Appl. 8 (2019) 11. [2] C. Gasser, J. Kilgus, M. Harasek, B. Lendl, M. Brandstetter, Enhanced mid-infrared multi-bounce ATR spectroscopy for online detection of hydrogen peroxide using a supercontinuum laser, Opt. Express 26 (9) (2018) 12169–12179. [3] J.B. Barria, S. Roux, D. Jean-Baptiste, M. Raybaut, M. Jean-Michel, A. Godard, M. Lefebvre, Microsecond fiber laser pumped, single-frequency optical parametric oscillator for trace gas detection, Opt. Lett. 38 (13) (2013) 2165–2167. [4] W. Gao, Z. Duan, K. Asano, T. Cheng, D. Deng, M. Matsumoto, T. Misumi, T. Suzuki, Y. Ohishi, Mid-infrared supercontinuum generation in a four-hole As2S5 chalcogenide microstructured optical fiber, Appl. Phys. B-Lasers. O 116 (2014) 847–853. [5] W. Gao, T. Cheng, X. Xue, L. Liu, L. Zhang, M. Liao, T. Suzuki, Y. Ohishi, Stimulated Raman scattering in AsSe2-As2S5 chalcogenide microstructured optical fiber with all-solid core, Opt. Express 24 (4) (2016) 3278–3293. [6] A. Herzog, A. Shamir, A.A. Ishaaya, Wavelength conversion of nanosecond pulses to the mid-IR in photonic crystal fibers, Opt. Lett. 37 (1) (2012) 82–84. [7] C.R. Petersen, P.M. Moselund, L. Huot, L. Hooper, O. Bang, Towards a table-top synchrotron based on supercontinuum generation, Infrared Phys. Tech. 91 (2018) 182–186. [8] C.R. Petersen, N. Prtljaga, M. Farries, J. Ward, B. Napier, G.R. Lloyd, J. Nallala, N. Stone, O. Bang, Mid-infrared multispectral tissue imaging using a chalcogenide fiber supercontinuum source, Opt. Lett. 43 (5) (2018) 999–1002. [9] S. Dupont, C. Petersen, J. Thøgersen, C. Agger, O. Bang, S.R. Keiding, IR Microscopy utilizing intense supercontinuum light source, Opt. Express 20 (5) (2012) 4887–4892. [10] D.G. Hall, Vector-beam solutions of Maxwell’s wave equation, Opt. Lett. 21 (1) (1996) 9–11. [11] Q. Zhan, Cylindrical vector beams: from mathematical concepts to applications, Adv. Opt. Photonics 1 (2009) 1–57. [12] Y. Zhao, J. Wang, High-base vector beam encoding/decoding for visible-light communications, Opt. Lett. 40 (21) (2015) 4843–4846. [13] Max-Planck-Research-Group for Optics, Information and Photonics, Sharper Focus for a Radially Polarized Light Beam, Phys. Rev. Lett. 91(23) (2003) 233901. [14] G.M. Lerman, A. Yanai, U. Levy, Demonstration of Nanofocusing by the use of Plasmonic Lens Illuminated with Radially Polarized Light, Nano Lett. 9 (5) (2009) 2139–2143. [15] L. Novotny, M.R. Beversluis, K.S. Youngworth, T.G. Brown, Longitudinal field modes probed by single molecules, Phys. Rev. Lett. 86 (23) (2001) 5251. [16] Yuichi Kozawa, Shunichi Sato, Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams, Opt. Express 18 (10) (2010) 10828–10833. [17] A.V. Nesterov, V.G. Niziev, Laser Beams with Axially Symmetric Polarization, J. Phys. D Appl. Phys. 33 (2000) 1817–1822. [18] S. Ramachandran, P. Kristensen, Optical vortices in fiber, P. Soc. Photo-Opt. Ins. 2 (2013) 455–474. [19] X. Wang, J. Ding, W. Ni, C. Guo, H. Wang, Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement, Opt. Lett. 32 (24) (2007) 3549–3551. [20] Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, Generation and propagation of a partially coherent vector beam with special correlation functions, Phys. Rev. A 89 (2014) 013801.
