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A design strategy of the circular photonic crystal fiber supporting good quality orbital angular momentum mode transmission
Optics Communications 397 (2017) 59–66
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A design strategy of the circular photonic crystal fiber supporting good quality orbital angular momentum mode transmission
MARK
⁎
Hu Zhanga,b, , Xiaoguang Zhanga, Hui Lia, Yifan Denga, Xia Zhanga, Lixia Xia, Xianfeng Tanga, Wenbo Zhanga,c a State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, 100876 Beijing, PR China b School of Ethnic Minority Education, Beijing University of Posts and Telecommunications, 100876 Beijing, PR China c School of Sciences, Beijing University of Posts and Telecommunications, 100876 Beijing, PR China
A R T I C L E I N F O
A BS T RAC T
Keywords: Fiber design and fabrication Orbital angular momentum Photonic crystal fibers Multiplexing
Based on 5 requirements which are essential for stable OAM mode transmission, we propose an OAM fiber family based on a structure of circular photonic crystal fiber (C-PCF). The proposed C-PCF in the family is made of pure silica, with a big round air hole at the center, several rings of air-hole array as the cladding, and a ring shaped silica area in between as the core where the OAM modes propagate. We also provide a design strategy with which the optimized C-PCF can be obtained with optimum number of high quality OAM modes (up to 42 OAM modes), large effective index separation for corresponding vector modes over a wide bandwidth, relative small and flat dispersion, and low nonlinear coefficient compared with a conventional single mode fiber. The designed fiber can be used in MDM communications and other OAM applications in fibers.
1. Introduction A light beam with the spiral phase front exp(ilφ) carries an orbital angular momentum (OAM), where, l is a integer called as topological charge, φ is azimuthal angle [1,2]. OAM beam has recently spurred tremendous interests because of its wide potential applications, in the field of optical manipulation [3], quantum information [4], imaging [5] and communications [6,7]. The orthogonality between different OAM states provides another dimension (mode division multiplexing, MDM) to be multiplexed to solve the currently urgent requirements for data capacity [7–10]. Terabit data transmission in a fiber carrying OAM modes has been demonstrated [7]. Since OAM modes in fibers can be properly constituted by the even and odd modes of a vector mode with the identical propagation constants, they will not undergo intermodal work-off effect. But the mode degeneracy between corresponding HE and EH eigenmodes will lead to strong modal coupling (couple into LP modes), subsequently break the stability of OAM modes transmission, and hence cause the serious crosstalk. To handle these problems, a specially designed optical fiber is required to possess high effective refractive index separation ( > 10−4) between different vector modes in the fiber [11,12] to avoid modal coupling and to be free from complex multi-input multi-output (MIMO) treatment at the receiver. Recently, a
ring shaped or annular shaped fiber was designed with a high index ring area as the ring core to hold OAM modes, and its central circle area and cladding are with lower indices [13–22]. The high index contrast between the ring core and cladding is achieved by up-doping in ring core area. So far, ring fibers can support 9 orders of OAM (34 OAM modes) or 36 information bearing states [18]. However, heavily updoping may result in higher fiber loss [21]. Compared with the conventional fibers, photonic crystal fibers (PCFs) [23,24] have more flexible structure design to provide unique fiber properties such as endlessly single mode [25], controllable nonlinearity and confinement loss [25,26], tailorable chromatic dispersion [26,27]. The structure of PCF used to realize the OAM modes transmission has potential application prospects in fiber communications. In Ref. [28] the ring photonic crystal (ring-PCF) with hexagonal air hole array as cladding was proposed which can only support less number of OAM modes (4 OAM modes) with bad mode quality. We find that the hexagonal structure of air holes is not a good design for its not good OAM mode quality and higher confinement loss. The reason is that hexagonal structure with noncircular symmetry is not consistent with ring shaped intensity of OAM modes. In order to gain more numbers of OAM mode and better mode field quality, we presented a new structure of circular shaped PCF (C-PCF)
⁎ Corresponding author at: State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, 100876 Beijing, PR China. E-mail address: [email protected] (H. Zhang).
http://dx.doi.org/10.1016/j.optcom.2017.03.075 Received 6 February 2017; Received in revised form 20 March 2017; Accepted 28 March 2017 0030-4018/ © 2017 Elsevier B.V. All rights reserved.
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Fig. 1. Cross-section and main parameters of the designed C-PCFs for (a) the fiber base, (b) C-PCF 1#, (c) C-PCF 2#, (d) C-PCF 3#.
conventional single mode fiber. Furthermore, it is proved that the ring thickness of the presented C-PCF has a relative large fabrication tolerance about 0.6 µm both to guarantee the good mode quality and suppress the high order radial modes. So this kind of fiber is suitable to be used in MDM systems for long distance.
[29] with circular symmetry which has a large air-hole in the center, and a circular shaped air-hole array rather than the hexagonal air holes as cladding. Compared with the ordinary ring fibers, the C-PCF has more advantages such as getting high index contrast without updoping, larger fiber bandwidth, good mode quality, lower confinement loss, and larger fabrication tolerance [30]. Recently, Ref. [31] presented a microstructure ring fiber (MRF) with the equally number of air holes in three different cladding rings to obtain the circular symmetry. The hole size of the proposed fiber is different in every ring. And the proposed MRF has narrow bandwidth and higher confinement loss. Ref. [32] adopted the similar fiber structure with that of Ref. [31], but the material of the fiber is the As2S3 glass. This kind of fiber has larger loss and high nonlinearity which is not suitable for the OAM mode transmission, while suitable for supercontinuum generation. In this paper, we summarize 5 essential requirements to ensure the OAM modes stable transmission in optical fiber, and based on which an OAM fiber family with the structure of ring circular PCF are proposed to realize MDM in optical communication systems. A detail fiber design strategy is presented under which the C-PCFs are designed to possess good features. Comparing the properties of three C-PCFs, we can find that C-PCF 3# has the maximum number of high quality OAM modes (11 OAM orders or 42 OAM modes), the relative wide bandwidth (460 nm from 1.25 µm to 1.71 µm) with large effective index separation of different vector modes, relative small and flat dispersion, low confinement loss and low nonlinear coefficient compared with a
2. Design strategy of the C-PCF supporting OAM modes OAM modes in fibers are defined as OAMl,m, where m is the number of concentric rings of the intensity profile of the mode in radial direction, l(l∈Z) is the topological charge. OAM modes in fibers can be regarded as the superposition of the vector eigenmodes by following relations: odd OAM ±± l, m = HEleven +1, m ± j HEl +1, m
(1)
odd OAM∓± l, m = EHleven −1, m ± j EHl −1, m
(2)
where the sign in superscript ‘ ± ’ denotes the right or left circular polarization, and the sign of ± l denotes the right or left wave front rotation direction, odd and even modes of HE or EH denotes a π/2 azimuthal rotation, j presents a π/2 phase shift. HE mode has the same directions of spin angular momentum (SAM) and OAM, while EH mode has the opposite directions of SAM and OAM. The OAM 0,±m mode composed by HE1,m with two circular polarizations can not be used as OAM mode for it carries no OAM. Moreover, the OAM∓± 1, m mode composed by azimuthally polarized TE0,m and radially polarized TM0,m 60
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where, λ is the wavelength in vacuum, nco is the index of the fiber core, ncl is the effective index of the cladding, ncl = 1.444(1 − f ) + f , f is the air filling fraction of the cladding, which is defined as