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Optical Fiber Technology journal homepage: www.elsevier.com/locate/yofte
Characteristics of vector beams in mid-infrared waveband in an As2Se3 photonic crystal fiber with small hollow core
T
Weiqing Gaoa, Xiu Zhanga, Wenhui Jianga, Zhengxiong Zhanga, Panyun Gaoa, Liang Chena, Peng Wanga, Wei Zhanga, Rui Wanga, Meisong Liaob, Takenobu Suzukic, Yasutake Ohishic, ⁎ Yong Zhoua, a
Department of Optical Engineering, School of Electronic Science & Applied Physics, Hefei University of Technology, Feicui Road 420, Hefei 230601, China R&D Center of High Power Laser Components, Shanghai Institute of Optics and Fine Mechanics, Shanghai 201800, China c Research Center for Advanced Photon Technology, Toyota Technological Institute, 2–12–1 Hisakata, Tempaku, Nagoya 468–8511, Japan b
A R T I C LE I N FO
A B S T R A C T
Keywords: Vector beam Mid-infrared waveband Chalcogenide fiber Photonic crystal fiber
Characteristics of vector beams in mid-infrared waveband are analyzed numerically in an As2Se3 photonic crystal fiber (PCF) with small central hollow core (SCHH), including the mode fields, confinement loss, effective refractive index and chromatic dispersion. In the PCFs with the SCHH diameter d0 of 0–1.5 μm, the confinement losses of the four vector beams (HE11, TM01, TE01, and HE21) are smaller than 1.0 dB/m for the wavelengths up to ~5 μm. For the SCHH diameter d0 of 0.5–1.5 μm, the field enhancement of the HE11 mode occurs in the SCHHs in the central area for the wavelengths ≥5 μm. This happens because the evanescent light penetrates into the SCHH. With d0 increasing, the effective refractive index separations (δneff) between the HE11 mode and highorder modes (TM01, TE01, and HE21) decreases, while the δneff among the high-order modes increases. At the wavelength of 5 μm, the δneff of HE11-TE01, TE01-HE21, and HE21-TM01 is 0.03098, 0.03960, and 0.06867, respectively, when d0 is 1.5 μm. The second ZDWs of TE01, HE21 and TM01 mode are blueshifted with the increase of d0, but they are longer than 5 μm. The negative dispersion regions of the high-order modes (TE01, HE21, and TM01) are much wider than the HE11 modes. With the increase of d0, the HE11 mode dispersion curves have no obvious change, while the high-order modes (TE01, HE21, and TM01) dispersion curves become steep at the longer wavelength, the TM01 mode changes most obviously, and the depression appears. The transmission property for the waveband of 5–10 μm could be improved by increasing the ring number or core diameter of the fiber. The fiber losses at 10 μm are 4.81, 7.65, 5.95 and 2.47 dB/m for the four kinds of vector beams, respectively, when the diameters of the central area and SCHH are increased to 15.09 and 5.25 μm. Our simulated results will be useful for the optical applications in mid-infrared waveband, such as the generation of cylindrical vector mode and orbital angular momentum (OAM) and the particle manipulation by optical fields.
1. Introduction Mid-infrared (MIR) fiber sources, especially those within the 2–10 μm waveband, have potential applications in various fields, such as optical coherence tomography [1], spectroscopy [2], and trace gas detection [3]. Recently, some optical fiber sources have been generated in the 2–10 μm waveband by supercontinuum generation (SCG) [4], stimulated Raman scattering [5], four-wave mixing [6], and so on. The brightness of supercontinuum sources was orders of magnitude larger than even square-mile large synchrotrons [7], which have been found important applications within multi-spectral bio-imaging [8] and microscopy [9].
⁎
Most of the optical sources in MIR wavebands only considered the intensity distribution. The electric field vector of light is important for applications in the MIR waveband. Vector beams constitute a solution of the Maxwell’s wave equations [10]. In addition, vector beams with spatially variant state of polarizations are expected to lead to new effects and phenomena in optical systems [11]. They have been encountered in numerous applications on optical communications [12], super-resolved focal spots [13], plasmonic nanofocusing [14], singlemolecule spectroscopy [15], optical trapping [16] and metal machining [17]. Vector beams can also be used to form the light carrying orbital angular momentum (OAM) to extend their applied fields [18]. The most typical method to generate vector beams used spatial light
Corresponding author. E-mail address: [email protected] (Y. Zhou).
https://doi.org/10.1016/j.yofte.2020.102152 Received 2 December 2019; Received in revised form 15 January 2020; Accepted 15 January 2020 1068-5200/ © 2020 Published by Elsevier Inc.
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vectorial mode solving technique. Fig. 2 shows the simulated evolution of the four vector beam fields (HE11, TE01, HE21, and TM01) as a function of the wavelength in the As2Se3 PCFs when the SCHH diameter changes from 0 to 1.5 μm. The parameters Λ and d are 1.5 and 1.125 μm, respectively. We defined the region surrounded by the air-hole cladding as the core area. The core diameter is 4.31 μm, which is defined as the diameter of the circle inscribed in the hexagonally arranged area. The confinement losses of the four vector beams are shown in Fig. 3(a)-3(d), and correspond to d0 values of 0, 0.5, 1.0, and 1.5 μm, respectively. The wavelength (λ1-dB) corresponding to the confinement loss of ~1 dB/m is used to express the transmission ability in MIR waveband of the vector modes. The confinement loss is part of the total fiber loss. We did not consider material loss in this simulation. According to Fig. 2, the four vector beam fields (HE11, TE01, HE21, and TM01) can be restrained effectively in the core area from 1 to 5 μm. From ~5 μm, the energy of the four vector beam will begin to appear in the air-hole cladding as the wavelength increases, and become very obvious at 10 μm. Table 1 shows the λ1-dB and λ10-dB of the four vector beams in different d0. When the value of confinement loss is equal to ~1 dB/m, HE11 mode will transmit ~6 μm, TE01, HE21, and TM01 modes will transmit ~5 μm. When the value of confinement loss is equal to ~10 dB/m, HE11 mode will transmit ~6–8 μm, TE01, HE21, and TM01 modes will transmit ~5–6 μm. It means that the four vector beams can be transmitted stably in mid-infrared waveband. The phenomenon of field enhancement is affected by the SCHH diameter [42]. When d0 is zero, there is no field enhancement at the central area of the PCF. When d0 is 0.5 μm, the light concentrates in the SCHH, and the field in this area is enhanced, as shown in Fig. 2(b). The field enhancement is also obvious when d0 increases to 1.0 and 1.5 μm, as shown in Fig. 2(c) and (d). Field enhancement occurs at longer wavelengths because the size of the SCHH (0.5–1.5 μm) is much smaller than the transmitted wavelength in the range of 5–10 μm, and is well below the diffraction limit. The evanescent light propagates into the SCHH because the refractive index of the SCHH is smaller than the As2Se3 glass around it. This effect enhances as the wavelength increases and causes field enhancements in the SCHH. However, for shorter wavelengths, the size of the SCHH is not much lower than the diffraction limit so that the field enhancement does not occur.