can also not be used as OAM mode because they have different propagation constants, and are unstable in the fiber. The OAM modes with the same l and m compose an OAM mode family, and each mode family supports 4 OAM modes except for OAM1,1 mode family (composed by even and odd of HE2,1). To design a good quality OAM mode supporting fiber, which can be used in MDM, following requirements should be considered: 1) This kind of fiber can hold OAM modes as many as possible and meanwhile avoid higher radial modes to be excited to simplify the multiplexing and the demultiplexing of OAM modes. 2) Large effective index separation between the corresponding HE and EH modes that belong to the same OAM mode family, should be achieved, in order to weaken the modal coupling. 3) All the supported OAM modes should possess good mode quality and low confinement loss. 4) Other features which the designed fiber would possess are flat and low dispersion over a large spectrum bandwidth, large mode area and low nonlinear coefficient. 5) All the requirements mentioned above should be valid covering a wide bandwidth, especially C+L band. For the reason that OAM modes have the circular symmetry of electric field, the desired fiber should possess circular shape which has a high refractive index ring area to hold OAM modes. Therefore we propose a design of the circular shaped photonic crystal fiber whose base material is made of pure silica (refractive index is 1.444 at the wavelength 1.55 µm) shown in Fig. 1. The circular photonic crystal fiber has a central air hole in the center and several ring air-hole arrays around it as the fiber cladding, in between a ring shaped area of base pure silica remains where OAM modes can be transmitted in this region. Roughly speaking, the region occupied by pure silica contributes to refractive index with the value 1.444, and the region occupied by air holes provides index with the value of 1, therefore the average refractive index of cladding area is the mean value of two regions weighted with the areas they occupy. Hence the larger air filling fraction is, the higher the index contrast between the OAM mode transmission area and cladding area is. But we have to consider the strong structure of the fiber simultaneously, and the diameters of the air-holes in the cladding should be a trade-off value. We find that larger number of cladding air-hole arrays makes a stronger mode constraint, but 4 arrays are enough to confine OAM modes in the fiber. Taking the shape of photonic crystal fiber shown Fig. 1(a) as the fiber base, we design a family of good quality circular photonic crystal fibers which can support OAM mode transmission. This fiber family consists of C-PCF 1#, C-PCF 2#, and C-PCF 3#, in which C-PCF n# fiber is constructed by removing off the first nth airhole ring of the base fiber (for example, C-PCF 1# corresponds to removing off the first one air-hole ring), and putting a central air hole at the center, as shown in Fig. 1(b), (c), and (d). The spatial positions of the air holes on the x–y plane are [33]
⎛ 2nπ ⎞ ⎛ 2nπ ⎞ x = ΛN cos ⎜ ⎟ , n = 1~6N ⎟ , y = ΛN sin ⎜ ⎝ 6N ⎠ ⎝ 6N ⎠
⎛ d ⎞2 6(4N − 6) ⎜ n ⎟ ⎛ d ⎞2 ⎝2⎠ f= = 3⎜ n ⎟ 2 2 ⎡⎛ ⎡⎛ ⎝ 2Λ ⎠ 1⎞ ⎤ 7⎞ ⎤ ⎢ ⎜N + ⎟ Λ ⎥ − ⎢ ⎜N − ⎟ Λ ⎥ 2⎠ ⎦ 2⎠ ⎦ ⎣⎝ ⎣⎝
where, f means the ratio of air hole area versus the total area in cladding. Roughly speaking, Veff determines the number of vector modes supported in the fiber, f makes an influence on ncl and hence the index contrast, ρ leads to the quality of the OAM modes held in the fiber. All the numerical analyses are performed using a full-vector finite element method (FEM) (COMSOL) and perfect matched layers (PMLs) are used as boundary condition. We used this model to run the analyses upon the above the C-PCFs in the fiber family with the optimized parameters of Λ=2.0 µm, dn/Λ=0.8, d2=d3=d4=d5=d6=d7=1.6 µm, which are convenient to fiber fabrication and can also guarantee the large index separation. r equals to 3.2 µm, 5.2 µm, and 7.2 µm for CPCF1#, C-PCF2#, and C-PCF3#, respectively. With the parameters above, f =0.48, ncl=1.231, which leads a high index contrast of 0.213. r0 determines the ring thickness of region that OAM modes are transmitted in, and needs to be optimized. We will discuss r0 in detail in Section 4. 3. The number of mode and fiber bandwidth OAM mode belongs to orthogonal eigenstates, and has infinite number of orthogonal modes theoretically. But the fiber can only support the limited number of OAM modes for the restriction of the fiber structure. So far, the most reported OAM mode supporting fibers are the ring fibers. For example, Ref. [19] reported a fiber which can support 9 OAM orders or 34 OAM modes. The number of OAM modes which can be stably transmitted in fiber is decided by two factors. One is the effective normalized frequency Veff parameter of the fiber, as we know that the number of mode increases with the increasing of Veff. Another one is degree of separation of mode effective index between different vector modes of the same OAM mode family. In order to guarantee the stable transmission of the OAM modes without coupling into LP modes, the effective index contrast should be larger than 10−4, which is the second requirement in Section 2. For C-PCF, we also need two-dimension parameters to describe the cut-off frequency, namely the effective normalized frequency Veff and the ratio between inner and outer radius ρ as similar as defined in Ref. [15]. From view of the Eq. (4) of the Veff parameter, there are two ways to increase the Veff for a given wavelength. One is to increase the radius of the effective mode field, another one is to enlarge the index contrast between the fiber core and the cladding. Due to the fact that C-PCF is made by the single material of silica and the cladding is made by 4 air-hole arrays, the more effective way to increase the Veff is to enlarge the radius of ring area r. As shown in Fig. 1, C-PCF 1# with Veff=9.79 constituted by changing the first one air-hole array into the ring core can support 4 OAM orders (l=1–4) or 14 OAM modes, C-PCF 2# with Veff=15.91 constituted by changing the first two air-hole arrays into the ring core can support 7 OAM orders (l=1–7) or 26 OAM modes, C-PCF 3# with Veff=22.04 constituted by changing the first three air-hole arrays into the ring core can support 11 OAM orders (l=1–11) or 42 OAM modes. We can get more OAM modes as the radius of the ring core goes on to increase by removing more arrays, but with the result of increasing fiber fabrication difficulty. In order for the proposed fiber to be used in WDM systems, we define a bandwidth of the C-PCF that is the range of the wavelength in which all the OAM modes can stably transmitted with the index separation of larger than 10−4 between corresponding HE and EH
(3)
where Λ and N denote the concentric spacing and number of concentric rings respectively, while r0 is the inner radius of the ring (also the radius of central air hole), r is the outer radius of the ring, d2 to d7 are the diameters of the cladding air holes. As mentioned above, the background material is silica with a refractive index of 1.444 at 1.55 µm, to make high index contrast without the need of up-doping. Unlike the conventional multi-mode fiber which need only one dimension parameters, normalized frequency V = (2π / λ ) r 2 2 , to analyze the quality of the mode, we define twoncore − ncladding dimension parameters, effective normalized frequency Veff and inner and outer radius ratio ρ, as follows
Veff =
ρ=
2π 2 r nco − ncl2 λ
r0 r
(6)
(4)
(5) 61
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Fig. 2. Effective index separation of the different OAM mode families as a function of wavelength for (a) C-PCF 1#, (b) C-PCF 2#, (c) C-PCF 3#.
requirement because it can support maximum OAM modes of the proposed fiber family.
modes belong to the same OAM mode family. This bandwidth is also determined by the cut-off wavelength of the highest order OAM mode. The effective index difference between corresponding HE and EH modes of the same OAM mode order increases with wavelength for the three C-PCFs as shown in Fig. 2 due to the increased mode area, where Δneff for l=1 denotes the effective index difference of HE2,1 mode and TE0,1 mode, the reason is that the effective index difference of HE2,1 mode and TM0,1 mode is larger than that of HE2,1 mode and TE0,1 mode. In Fig. 2, all the fiber structure parameters take the same values as mentioned in Section 2, and r0=1.2 µm for C-PCF 1#, r0=3.5 µm for C-PCF 2#, r0=5.3 µm for C-PCF 3#. Fig. 2 also illustrates a very good feature of large separation of the effective indices of the modes. The index difference of all OAM orders is larger than 1×10−4. From Fig. 2, we can find that the common bandwidth of the four OAM mode families covers 560 nm (1.25–1.81 µm) for C-PCF 1#, the common bandwidth of the seven OAM mode families covers 480 nm (1.25– 1.73 µm) for C-PCF 2#, and the common bandwidth of the eleven OAM mode families covers 460 nm (1.25–1.71 µm) for C-PCF 3#. They all do cover all bands of optical fiber communications (O, E, S, C, L, U, from 1.26 µm to 1.675 µm). Apparently, the bandwidth of the proposed C-PCF is wider than that of the ordinary ring fibers [13,14,16–20] and the MRFs [32], which makes the C-PCF be good MDM fiber in WDM optical fiber communications. Up to now, we see that all three C-PCFs meet the 2nd and 5th requirements mentioned above, but C-PCF 3# also satisfies the 1st
4. Mode quality Apart from pursuing more number of OAM modes supported in the fiber, we should also pay attention to the mode quality in the fiber, because only good quality OAM modes can ensure the stable transmission in the fiber. Therefore, we introduce the light intensity overlap factor to qualitatively evaluate the mode quality
⎯→ ⎯ 2 ∬ring E dxdy Ir η= = ⎯→ ⎯ 2 Ic ∬cross −section E dxdy
(7)
where, Ir is the average mode intensity in ring area where all the OAM modes should be confined and Ic is the average mode intensity in the whole cross-section area of the C-PCF. In Section 3, we find that C-PCF 1#, C-PCF 2#, and C-PCF 3# can support 14, 26, and 42 OAM modes. Now we will discuss in order to meet the 3rd requirement what is the optimized thickness (namely optimized radius r0 of the central air hole). We take C-PCF 2# as an example. The electric field intensity distribution of HE3,1 mode in CPCF 2# with r0=3.2 µm and r0=4.0 µm at 1.55 µm are shown in Fig. 3(a) and (b) respectively. C-PCF 2# with r0=3.2 µm confines more mode energy in the ring shaped core and shows better mode quality 62
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Fig. 3. The electric field intensity distribution of HE3,1 mode in C-PCF 2# with (a) r0=3.2 µm and (b) r0=4.0 µm at 1.55 µm.