modulators [19]. Chen et al. researched the generation and propagation of a partially coherent vector beam with special correlation functions theoretically and experimentally [20]. Yue et al. proposed and experimentally demonstrated an approach to generate vector vortex beams with a single metasurface by locally tailoring phase and transverse polarization distributions [21]. Recently, several other methods have been reported to generate vector beams. Matsuba et al. generated vector beams by superposing two optical vortex beams from helical undulators [22]. Kozawa et al. obtained the vector beam output from vertical cavity, surface-emitting lasers, using a birefringent lens inside the cavity to select the single transverse mode of the vector beam [23]. Li et al. generated a perfect optical vortex and elliptic perfect vector beam by modulating the dynamic and geometric phases [24]. Vector beams could also be generated in optical fibers, such as in the cases where only few-mode fibers (FMFs) are used [25] by adjusting the misalignment in the angle and transverse dimensions between fibers [26], by applying acoustic flexural waves on the fibers [27], or by using mode selective couplers [28] and long-period fiber gratings [29]. The pulsed vector beams could also be generated in fibers [30]. To extend the application to MIR wavebands, the generation of vector beams in soft-glass fibers is attractive. Compared to silica fibers, soft-glass fibers have the advantages of high nonlinearity and wide transmission windows within the MIR waveband [31]. In 2006, Littler et al. first reported the acousto-optic resonances in non-silica fibers, which could be used for the generation of vector beams in soft-glass fibers [32]. In 2012, Yue et al. demonstrated the SCG using different vortex modes in As2S3 chalcgenide fibers [33]. In 2016, Ji et al. presented numerically the propagation of radially and azimuthally polarized cylindrical vector modes in tellurite fibers [34,35]. To generate the light source covering the entire spectral range of 2–10 μm with high efficiency, it is important to design the fiber so that the optical modes inside the core area can be confined for wide spectral ranges. The fibers transmitting the light in the MIR waveband are developed with different lateral microstructures, including photonic crystal [36,37], hollow-core [38], negative curvature [39], and twisted fibers [40]. It is significant to develop vector beams in MIR wavebands to extend the applications and generate new effects in the region. The first key problem is to develop the fibers with the structures that can transmit vector beams covering MIR wavebands using single fiber. In this work, the characteristics in the MIR waveband of four vector beams (HE11, TM01, TE01, and HE21) in an As2Se3 PCF with SCHH, are analyzed versus the diameter of the SCHH, including the effective refractive index, chromatic dispersion, confinement loss, and vector beam field. As the SCHH diameter and wavelength increase, the energy distribution changes greatly. With the SCHH diameter increasing, the effective refractive index separations (δneff) between the HE11 mode and high-order modes (TM01, TE01, and HE21) decreases, while the δneff among the high-order modes increases. At the wavelength of 5 μm, the δneff between each mode can be greater than 0.03. This can reduce the probability of mode coupling efficiently and enable good differentiation among different vector modes. The confinement loss increases with the SCHH diameter increasing, but it is also can transmit in the spectral range up to approximately 5 μm with the loss lower than 1 dB/m.
3. Effective refractive index and chromatic dispersion The effective refractive index and chromatic dispersion profiles of the As2Se3 PCF are shown as in Fig. 4. When d0 is zero, the effective refractive indices of the four vector beams decrease as the wavelength increases, as shown in Fig. 4(a). The effective refractive index separation (δneff) between the HE11 and high-order modes (TE01, HE21, and TM01) is obvious, but the δneff among the high-order modes (TE01, HE21, and TM01) is smaller. The δneff between the HE21 and TM01 mode is the smallest. For the same wavelength, the HE11 mode has the highest effective refractive index, followed by the TE01 mode. When the d0 are 0.5, 1.0, and 1.5 μm, the effective refractive indices of the four vector beams also decrease with the wavelength increasing, as shown in Fig. 4(b)-(d). In Fig. 4(b) and (c), the δneff between the HE11 mode and the higher order modes is obvious, but the δneff among the higher order modes are still small. At the same wavelength, the HE11 mode has the highest refractive index, followed by the TE01 mode. In Fig. 4(d), the δneff between the HE11 mode and the higher order modes is obvious, and the δneff among the high-order modes is also obvious. In Ref. [43], a high index ring was added around the fiber core to realize the separated coupling from the fundamental mode to the constitutive vector modes of the LP11 group by an effective refractive index separation δneff is 1.8 × 10−4. Patrick et al. demonstrated that an air core applied into the high index ring fiber was helpful to support 12 OAM modes with δneff is 1 × 10−4 [44]. A FMF was characterized by an inverse-parabolic graded-refractive-index profile to realize the effective
2. Vector beam fields in the As2Se3 PCFs with different SCHH diameters The As2Se3 PCF has a SCHH and three periods of hexagonally arranged holes, as shown in Fig. 1(a). The parameters Λ, d, and d0, represent the pitch distance, air-hole diameter and SCHH diameter, respectively. Fig. 1(b) shows the material refractive index of the As2Se3 glass calculated according to Sellmeier’s equation [41]. The finite element method (FEM) was used to solve Maxwell’s equations in combination with the initial conditions and boundary conditions to achieve the distribution of the vector beam fields. The vector beams of HE11, TE01, HE21 and TM01 were simulated by the full2
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Fig. 1. (a) Cross-section of the As2Se3 PCF. (b) Refractive index profile of material As2Se3 glass.
Fig. 2. Simulated evolution of mode fields in As2Se3 PCF. (a) d0 = 0 μm, (b) d0 = 0.5 μm, (c) d0 = 1.0 μm, and (d) d0 = 1.5 μm.
index separations by δneff is 2.1 × 10-4 [45]. Fig. 5(a)-(b) shows the effective refractive indices of the four vector beams at the wavelengths of 3 and 5 μm for hollow core diameters d0 in the range of 0–1.5 μm. According to Fig. 5(a)–(b), the neff of the four vector beams decreases as d0 increases. The δneff between the HE11 mode and the high-order modes decreases as d0 increases, but it still arrives 0.00854 at the wavelength of 3 μm and 0.03098 at the wavelength of 5 μm when d0 is 1.5 μm. The δneff among the high-order modes increases as d0 increases. As shown in Table 2, for a d0 of 1.5 μm, the δneff of HE11-TE01, TE01-HE21, and HE21-
TM01, are equal to 0.03098, 0.03960, and 0.06867, respectively. The δneff among all the vector modes is high compared to the common fibers. The increased δneff can reduce effectively the possibility of mode coupling between the vector modes, and avoids vector mode degeneration into linearly polarized (LP) modes. It can also suppress mode disorder. The As2Se3 PCF then becomes beneficial for the transmission of vector beams for wavelengths up to 5 μm. The chromatic dispersion profiles are shown by the dashed lines in Fig. 4(a)-(d). The first zero-dispersion wavelengths (ZDWs) and the 3
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it decreases when d0 is 1.5 μm. When d0 is 0, 0.5 μm, 1.0 μm, and 1.5 μm, the first ZDW of the TE01 modes are redshifted gradually, while the first ZDW of TM01 modes are blueshifted. The second ZDWs of TE01, HE21 and TM01 mode are blueshifted with the increase of d0, but they are longer than 5 μm. The negative dispersion regions of the high-order modes (TE01, HE21, and TM01) are much wider than the HE11 modes. Obviously, the dispersion curve of HE11 mode is flatter than that of the higher-order modes (TE01, HE21, and TM01). With the increase of d0, the HE11 mode dispersion curves have no obvious change, while the highorder modes (TE01, HE21, and TM01) dispersion curves become steep at the longer wavelength, the TM01 mode changes most obviously, and the depression appears. 4. Improvement of the transmission property To transmit the light at 10 μm waveband with low loss, we have made some attempts to improve the transmission property of the proposed fiber. The transmission loss is affected by the limitation extent of light into the central area, which is dominated chiefly by the ring number of cladding and core diameter. Firstly, the ring number was increased to four and five, respectively, and the other fiber parameters were kept the same as in Fig. 3 with the d0 of 1.5 μm. The simulated loss curves for the four kinds of vector beams are shown as in Fig. 7. For the HE11, TE01, HE21 and TM01 mode, λ1-dB are 6.6, 6.2, 6.1 and 5.6 μm in the PCF with four rings, respectively, while λ1-dB are 7.7, 7.0, 6.7 and 6.3 μm in the PCF with five rings, respectively. We can see that the loss is decreased obviously for the wavelengths longer than 5 μm compared to three rings in Fig. 3. But the losses at 10 μm are