(ρVeff=12.24, ρ=0.77, Veff=15.91) can suppress the high order radial modes the reason is that for TE0,2, Vcut=16.51 > Veff=15.91, which will not be excited. So the appropriate inner radius r0 should be optimized to make trade-off between avoiding the high radial order modes and achieving the good quality of OAM modes. When r0 < 3.6 µm (ρVeff=11.02, ρ=0.69) the intensity overlap factors for all the modes supported in C-PCF 2# are above 82%, and corresponding confinement losses take relative small values. When r0 > 3.8 µm the intensity overlap factors show sharp decrease (accordingly sharp increase of confinement losses) for larger order modes. The thickness of the ring core can also influence on the effective index separation. As shown in Fig. 5, the lower value of r0 (larger thickness) is, the smaller the effective index separation (at 1.55 µm) is. For the cases that r0=3.0 µm and r0=3.2 µm, we find that there are one OAM order modes (l=5 for r0=3.0 µm and l=6 for r0 =3.2 µm) whose effective index separation is less than or nearly equal to 1×10−4. Considering the trade-off among mode quality, high order radial modes and effective index separation, we can take the value of r0 to be larger than 3.2 µm and smaller than 3.8 µm for C-PCF 2#, which provides a range of r0 about 0.6 µm. For C-PCF 2#, the optimized range of the inner radius r0 is from 3.2 µm (ρVeff=9.79, ρ=0.62) to 3.8 µm (ρVeff=11.63, ρ=0.76) to satisfy the 2nd, 3rd, and 5th requirements of the fiber design in Section 2. Also, for C-PCF 1# and C-PCF 3#, the same procedure in this section can be used to optimize the fiber structure parameters.
which means less energy leaked into the cladding than that of C-PCF 2# with r0=4.0 µm. Accordingly, the light intensity overlap factors corresponding to the case of r0=3.2 µm and r0=4.0 µm are 96% and 80% respectively, which proves that the light intensity overlap factor can be used to evaluate the mode quality. Furthermore, the confinement losses corresponding to the case of r0=3.2 µm and r0=4.0 µm are 2.17×10−10 dB/m and 1.58×10−6 dB/m, respectively. Therefore, the larger intensity overlap factor leads to better mode quality and lower confinement loss which is more conductive to long distance OAM mode propagation. From Fig. 4.(a) we can find that as the inner radius r0 increases (the ring thickness decreases), the light intensity overlap factors for all 7 orders of OAM modes supported in C-PCF 2# decrease at 1.55 µm, which means that the mode quality decline gradually. Fig. 4.(a) also depicts that when r0 is less than 3.6 µm all 7 orders of OAM mode can be well supported in C-PCF 2#. Larger light intensity overlap factor means stronger energy confinement in the ring shaped fiber core, and hence leads to lower confinement loss of the fiber. Fig. 4. (b) shows that confinement loss of different vector modes at 1.55 µm increases with increasing of r0 (decreasing of thickness), which is consistent with the trend of intensity overlap factor. The smaller the inner radius of C-PCF 2# is, the smaller the confinement loss is. But C-PCF 2# with too small inner radius (too large thickness) will excite the high order radial modes. As shown in Fig. 4.(b), at the right side of the figure, apart from EH7,1 mode, the high order radial modes TE0,2, HE1,2, and HE2,2 will excited for the case of r0=3.0 µm (ρVeff=9.18, ρ=0.58) which are not desired for the reason that it will bring difficulty to carry out demultiplexing in optical communications. C-PCF 2# with other inner radius range from 3.2 µm (ρVeff=9.79, ρ=0.62, Veff=15.91) to 4.0 µm
5. Transmission properties In order for the designed fiber to hold the OAM modes realizing
Fig. 4. (a)The light intensity overlap factor and (b) the confinement loss of different OAM modes of C-PCF 2# with different inner radius at 1.55 µm.
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long distance OAM modes propagation. Through optimization under the consideration of 5 requirements in Section 2, we obtain the accepted inner radius r0 to be from 5.0 µm (ρVeff=15.30, ρ=0.69) to 5.6 µm (ρVeff=17.14, ρ=0.78) (corresponding to the ring thickness from 1.6 µm to 2.2 µm), which also provides a range of r0 about 0.6 µm. The total loss of the C-PCF is determined by material loss (including Rayleigh scattering and absorption induced by impurities) and confinement loss. Because the proposed C-PCF is pure silica based, it reduces the risk of absorption induced by impurities. In C+L band, the confinement loss had better take the value less than 1×10−4 dB/m (1×10−1 dB/km) which is about the value of Rayleigh scattering. Fig. 7 shows the confinement losses take the increasing trend with the wavelength due to reason that the mode size is larger with larger wavelength leading to more chance of energy leakage through the cladding. C-PCF 3# shows relative low confinement loss from 1.25 µm to 2.0 µm. In C+L band, the confinement losses of the modes are less than 1×10−4 dB/m except for HE11,1 and EH9,1(4 OAM ± 10,1),HE12,1 and EH10,1(4 OAM ± 11,1) modes. Fig. 8 shows the dispersion property of different orders OAM modes in the designed C-PCF 3# as the function of wavelength. All the OAM modes have a relatively flat dispersion and low dispersion value. In C +L band over 110 nm, total dispersion variations for all orders OAM modes are less than 19.2 ps nm−1 km−1. Fig. 9 depicts the property of nonlinear coefficient of the designed C-PCF 3#. For all the modes supported in the fiber, the nonlinear coefficient decreases with the increasing of wavelength, mainly due to the fact that larger wavelength leads to larger effective mode area. At 1.55 µm, the HE11,1 mode has the largest nonlinear coefficient of 2.18 W−1 km−1, while the HE1,1 mode has the least nonlinear coefficient of 1.09 W−1 km−1, which are the same order as that of single mode fiber. We can conclude that C-PCF 3# possesses good properties which satisfy 5 requirements in Section 2.
Fig. 5. The effective index separation in C-PCF 2# from l=1 to l=8 with different inner radius at 1.55 µm.
long distance propagation, the proposed C-PCFs should meet the 4th requirement mentioned in Section 2. A Low dispersion value and its flatness over a wide wavelength range are very important, which can ensure the optical signals stable transmission modulated on the OAM modes and free from dispersion compensation. In addition, low nonlinear coefficient can avoid signal distortion induced by the nonlinear effects. Table 1 makes a comparison of the three C-PCFs with their number of mode, bandwidth and their performance features. The C-PCF 1# has the minimum number of mode and the maximum bandwidth, while C-PCF 3# has the maximum number of mode and the minimum bandwidth among three fibers as shown in Table 1. Three C-PCFs all cover entire band of optical fiber communications. Take HE4,1 mode as example, we can find that C-PCF 3# has the best performance features such as dispersion value, dispersion difference ΔD in C band, confinement loss and nonlinear coefficient, with the values of 111.6 ps/(km nm), 1.5 ps/(km nm), 9.52×10−9 dB/ m and 1.544 W−1/km, respectively. As mentioned in Section 2, from CPCF 1# to C-PCF 3#, the design strategy is to change larger and larger number order of air-hole array into ring shaped core to hold the OAM modes. From C-PCF 1# to C-PCF 3#, the larger number order of airhole array, the lower values of dispersion, confinement loss, and nonlinear coefficient are, and more flat over wavelength range the dispersion is. The parts of reasons are the fact that larger number order of air-hole array provides larger area for OAM mode's transmission, subsequently bigger light intensity overlap factor, and hence less chance for mode energy leakage, and also larger effective mode area leads to smaller nonlinear coefficient. Therefore, C-PCF 3# may be the best choice among three. Now we focus our attention to C-PCF 3# to evaluate its transmission properties. Fig. 6 shows that in C-PCF 3# with r0=5.3 µm, mode intensity distributions of the OAM modes are well controlled within the fiber ring shaped region, which means that the fiber well supports the 11 orders of OAM modes with good mode quality and is suitable for
6. Conclusions In order to design an OAM supporting fiber for long distance and stable OAM modes transmission in MDM systems, we present a detail design strategy by considering 5 essential requirements to optimize the structure parameters of the C-PCF. It is an effective method to increase the number of the OAM modes by changing inner air-hole arrays into ring core area, and hence enlarging the radius of ring area r, by which we design an OAM family and find that C-PCF 1#, C-PCF 2#, and CPCF 3# can support 14, 26, and 42 OAM modes with high mode quality. By optimization we can find that for C-PCF 3# thickness of the ring should be within 1.6 µm to 2.2 µm to cope with the trade-off between possessing good OAM mode quality and avoiding excitation of the radial higher order modes. This optimal thickness range 1.6 µm to 2.2 µm corresponds to the inner radius 5.0 µm (ρVeff=15.30, ρ=0.69) to 5.6 µm (ρVeff=17.14, ρ=0.78) with the unchanged outer radius 7.2 µm (Veff=22.04), and also provides a large fabrication tolerance of 0.6 µm. Through the comparison among C-PCF 1#, C-PCF 2# and C-PCF 3#, we can find that C-PCF 1# and C-PCF 2# meet the 2nd, 3rd and 5th
Table 1 Characteristics of three C-PCFs.
Number of mode Bandwidth Δneff (l=3,at 1.55 µm) D (HE41, at 1.55 µm) ΔD (HE4,1 at C band(1530–1565 µm)) Confinement loss (HE4,1, at 1.55 µm) γ（HE4,1, at 1.55 µm）
C-PCF 1#
C-PCF 2#
C-PCF 3#
14 560 nm(1.25–1.81 µm) 4.345×10−4 269.1 ps/(km nm) 6.1 ps/(km nm) 1.24×10−6 dB/m 4.04 W−1/km
26 480 nm (1.25–1.73 µm) 2.538×10−3 128.7 ps/(km nm) 2.1 ps/(km nm) 1.77×10−8 dB/m 2.46 W−1/km
42 460 nm(1.25–1.71 µm) 2.067×10−3 111.6 ps/(km nm) 1.5 ps/(km nm) 9.52×10−9 dB/m 1.544 W−1/km
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Fig. 6. (a) Intensity distributions of part vortex modes, and (b) normalized intensity distributions of different vector modes in the high-index ring region.
Fig. 9. Nonlinear as a function of wavelength for different modes in the designed C-PCF 3# over C+L band. Fig. 7. Confienment loss as a function of wavelength for different vector modes in the designed C-PCF 3#.
propagation in C+L band. So C-PCF 3# is the optimal choice among three C-PCFs that can be used in MDM communications and other OAM applications in fibers. Funding National Natural Science Foundation of China (NSFC) (61571057, 61501214, 61527820, and 61575082); Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications) (IPOC2016ZZ04). Acknowledgements The authors wish to thank the anonymous reviewers for their valuable suggestions. References [1] L. Allen, M.W. Beijersbergen, R. Spreeuw, J. Woerdman, Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 45 (11) (1992) 8185–8189. [2] A.M. Yao, M.J. Padgett, Orbital angular momentum: origins, behavior and applications, Adv. Opt. Photon 3 (2011) 161–204. [3] D.G. Grier, A revolution in optical manipulation, Nature 424 (6950) (2003) 810–816. [4] A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, J. Laurat, A quantum memory for orbital angular momentum photonic qubits, Nat. Photon. 8 (3) (2014) 234–238. [5] S. Fürhapter, A. Jesacher, S. Bernet, M. Ritsch-Marte, Spiral phase contrast imaging in microscopy, Opt. Express 13 (3) (2005) 689–694.
Fig. 8. Dispersion as a function of wavelength for different vector modes in the designed C-PCF 3# over C+L band.
requirements in Section 2, while C-PCF 3# meets all 5 requirements. As a result that C-PCF 3# shows the good properties of flat dispersion (total dispersion variation less than 19.2 ps nm−1km−1), low confinement loss ( < 10−4 dB/m except for modes HE11,1, EH9,1, HE12,1 and EH10,1) and low nonlinear coefficient (≤2.18 W−1 m−1 for all supported OAM modes), which are required for longer distance OAM modes 65
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fiber designs for OAM multiplexing, Opt. Express 23 (3) (2015) 3721–3730. [20] P. Gregg, P. Kristensen, S. Ramachandran, Conservation of orbital angular momentum in air core optical fibers, Optica 2 (3) (2015) 2334–2536. [21] C. Brunet, L.A. Rusch, Optical fibers for the transmission of orbital angular momentum modes, Opt. Fiber Technol. 31 (2016) 172–177. [22] J. Ye, Y. Li, Y. Han, D. Deng, Z. Guo, J. Gao, Q. Sun, Y. Liu, S. Qu, Excitation and separation of vortex modes in twisted air-core fiber, Opt. Express 24 (8) (2016) 8310–8316. [23] P. St.J. Russell, Photonic crystal fibers, J. Light. Technol. 24 (12) (2006) 4729–4749. [24] F. Benabid, P.S.J. Russell, Hollow core photonic crystal fibers: progress and prospects, Proc. SPIE 5733 (2005) 176–189. [25] N.A. Mortensen, M.D. Nielsen, J.R. Folkenberg, A. Petersson, H.R. Simonsen, Improved large-mode-area endlessly single-mode photonic crystal fibers, Opt. Lett. 28 (6) (2003) 393–395. [26] R. Amezcua-Correa, N.G. Broderick, M.N. Petrovich, F. Poletti, D.J. Richardson, Design of 7 and 19 cells core air-guiding photonic crystal fibers for low-loss, wide bandwidth and dispersion controlled operation, Opt. Express 15 (26) (2007) 17577–17586. [27] J. Lægsgaard, P.J. Roberts, M. Bache, Tailoring the dispersion properties of photonic crystal fibers, Opt. Quantum Electron. 39 (12–13) (2007) 995–1008. [28] Y. Yue, L. Zhang, Y. Yan, N. Ahmed, J. Yang, H. Huang, Y. Ren, S. Dolinar, M. Tur, A.E. Willner, Octave-spanning supercontinuum generation of vortices in an As2S3 ring photonic crystal fiber, Opt. Lett. 37 (11) (2012) 1889–1891. [29] H. Zhang, W. Zhang, L. Xi, X. Tang, W. Tian, X. Zhang, X. Zhang, Design of a Circular Photonic Crystal Fiber Supporting OAM Modes, in: Asia Communications and Photonics Conference: 2015 OSA Technical Digest (online) (Optical Society of America, 2015), paper ASu2A.54. [30] H. Zhang, W. Zhang, L. Xi, X. Tang, X. Zhang, X. Zhang, A new type circular photonic crystal fiber for orbital angular momentum mode transmission, IEEE Photon. Technol. Lett. 28 (13) (2016) 1426–1429. [31] G. Zhou, G. Zhou, C. Chen, M. Xu, C. Xia, Z. Hou, Design and analysis of a microstructure ring fiber for orbital angular momentum transmission, IEEE Photon. J. 8 (2) (2016) (Art. no. 7802512). [32] Z. Hu, Y. Huang, A. Luo, H. Cui, Z. Luo, W. Xu, Photonic crystal fiber for supporting 26 orbital angular momentum modes, Opt. Express 24 (15) (2016) 17285–17291. [33] P. Lee, T. Lu, J. Fan, F. Tsai, High quality factor microcavity lasers realized by circular photonic crystal with isotropic photonic band gap effect, Appl. Phys. Lett. 90 (90) (2007) (Art. no. 151125).