still higher than 100 dB/m. Secondly, the Λ and d were kept the same in the PCF with three rings as in Fig. 3, but the diameters of the central area and SCHH were increased to 15.09 and 5.25 μm, respectively, with the same ratio of 3.5. The simulated loss curves are shown as in Fig. 8. For the HE11, TE01, HE21 and TM01 mode, λ1-dB are 8.9, 8.6, 8.8, and 9.4 μm, respectively, which are decreased greatly compared to Fig. 7. In Fig. 8, the fiber losses at 10 μm are 4.81, 7.65, 5.95 and 2.47 dB/m for the four kinds of vector beams, respectively, which are much lower than those in Fig. 3. Then the As2Se3 PCF with the transmission property as in Fig. 8 can transmit the MIR wavelengths close to 10 μm. In brief, increasing ring number or core diameter can improve the transmission property for the waveband of 5–10 μm. The increase of ring number will increase the difficulty of fiber fabrication, especially for the soft-glass PCFs. While increasing core diameter will not bring additional requirements for fiber fabrication. With the development of fiber fabrication techniques in the future, the two ways could be combined together to improve the transmission at MIR waveband of 10 μm or longer. Then the MIR fibers with more excellent transmission characteristics for vector beams could be realized.
Fig. 3. Loss curves of the As2Se3 PCF with the following hollow core diameters: (a) d0 = 0 μm, (b) d0 = 0.5 μm, (c) d0 = 1.0 μm, and (d) d0 = 1.5 μm. Unit of vertical axis of the partial enlarged view is dB/m.
Table 1 Values of λ1-dB and λ10-dB, of the four vector beams in different d0. d0 (μm)
0
0.5
modes
HE11
TE01
HE21
TM01
HE11
TE01
HE21
TM01
λ1-dB (μm) λ10-dB (μm)
6.6 7.6
5.4 6.1
5.4 6.0
5.4 6.1
6.3 7.3
5.4 6.1
5.4 6.0
5.4 5.9
d0 (μm)
1.0
modes
HE11
TE01
HE21
TM01
HE11
TE01
HE21
TM01
λ1-dB (μm) λ10-dB (μm)
5.8 6.7
5.3 6.0
5.3 5.9
5.1 5.7
5.3 6.1
5.2 5.8
5.0 5.6
4.8 5.3
1.5
5. Conclusions The characteristics of the optical field distribution, confinement loss, effective refractive index, and chromatic dispersion, were analyzed numerically for four types of vector beams for the HE11, TE01, HE21, and TM01 modes in the As2Se3 PCFs with different SCHH diameters. For SCHH diameters in 0, 0.5, 1.0, and 1.5 μm, the HE11 mode could transmit within the MIR waveband at wavelengths up to approximately 6.6, 6.3, 5.8, and 5.3 μm, respectively. For the high-order modes, the maximum wavelength that can be transmitted is approximately 5 μm. The field enhancement of the HE11 mode in the central area occurred for wavelengths ≥5 μm for SCHH diameters in 0.5, 1.0, and 1.5 μm. This was caused because the evanescent light propagated into the SCHH. The profiles of the refractive index and chromatic dispersion were analyzed for the four types of vector modes. The effective refractive index separations between the HE11 mode and the high-order modes decreased with increased SCHH diameters, while the effective
Fig. 4. Effective refractive index (solid lines) and chromatic dispersion (dotted lines) curves of the As2Se3 PCF with the following hollow core diameters: (a) d0 = 0 μm, (b) d0 = 0.5 μm, (c) d0 = 1.0 μm, and (d) d0 = 1.5 μm.