[6] J. Wang, J.Y. Yang, I.M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, A.E. Willner, Terabit free-space data transmission employing orbital angular momentum multiplexing, Nat. Photonics 6 (7) (2012) 488–496. [7] N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A.E. Willner, S. Ramachandran, Terabit scale orbital angular momentum mode division multiplexing in fibers, Science 340 (2013) 1545–1548. [8] Y. Yan, Y. Yue, H. Huang, J. Yang, M.R. Chitgarha, N. Ahmed, M. Tur, S.J. Dolinar, A.E. Willner, Efficient generation and multiplexing of optical orbital angular momentum modes in a ring fiber by using multiple coherent inputs, Opt. Lett. 37 (17) (2012) 3645–3647. [9] D.J. Richardson, J.M. Fini, L.E. Nelson, Space-division multiplexing in optical fibres, Nat. Photon. 7 (5) (2013) 354–362. [10] S. Li, J. Wang, Multi-orbital-angular-momentum multi-ring fiber for high-density space-division multiplexing, IEEE Photon. J. 5 (5) (2013) (Art. no. 7101007). [11] S. Ramachandran, P. Kristensen, Optical vortices in fiber, Nanophotonics 2 (5–6) (2013) 455–474. [12] S. Ramachandran, P. Kristensen, M.F. Yan, Generation and propagation of radially polarized beams in optical fibers, Opt. Lett. 34 (16) (2009) 2525–2527. [13] Y. Yue, Y. Yan, N. Ahmed, J. Yang, L. Zhang, Y. Ren, H. Huang, K.M. Birnbaum, B.I. Erkmen, S. Dolinar, M. Tur, A.E. Willner, Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber, IEEE Photon. J. 4 (2) (2012) 535–543. [14] C. Brunet, B. Ung, Y. Messaddeq, S. LaRochelle, E. Bernier, L.A. Rusch, Design of an optical fiber supporting 16 OAM modes, in: Optical Fiber Communication Conference: 2014 OSA Technical Digest (online) (Optical Society of America, 2014) paper Th2A.24. [15] C. Brunet, B. Ung, P.A. Bélanger, Y. Messaddeq, S. LaRochelle, L.A. Rusch, Vector mode analysis of ring-core fibers: design tools for spatial division multiplexing, J. Light. Technol. 32 (23) (2014) 4046–4057. [16] P. Gregg, P. Kristensen, S. Golowich, J. Olsen, P. Steinvurzel, S. Ramachandran, Stable transmission of 12 oam states in air-core fiber, in: Conference on Lasers and Electro-Optics: 2013 OSA Technical Digest (online) (Optical Society of America, 2013), paper CTu2K.2. [17] C. Brunet, B. Ung, P. Vaity, L. Wang, Y. Messaddeq, S. LaRochelle, L.A. Rusch, Design of a family of ring-core fibers for OAM transmission studies, Opt. Express 23 (8) (2015) 10553–10563. [18] C. Brunet, P. Vaity, Y. Messaddeq, S. LaRochelle, L.A. Rusch, Design, fabrication and validation of an OAM fiber supporting 36 states, Opt. Express 22 (21) (2014) 26117–26127. [19] S. Ramachandran, P. Gregg, P. Kristensen, S.E. Golowich, On the scalability of ring
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
A design strategy of the circular photonic crystal fiber supporting good quality orbital angular momentum mode transmission
MARK
⁎
Hu Zhanga,b, , Xiaoguang Zhanga, Hui Lia, Yifan Denga, Xia Zhanga, Lixia Xia, Xianfeng Tanga, Wenbo Zhanga,c a State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, 100876 Beijing, PR China b School of Ethnic Minority Education, Beijing University of Posts and Telecommunications, 100876 Beijing, PR China c School of Sciences, Beijing University of Posts and Telecommunications, 100876 Beijing, PR China
A R T I C L E I N F O
A BS T RAC T
Keywords: Fiber design and fabrication Orbital angular momentum Photonic crystal fibers Multiplexing
Based on 5 requirements which are essential for stable OAM mode transmission, we propose an OAM fiber family based on a structure of circular photonic crystal fiber (C-PCF). The proposed C-PCF in the family is made of pure silica, with a big round air hole at the center, several rings of air-hole array as the cladding, and a ring shaped silica area in between as the core where the OAM modes propagate. We also provide a design strategy with which the optimized C-PCF can be obtained with optimum number of high quality OAM modes (up to 42 OAM modes), large effective index separation for corresponding vector modes over a wide bandwidth, relative small and flat dispersion, and low nonlinear coefficient compared with a conventional single mode fiber. The designed fiber can be used in MDM communications and other OAM applications in fibers.
1. Introduction A light beam with the spiral phase front exp(ilφ) carries an orbital angular momentum (OAM), where, l is a integer called as topological charge, φ is azimuthal angle [1,2]. OAM beam has recently spurred tremendous interests because of its wide potential applications, in the field of optical manipulation [3], quantum information [4], imaging [5] and communications [6,7]. The orthogonality between different OAM states provides another dimension (mode division multiplexing, MDM) to be multiplexed to solve the currently urgent requirements for data capacity [7–10]. Terabit data transmission in a fiber carrying OAM modes has been demonstrated [7]. Since OAM modes in fibers can be properly constituted by the even and odd modes of a vector mode with the identical propagation constants, they will not undergo intermodal work-off effect. But the mode degeneracy between corresponding HE and EH eigenmodes will lead to strong modal coupling (couple into LP modes), subsequently break the stability of OAM modes transmission, and hence cause the serious crosstalk. To handle these problems, a specially designed optical fiber is required to possess high effective refractive index separation ( > 10−4) between different vector modes in the fiber [11,12] to avoid modal coupling and to be free from complex multi-input multi-output (MIMO) treatment at the receiver. Recently, a
ring shaped or annular shaped fiber was designed with a high index ring area as the ring core to hold OAM modes, and its central circle area and cladding are with lower indices [13–22]. The high index contrast between the ring core and cladding is achieved by up-doping in ring core area. So far, ring fibers can support 9 orders of OAM (34 OAM modes) or 36 information bearing states [18]. However, heavily updoping may result in higher fiber loss [21]. Compared with the conventional fibers, photonic crystal fibers (PCFs) [23,24] have more flexible structure design to provide unique fiber properties such as endlessly single mode [25], controllable nonlinearity and confinement loss [25,26], tailorable chromatic dispersion [26,27]. The structure of PCF used to realize the OAM modes transmission has potential application prospects in fiber communications. In Ref. [28] the ring photonic crystal (ring-PCF) with hexagonal air hole array as cladding was proposed which can only support less number of OAM modes (4 OAM modes) with bad mode quality. We find that the hexagonal structure of air holes is not a good design for its not good OAM mode quality and higher confinement loss. The reason is that hexagonal structure with noncircular symmetry is not consistent with ring shaped intensity of OAM modes. In order to gain more numbers of OAM mode and better mode field quality, we presented a new structure of circular shaped PCF (C-PCF)
⁎ Corresponding author at: State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, 100876 Beijing, PR China. E-mail address: [email protected] (H. Zhang).
http://dx.doi.org/10.1016/j.optcom.2017.03.075 Received 6 February 2017; Received in revised form 20 March 2017; Accepted 28 March 2017 0030-4018/ © 2017 Elsevier B.V. All rights reserved.
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Fig. 1. Cross-section and main parameters of the designed C-PCFs for (a) the fiber base, (b) C-PCF 1#, (c) C-PCF 2#, (d) C-PCF 3#.
conventional single mode fiber. Furthermore, it is proved that the ring thickness of the presented C-PCF has a relative large fabrication tolerance about 0.6 µm both to guarantee the good mode quality and suppress the high order radial modes. So this kind of fiber is suitable to be used in MDM systems for long distance.
[29] with circular symmetry which has a large air-hole in the center, and a circular shaped air-hole array rather than the hexagonal air holes as cladding. Compared with the ordinary ring fibers, the C-PCF has more advantages such as getting high index contrast without updoping, larger fiber bandwidth, good mode quality, lower confinement loss, and larger fabrication tolerance [30]. Recently, Ref. [31] presented a microstructure ring fiber (MRF) with the equally number of air holes in three different cladding rings to obtain the circular symmetry. The hole size of the proposed fiber is different in every ring. And the proposed MRF has narrow bandwidth and higher confinement loss. Ref. [32] adopted the similar fiber structure with that of Ref. [31], but the material of the fiber is the As2S3 glass. This kind of fiber has larger loss and high nonlinearity which is not suitable for the OAM mode transmission, while suitable for supercontinuum generation. In this paper, we summarize 5 essential requirements to ensure the OAM modes stable transmission in optical fiber, and based on which an OAM fiber family with the structure of ring circular PCF are proposed to realize MDM in optical communication systems. A detail fiber design strategy is presented under which the C-PCFs are designed to possess good features. Comparing the properties of three C-PCFs, we can find that C-PCF 3# has the maximum number of high quality OAM modes (11 OAM orders or 42 OAM modes), the relative wide bandwidth (460 nm from 1.25 µm to 1.71 µm) with large effective index separation of different vector modes, relative small and flat dispersion, low confinement loss and low nonlinear coefficient compared with a
2. Design strategy of the C-PCF supporting OAM modes OAM modes in fibers are defined as OAMl,m, where m is the number of concentric rings of the intensity profile of the mode in radial direction, l(l∈Z) is the topological charge. OAM modes in fibers can be regarded as the superposition of the vector eigenmodes by following relations: odd OAM ±± l, m = HEleven +1, m ± j HEl +1, m
(1)
odd OAM∓± l, m = EHleven −1, m ± j EHl −1, m
(2)
where the sign in superscript ‘ ± ’ denotes the right or left circular polarization, and the sign of ± l denotes the right or left wave front rotation direction, odd and even modes of HE or EH denotes a π/2 azimuthal rotation, j presents a π/2 phase shift. HE mode has the same directions of spin angular momentum (SAM) and OAM, while EH mode has the opposite directions of SAM and OAM. The OAM 0,±m mode composed by HE1,m with two circular polarizations can not be used as OAM mode for it carries no OAM. Moreover, the OAM∓± 1, m mode composed by azimuthally polarized TE0,m and radially polarized TM0,m 60
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where, λ is the wavelength in vacuum, nco is the index of the fiber core, ncl is the effective index of the cladding, ncl = 1.444(1 − f ) + f , f is the air filling fraction of the cladding, which is defined as
can also not be used as OAM mode because they have different propagation constants, and are unstable in the fiber. The OAM modes with the same l and m compose an OAM mode family, and each mode family supports 4 OAM modes except for OAM1,1 mode family (composed by even and odd of HE2,1). To design a good quality OAM mode supporting fiber, which can be used in MDM, following requirements should be considered: 1) This kind of fiber can hold OAM modes as many as possible and meanwhile avoid higher radial modes to be excited to simplify the multiplexing and the demultiplexing of OAM modes. 2) Large effective index separation between the corresponding HE and EH modes that belong to the same OAM mode family, should be achieved, in order to weaken the modal coupling. 3) All the supported OAM modes should possess good mode quality and low confinement loss. 4) Other features which the designed fiber would possess are flat and low dispersion over a large spectrum bandwidth, large mode area and low nonlinear coefficient. 5) All the requirements mentioned above should be valid covering a wide bandwidth, especially C+L band. For the reason that OAM modes have the circular symmetry of electric field, the desired fiber should possess circular shape which has a high refractive index ring area to hold OAM modes. Therefore we propose a design of the circular shaped photonic crystal fiber whose base material is made of pure silica (refractive index is 1.444 at the wavelength 1.55 µm) shown in Fig. 1. The circular photonic crystal fiber has a central air hole in the center and several ring air-hole arrays around it as the fiber cladding, in between a ring shaped area of base pure silica remains where OAM modes can be transmitted in this region. Roughly speaking, the region occupied by pure silica contributes to refractive index with the value 1.444, and the region occupied by air holes provides index with the value of 1, therefore the average refractive index of cladding area is the mean value of two regions weighted with the areas they occupy. Hence the larger air filling fraction is, the higher the index contrast between the OAM mode transmission area and cladding area is. But we have to consider the strong structure of the fiber simultaneously, and the diameters of the air-holes in the cladding should be a trade-off value. We find that larger number of cladding air-hole arrays makes a stronger mode constraint, but 4 arrays are enough to confine OAM modes in the fiber. Taking the shape of photonic crystal fiber shown Fig. 1(a) as the fiber base, we design a family of good quality circular photonic crystal fibers which can support OAM mode transmission. This fiber family consists of C-PCF 1#, C-PCF 2#, and C-PCF 3#, in which C-PCF n# fiber is constructed by removing off the first nth airhole ring of the base fiber (for example, C-PCF 1# corresponds to removing off the first one air-hole ring), and putting a central air hole at the center, as shown in Fig. 1(b), (c), and (d). The spatial positions of the air holes on the x–y plane are [33]
⎛ 2nπ ⎞ ⎛ 2nπ ⎞ x = ΛN cos ⎜ ⎟ , n = 1~6N ⎟ , y = ΛN sin ⎜ ⎝ 6N ⎠ ⎝ 6N ⎠
⎛ d ⎞2 6(4N − 6) ⎜ n ⎟ ⎛ d ⎞2 ⎝2⎠ f= = 3⎜ n ⎟ 2 2 ⎡⎛ ⎡⎛ ⎝ 2Λ ⎠ 1⎞ ⎤ 7⎞ ⎤ ⎢ ⎜N + ⎟ Λ ⎥ − ⎢ ⎜N − ⎟ Λ ⎥ 2⎠ ⎦ 2⎠ ⎦ ⎣⎝ ⎣⎝
where, f means the ratio of air hole area versus the total area in cladding. Roughly speaking, Veff determines the number of vector modes supported in the fiber, f makes an influence on ncl and hence the index contrast, ρ leads to the quality of the OAM modes held in the fiber. All the numerical analyses are performed using a full-vector finite element method (FEM) (COMSOL) and perfect matched layers (PMLs) are used as boundary condition. We used this model to run the analyses upon the above the C-PCFs in the fiber family with the optimized parameters of Λ=2.0 µm, dn/Λ=0.8, d2=d3=d4=d5=d6=d7=1.6 µm, which are convenient to fiber fabrication and can also guarantee the large index separation. r equals to 3.2 µm, 5.2 µm, and 7.2 µm for CPCF1#, C-PCF2#, and C-PCF3#, respectively. With the parameters above, f =0.48, ncl=1.231, which leads a high index contrast of 0.213. r0 determines the ring thickness of region that OAM modes are transmitted in, and needs to be optimized. We will discuss r0 in detail in Section 4. 3. The number of mode and fiber bandwidth OAM mode belongs to orthogonal eigenstates, and has infinite number of orthogonal modes theoretically. But the fiber can only support the limited number of OAM modes for the restriction of the fiber structure. So far, the most reported OAM mode supporting fibers are the ring fibers. For example, Ref. [19] reported a fiber which can support 9 OAM orders or 34 OAM modes. The number of OAM modes which can be stably transmitted in fiber is decided by two factors. One is the effective normalized frequency Veff parameter of the fiber, as we know that the number of mode increases with the increasing of Veff. Another one is degree of separation of mode effective index between different vector modes of the same OAM mode family. In order to guarantee the stable transmission of the OAM modes without coupling into LP modes, the effective index contrast should be larger than 10−4, which is the second requirement in Section 2. For C-PCF, we also need two-dimension parameters to describe the cut-off frequency, namely the effective normalized frequency Veff and the ratio between inner and outer radius ρ as similar as defined in Ref. [15]. From view of the Eq. (4) of the Veff parameter, there are two ways to increase the Veff for a given wavelength. One is to increase the radius of the effective mode field, another one is to enlarge the index contrast between the fiber core and the cladding. Due to the fact that C-PCF is made by the single material of silica and the cladding is made by 4 air-hole arrays, the more effective way to increase the Veff is to enlarge the radius of ring area r. As shown in Fig. 1, C-PCF 1# with Veff=9.79 constituted by changing the first one air-hole array into the ring core can support 4 OAM orders (l=1–4) or 14 OAM modes, C-PCF 2# with Veff=15.91 constituted by changing the first two air-hole arrays into the ring core can support 7 OAM orders (l=1–7) or 26 OAM modes, C-PCF 3# with Veff=22.04 constituted by changing the first three air-hole arrays into the ring core can support 11 OAM orders (l=1–11) or 42 OAM modes. We can get more OAM modes as the radius of the ring core goes on to increase by removing more arrays, but with the result of increasing fiber fabrication difficulty. In order for the proposed fiber to be used in WDM systems, we define a bandwidth of the C-PCF that is the range of the wavelength in which all the OAM modes can stably transmitted with the index separation of larger than 10−4 between corresponding HE and EH
(3)
where Λ and N denote the concentric spacing and number of concentric rings respectively, while r0 is the inner radius of the ring (also the radius of central air hole), r is the outer radius of the ring, d2 to d7 are the diameters of the cladding air holes. As mentioned above, the background material is silica with a refractive index of 1.444 at 1.55 µm, to make high index contrast without the need of up-doping. Unlike the conventional multi-mode fiber which need only one dimension parameters, normalized frequency V = (2π / λ ) r 2 2 , to analyze the quality of the mode, we define twoncore − ncladding dimension parameters, effective normalized frequency Veff and inner and outer radius ratio ρ, as follows
Veff =
ρ=
2π 2 r nco − ncl2 λ
r0 r
(6)
(4)
(5) 61
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Fig. 2. Effective index separation of the different OAM mode families as a function of wavelength for (a) C-PCF 1#, (b) C-PCF 2#, (c) C-PCF 3#.