second ZDWs (ZDW2) of the modes HE11, TE01, HE21, and TM01, are listed as in Table 3. When the values of d0 are equal to 0 and 0.5 μm, the HE11 mode has only one ZDW in the range of 1–10 μm. The first ZDW (ZDW1) of the HE11 mode is larger than those for high-order modes. As shown in Fig. 6, when d0 is 0.5, 1.0, and 1.5 μm, the first ZDW of the fundamental mode are blueshifted gradually. The first ZDW of HE21 mode is redshifted when d0 becomes equal to 0, 0.5, and 1.0 μm, while 4
Optical Fiber Technology 55 (2020) 102152
W. Gao, et al.
Fig. 5. (a) Effective refractive index (neff) of the four vector beams at 3 μm. (b) Corresponding neff of the four vector beams at 5 μm. Table 2 δneff among the four vector beams at the wavelength of 5 μm in the As2Se3 PCF. δneffd0 d0 d0 d0 d0
= = = =
0 μm 0.5 μm 1.0 μm 1.5 μm
δneff (HE11-TE01)
δneff (TE01-HE21)
δneff (HE21-TM01)
0.15487 0.11283 0.06516 0.03098
0.02065 0.02146 0.02780 0.03960
0.00495 0.01801 0.04218 0.06867
Fig. 7. Loss curves of the As2Se3 PCF with different cladding number for the four kinds of vector beams: (a) HE11, (b) TE01, (c) HE21, and (d) TM01.
Table 3 ZDWs of different optical modes in As2Se3 PCF. d0 (μm)
d0 = 0 μm d0 = 0.5 μm d0 = 1.0 μm d0 = 1.5 μm
ZDW (μm)
ZDW1 ZDW2 ZDW1 ZDW2 ZDW1 ZDW2 ZDW1 ZDW2
(μm) (μm) (μm) (μm) (μm) (μm) (μm) (μm)
MODES HE11
TE01
HE21
TM01
3.304 – 3.821 – 3.324 4.580 2.950 4.361
2.636 8.251 2.644 8.248 2.698 8.169 2.742 7.765
2.546 7.730 2.558 7.725 2.590 7.612 2.563 7.092
2.528 6.798 2.499 6.592 2.430 6.091 2.326 5.407
Fig. 8. Loss curves of the As2Se3 PCF with the diameters of the central area and SCHH of 15.09 and 5.25 μm, respectively, for the four kinds of vector beams.
TM01) dispersion curves become steep at the longer wavelength, the TM01 mode changes most obviously, and the depression appears. Increasing the ring number or core diameter of the fiber could improve the transmission property for the waveband of 5–10 μm. With the increase of core diameter to 15.09 μm, the fiber losses could be lower than 10 dB/m for the four kinds of vector beams. The MIR fibers with more excellent transmission characteristics for vector beams could be realized with the development of fiber fabrication techniques. Our simulated results would be helpful and instructive for experimental research in the future.
CRediT authorship contribution statement Weiqing Gao: Conceptualization, Funding acquisition, Project administration. Xiu Zhang: Data curation, Writing - original draft. Wenhui Jiang: Data curation. Zhengxiong Zhang: Data curation. Panyun Gao: Resources. Liang Chen: Resources. Peng Wang: Resources. Wei Zhang: Investigation. Rui Wang: Investigation. Meisong Liao: Conceptualization. Takenobu Suzuki: Writing - review & editing. Yasutake Ohishi: Writing - review & editing. Yong Zhou: Conceptualization, Software, Funding acquisition.
Fig. 6. ZDW1 and ZDW2 curve of the As2Se3 PCF with the following modes: (a) HE11, (b) TE01, (c) HE21, and (d) TM01.
refractive index separations among the high-order modes increased at increased SCHH diameters. At the wavelength of 5 μm, the δneff values of HE11-TE01, TE01-HE21, and HE21-TM01, were 0.03098, 0.03960, and 0.06867, respectively, when d0 was 1.5 μm. The dispersion curve of HE11 mode is flatter than that of the higher-order modes (TE01, HE21, and TM01). With the increase of d0, the HE11 mode dispersion curves have no obvious change, while the high-order modes (TE01, HE21, and
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.
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Acknowledgments
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