requirement because it can support maximum OAM modes of the proposed fiber family.
modes belong to the same OAM mode family. This bandwidth is also determined by the cut-off wavelength of the highest order OAM mode. The effective index difference between corresponding HE and EH modes of the same OAM mode order increases with wavelength for the three C-PCFs as shown in Fig. 2 due to the increased mode area, where Δneff for l=1 denotes the effective index difference of HE2,1 mode and TE0,1 mode, the reason is that the effective index difference of HE2,1 mode and TM0,1 mode is larger than that of HE2,1 mode and TE0,1 mode. In Fig. 2, all the fiber structure parameters take the same values as mentioned in Section 2, and r0=1.2 µm for C-PCF 1#, r0=3.5 µm for C-PCF 2#, r0=5.3 µm for C-PCF 3#. Fig. 2 also illustrates a very good feature of large separation of the effective indices of the modes. The index difference of all OAM orders is larger than 1×10−4. From Fig. 2, we can find that the common bandwidth of the four OAM mode families covers 560 nm (1.25–1.81 µm) for C-PCF 1#, the common bandwidth of the seven OAM mode families covers 480 nm (1.25– 1.73 µm) for C-PCF 2#, and the common bandwidth of the eleven OAM mode families covers 460 nm (1.25–1.71 µm) for C-PCF 3#. They all do cover all bands of optical fiber communications (O, E, S, C, L, U, from 1.26 µm to 1.675 µm). Apparently, the bandwidth of the proposed C-PCF is wider than that of the ordinary ring fibers [13,14,16–20] and the MRFs [32], which makes the C-PCF be good MDM fiber in WDM optical fiber communications. Up to now, we see that all three C-PCFs meet the 2nd and 5th requirements mentioned above, but C-PCF 3# also satisfies the 1st
4. Mode quality Apart from pursuing more number of OAM modes supported in the fiber, we should also pay attention to the mode quality in the fiber, because only good quality OAM modes can ensure the stable transmission in the fiber. Therefore, we introduce the light intensity overlap factor to qualitatively evaluate the mode quality
⎯→ ⎯ 2 ∬ring E dxdy Ir η= = ⎯→ ⎯ 2 Ic ∬cross −section E dxdy
(7)
where, Ir is the average mode intensity in ring area where all the OAM modes should be confined and Ic is the average mode intensity in the whole cross-section area of the C-PCF. In Section 3, we find that C-PCF 1#, C-PCF 2#, and C-PCF 3# can support 14, 26, and 42 OAM modes. Now we will discuss in order to meet the 3rd requirement what is the optimized thickness (namely optimized radius r0 of the central air hole). We take C-PCF 2# as an example. The electric field intensity distribution of HE3,1 mode in CPCF 2# with r0=3.2 µm and r0=4.0 µm at 1.55 µm are shown in Fig. 3(a) and (b) respectively. C-PCF 2# with r0=3.2 µm confines more mode energy in the ring shaped core and shows better mode quality 62
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Fig. 3. The electric field intensity distribution of HE3,1 mode in C-PCF 2# with (a) r0=3.2 µm and (b) r0=4.0 µm at 1.55 µm.
(ρVeff=12.24, ρ=0.77, Veff=15.91) can suppress the high order radial modes the reason is that for TE0,2, Vcut=16.51 > Veff=15.91, which will not be excited. So the appropriate inner radius r0 should be optimized to make trade-off between avoiding the high radial order modes and achieving the good quality of OAM modes. When r0 < 3.6 µm (ρVeff=11.02, ρ=0.69) the intensity overlap factors for all the modes supported in C-PCF 2# are above 82%, and corresponding confinement losses take relative small values. When r0 > 3.8 µm the intensity overlap factors show sharp decrease (accordingly sharp increase of confinement losses) for larger order modes. The thickness of the ring core can also influence on the effective index separation. As shown in Fig. 5, the lower value of r0 (larger thickness) is, the smaller the effective index separation (at 1.55 µm) is. For the cases that r0=3.0 µm and r0=3.2 µm, we find that there are one OAM order modes (l=5 for r0=3.0 µm and l=6 for r0 =3.2 µm) whose effective index separation is less than or nearly equal to 1×10−4. Considering the trade-off among mode quality, high order radial modes and effective index separation, we can take the value of r0 to be larger than 3.2 µm and smaller than 3.8 µm for C-PCF 2#, which provides a range of r0 about 0.6 µm. For C-PCF 2#, the optimized range of the inner radius r0 is from 3.2 µm (ρVeff=9.79, ρ=0.62) to 3.8 µm (ρVeff=11.63, ρ=0.76) to satisfy the 2nd, 3rd, and 5th requirements of the fiber design in Section 2. Also, for C-PCF 1# and C-PCF 3#, the same procedure in this section can be used to optimize the fiber structure parameters.
which means less energy leaked into the cladding than that of C-PCF 2# with r0=4.0 µm. Accordingly, the light intensity overlap factors corresponding to the case of r0=3.2 µm and r0=4.0 µm are 96% and 80% respectively, which proves that the light intensity overlap factor can be used to evaluate the mode quality. Furthermore, the confinement losses corresponding to the case of r0=3.2 µm and r0=4.0 µm are 2.17×10−10 dB/m and 1.58×10−6 dB/m, respectively. Therefore, the larger intensity overlap factor leads to better mode quality and lower confinement loss which is more conductive to long distance OAM mode propagation. From Fig. 4.(a) we can find that as the inner radius r0 increases (the ring thickness decreases), the light intensity overlap factors for all 7 orders of OAM modes supported in C-PCF 2# decrease at 1.55 µm, which means that the mode quality decline gradually. Fig. 4.(a) also depicts that when r0 is less than 3.6 µm all 7 orders of OAM mode can be well supported in C-PCF 2#. Larger light intensity overlap factor means stronger energy confinement in the ring shaped fiber core, and hence leads to lower confinement loss of the fiber. Fig. 4. (b) shows that confinement loss of different vector modes at 1.55 µm increases with increasing of r0 (decreasing of thickness), which is consistent with the trend of intensity overlap factor. The smaller the inner radius of C-PCF 2# is, the smaller the confinement loss is. But C-PCF 2# with too small inner radius (too large thickness) will excite the high order radial modes. As shown in Fig. 4.(b), at the right side of the figure, apart from EH7,1 mode, the high order radial modes TE0,2, HE1,2, and HE2,2 will excited for the case of r0=3.0 µm (ρVeff=9.18, ρ=0.58) which are not desired for the reason that it will bring difficulty to carry out demultiplexing in optical communications. C-PCF 2# with other inner radius range from 3.2 µm (ρVeff=9.79, ρ=0.62, Veff=15.91) to 4.0 µm
5. Transmission properties In order for the designed fiber to hold the OAM modes realizing
Fig. 4. (a)The light intensity overlap factor and (b) the confinement loss of different OAM modes of C-PCF 2# with different inner radius at 1.55 µm.
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long distance OAM modes propagation. Through optimization under the consideration of 5 requirements in Section 2, we obtain the accepted inner radius r0 to be from 5.0 µm (ρVeff=15.30, ρ=0.69) to 5.6 µm (ρVeff=17.14, ρ=0.78) (corresponding to the ring thickness from 1.6 µm to 2.2 µm), which also provides a range of r0 about 0.6 µm. The total loss of the C-PCF is determined by material loss (including Rayleigh scattering and absorption induced by impurities) and confinement loss. Because the proposed C-PCF is pure silica based, it reduces the risk of absorption induced by impurities. In C+L band, the confinement loss had better take the value less than 1×10−4 dB/m (1×10−1 dB/km) which is about the value of Rayleigh scattering. Fig. 7 shows the confinement losses take the increasing trend with the wavelength due to reason that the mode size is larger with larger wavelength leading to more chance of energy leakage through the cladding. C-PCF 3# shows relative low confinement loss from 1.25 µm to 2.0 µm. In C+L band, the confinement losses of the modes are less than 1×10−4 dB/m except for HE11,1 and EH9,1(4 OAM ± 10,1),HE12,1 and EH10,1(4 OAM ± 11,1) modes. Fig. 8 shows the dispersion property of different orders OAM modes in the designed C-PCF 3# as the function of wavelength. All the OAM modes have a relatively flat dispersion and low dispersion value. In C +L band over 110 nm, total dispersion variations for all orders OAM modes are less than 19.2 ps nm−1 km−1. Fig. 9 depicts the property of nonlinear coefficient of the designed C-PCF 3#. For all the modes supported in the fiber, the nonlinear coefficient decreases with the increasing of wavelength, mainly due to the fact that larger wavelength leads to larger effective mode area. At 1.55 µm, the HE11,1 mode has the largest nonlinear coefficient of 2.18 W−1 km−1, while the HE1,1 mode has the least nonlinear coefficient of 1.09 W−1 km−1, which are the same order as that of single mode fiber. We can conclude that C-PCF 3# possesses good properties which satisfy 5 requirements in Section 2.
Fig. 5. The effective index separation in C-PCF 2# from l=1 to l=8 with different inner radius at 1.55 µm.
long distance propagation, the proposed C-PCFs should meet the 4th requirement mentioned in Section 2. A Low dispersion value and its flatness over a wide wavelength range are very important, which can ensure the optical signals stable transmission modulated on the OAM modes and free from dispersion compensation. In addition, low nonlinear coefficient can avoid signal distortion induced by the nonlinear effects. Table 1 makes a comparison of the three C-PCFs with their number of mode, bandwidth and their performance features. The C-PCF 1# has the minimum number of mode and the maximum bandwidth, while C-PCF 3# has the maximum number of mode and the minimum bandwidth among three fibers as shown in Table 1. Three C-PCFs all cover entire band of optical fiber communications. Take HE4,1 mode as example, we can find that C-PCF 3# has the best performance features such as dispersion value, dispersion difference ΔD in C band, confinement loss and nonlinear coefficient, with the values of 111.6 ps/(km nm), 1.5 ps/(km nm), 9.52×10−9 dB/ m and 1.544 W−1/km, respectively. As mentioned in Section 2, from CPCF 1# to C-PCF 3#, the design strategy is to change larger and larger number order of air-hole array into ring shaped core to hold the OAM modes. From C-PCF 1# to C-PCF 3#, the larger number order of airhole array, the lower values of dispersion, confinement loss, and nonlinear coefficient are, and more flat over wavelength range the dispersion is. The parts of reasons are the fact that larger number order of air-hole array provides larger area for OAM mode's transmission, subsequently bigger light intensity overlap factor, and hence less chance for mode energy leakage, and also larger effective mode area leads to smaller nonlinear coefficient. Therefore, C-PCF 3# may be the best choice among three. Now we focus our attention to C-PCF 3# to evaluate its transmission properties. Fig. 6 shows that in C-PCF 3# with r0=5.3 µm, mode intensity distributions of the OAM modes are well controlled within the fiber ring shaped region, which means that the fiber well supports the 11 orders of OAM modes with good mode quality and is suitable for
6. Conclusions In order to design an OAM supporting fiber for long distance and stable OAM modes transmission in MDM systems, we present a detail design strategy by considering 5 essential requirements to optimize the structure parameters of the C-PCF. It is an effective method to increase the number of the OAM modes by changing inner air-hole arrays into ring core area, and hence enlarging the radius of ring area r, by which we design an OAM family and find that C-PCF 1#, C-PCF 2#, and CPCF 3# can support 14, 26, and 42 OAM modes with high mode quality. By optimization we can find that for C-PCF 3# thickness of the ring should be within 1.6 µm to 2.2 µm to cope with the trade-off between possessing good OAM mode quality and avoiding excitation of the radial higher order modes. This optimal thickness range 1.6 µm to 2.2 µm corresponds to the inner radius 5.0 µm (ρVeff=15.30, ρ=0.69) to 5.6 µm (ρVeff=17.14, ρ=0.78) with the unchanged outer radius 7.2 µm (Veff=22.04), and also provides a large fabrication tolerance of 0.6 µm. Through the comparison among C-PCF 1#, C-PCF 2# and C-PCF 3#, we can find that C-PCF 1# and C-PCF 2# meet the 2nd, 3rd and 5th
Table 1 Characteristics of three C-PCFs.
Number of mode Bandwidth Δneff (l=3,at 1.55 µm) D (HE41, at 1.55 µm) ΔD (HE4,1 at C band(1530–1565 µm)) Confinement loss (HE4,1, at 1.55 µm) γ（HE4,1, at 1.55 µm）
C-PCF 1#
C-PCF 2#
C-PCF 3#
14 560 nm(1.25–1.81 µm) 4.345×10−4 269.1 ps/(km nm) 6.1 ps/(km nm) 1.24×10−6 dB/m 4.04 W−1/km
26 480 nm (1.25–1.73 µm) 2.538×10−3 128.7 ps/(km nm) 2.1 ps/(km nm) 1.77×10−8 dB/m 2.46 W−1/km
42 460 nm(1.25–1.71 µm) 2.067×10−3 111.6 ps/(km nm) 1.5 ps/(km nm) 9.52×10−9 dB/m 1.544 W−1/km
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Fig. 6. (a) Intensity distributions of part vortex modes, and (b) normalized intensity distributions of different vector modes in the high-index ring region.
Fig. 9. Nonlinear as a function of wavelength for different modes in the designed C-PCF 3# over C+L band. Fig. 7. Confienment loss as a function of wavelength for different vector modes in the designed C-PCF 3#.
propagation in C+L band. So C-PCF 3# is the optimal choice among three C-PCFs that can be used in MDM communications and other OAM applications in fibers. Funding National Natural Science Foundation of China (NSFC) (61571057, 61501214, 61527820, and 61575082); Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications) (IPOC2016ZZ04). Acknowledgements The authors wish to thank the anonymous reviewers for their valuable suggestions. References [1] L. Allen, M.W. Beijersbergen, R. Spreeuw, J. Woerdman, Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 45 (11) (1992) 8185–8189. [2] A.M. Yao, M.J. Padgett, Orbital angular momentum: origins, behavior and applications, Adv. Opt. Photon 3 (2011) 161–204. [3] D.G. Grier, A revolution in optical manipulation, Nature 424 (6950) (2003) 810–816. [4] A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, J. Laurat, A quantum memory for orbital angular momentum photonic qubits, Nat. Photon. 8 (3) (2014) 234–238. [5] S. Fürhapter, A. Jesacher, S. Bernet, M. Ritsch-Marte, Spiral phase contrast imaging in microscopy, Opt. Express 13 (3) (2005) 689–694.
Fig. 8. Dispersion as a function of wavelength for different vector modes in the designed C-PCF 3# over C+L band.
requirements in Section 2, while C-PCF 3# meets all 5 requirements. As a result that C-PCF 3# shows the good properties of flat dispersion (total dispersion variation less than 19.2 ps nm−1km−1), low confinement loss ( < 10−4 dB/m except for modes HE11,1, EH9,1, HE12,1 and EH10,1) and low nonlinear coefficient (≤2.18 W−1 m−1 for all supported OAM modes), which are required for longer distance OAM modes 65
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