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Numerical analysis of birefringence and coupling length on dual-core photonics crystal fiber with complex air holes
Optik 124 (2013) 5941–5944
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Numerical analysis of birefringence and coupling length on dual-core photonics crystal fiber with complex air holes Huimei He a,b,∗ , Li Wang a a b
College of Applied Sciences, Beijing University of Technology, Beijing 100124, China College of Science, Beijing University of Chemical Technology, Beijing 100029, China
a r t i c l e
i n f o
Article history: Received 21 November 2012 Accepted 18 April 2013
Keywords: Dual-core photonic crystal fibers Birefringence Coupling length
a b s t r a c t A type of high birefringence dual-core photonic crystal fibers (DC-PCFs) with a central row of elliptical air holes have been proposed. The transverse electric field vector distributions of the two modes are evaluated, the birefringence or coupling length with the different parameters is numerically analyzed based on finite-element method. The numerical results show values for the birefringence of 8.247 × 10−3 (for wavelength, = 1.5 m and lattice length, = 1.3 m), and for the coupling lengths about 3.1 mm and 2.6 mm ( = 1.5 m and = 1.5 m) to modes of x and y polarized, respectively. With the increasing of the air-filling fraction in proposed DC-PCF, the coupling length becomes longer and the birefringence becomes higher. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Since photonic crystal fibers (PCFs) which consist of a central defect region in a periodic array of air holes were fabricated [1], it has been extensively researched due to their unique properties, such as endlessly single-mode propagation capability [2,3], dispersion management [4], large mode area [5,6], high birefringence [7,8], and the remarkable flexibility in the design of PCF structure. One of these special characteristics of PCFs is their high birefringence. These highly birefringent PCFs have a great deal potential of practical use in high performance optical devices, such as fiber optical gyroscope, couplers, and polarization maintaining photonic devices. Many methods have been recommended to induce high birefringence in index-guiding PCFs, such as, using different air hole sizes or elliptical air holes in the cladding instead of circular ones [9–11], squeezing the air holes lattice [12,13], or using asymmetric core [14]. Destroying the symmetry of the PCFs structure is the key point of all methods to obtain higher birefringence in PCFs. Recently, dual-core photonic crystal fibers (DC-PCFs) which introduced two defects in the central region of PCFs have been widely researched and applied in optical fiber coupler [15–20]. For example, model interference in DC-PCFs has been studied by Ren [15]. The coupling characteristics of DC-PCFs and its application to a multiplexer–demultiplexer have been investigated by Saitoh [16]. The nonlinear phenomenon in DC-PCF couplers has been discussed
∗ Corresponding author at: College of Applied Sciences, Beijing University of Technology, Beijing 100124, China. E-mail address: [email protected] (H. He). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.04.124
by Salgueiro Jose and Kivshar Yuri [17]. Highly birefringent DC-PCFs can also be used as polarization splitters. For example, a DCPCFs polarization splitter is demonstrated by Zhang [21], and highly birefringent DC-PCFs polarization splitters with diverse structures are reported by Wen [22] and Chen [23]. In this paper, a new design of high birefringence DC-PCFs combining circular and elliptical air holes is proposed. The guided modes, birefringence and coupling length are analyzed numerically using the perpendicular wave module of the commercially available COMSOL MULTIPHYSICS software. 2. Structure and theory Fig. 1 shows the structure of our proposed DC-PCFs. In the cladding of DC-PCFs, there are triangular lattice of circular and elliptical air holes, and the central row is formed by the elliptical air holes. DC-PCFs cores which are separated by one air hole are constituted by the omission of three neighboring air holes. The structure of our proposed DC-PCFs is combining both asymmetries of the core and the cladding. The DC-PCFs is characterized by lattice length (), the circular air hole’s radius (r), the elliptical air hole’s half length along x direction (c) and along y direction (b). The elliptical ratio is c/b. The relationship between the elliptical and circular air holes is determined by b/r. The refractive index of the silica cladding and the air is 1.444 (at 1550 nm) and 1.0, respectively. In DC-PCFs, there are four modes, which include the even y x , E y . Their propagation x modes Eeven , Eeven and the odd modes Eodd odd y y constants and effective refractive indices are ˇex , ˇe , ˇox , ˇo , and nxe , y y x ne , no , no , respectively.
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H. He, L. Wang / Optik 124 (2013) 5941–5944
Fig. 1. Cross-section of the proposed DC-PCF.
According to the theory of mode coupling, the coupling of a DCPCF can be described by the use of the odd and even modes. The coupling length L is the distance along the fiber in which the optical power in the PCF will be entirely transferred from one fiber core to the other. In DC-PCF the coupling length is defined as [20] Li =
ˇei − ˇoi
=
2(nie − nio )
,
i = x, y
(1)
where is the operation wavelength, ˇei and ˇoi are the propagation constants, nie and nio are the effective indices, corresponding to even mode and odd mode in i-polarized state, respectively. In general, optical fiber birefringence is mostly described by the model birefringence, and its defined formula is y
B = neff − nxeff
(2) y
where nxeff and neff are the fundamental mode effective indices of two orthogonal polarization state. In this paper, the guided modes and the effective refractive indices of the proposed DC-PCFs were calculated using the perpendicular wave module of commercially available COMSOL MULTIPHYSICS software, and then the coupling length was obtained by Eq. (1). The even and odd transmission modes of the proposed DC-PCFs are evaluated, their x-polarized electric field distributions and vectors are shown in Fig. 2, where the = 1.5 m, r = 0.5 m, b = 0.5 m and c = 0.375 m. 3. Results The influence of the lattice length () on the fiber birefringence and coupling length, with fixed r = 0.5 m, b = 0.5 m and c = 0.375 m, is illustrated in Figs. 3 and 4, respectively.
Fig. 2. Amplitude of the electric field for (a) even mode and (b) odd mode of the DC-PCF. Arrow diagrams show corresponding electric field vectors.
Fig. 3. Birefringence of the fundamental mode as a function of wavelength with different (r = 0.5 m, b = 0.5 m, c = 0.375 m).
Fig. 3 shows the relationship between the birefringence and operating wavelength with different lattice length = 1.3, 1.5, 1.6, 1.9 and 2.2 m. For certain lattice length , the birefringence increases as the wavelength increases. When keeping constant, the values of the birefringence increases as decrease, it is because that a larger will make the DC-PCFs structure more asymmetric, and then it results the effective refractive index in x-polarized reduced less than in y-polarized. When = 1.3 m, the birefringence is 8.247 × 10−3 at 1.55 m operating wavelength, which is higher than that of previous structure [24]. In Fig. 4, we have simulated the coupling length L for both polarization orientations with different lattice length = 1.5, 1.6, 1.9 and 2.2 m. The coupling length increases as the wavelength decreases, because the mode field extends a little to the cladding region and the model coupling from one core to another becomes difficult with the wavelength decreased. Moreover, the coupling length increases when is squeezed, it is because that the core compressed degree in horizontal direction is higher than vertical compression with the decreases of , so the mode field is extending in vertical direction, and the model coupling between two cores becomes difficult. When keeping and constant, the coupling lengths of xpolarized and y-polarized are different, because there is different of the effective refractive index between the even and odd mode. When = 1.5 m, the coupling lengths are 3.1 mm and 2.63 mm at wavelength = 1.55 m for x-polarized and y-polarized state, respectively. To illustrate the relative of birefringence and coupling length to the central row air holes of our proposed fiber with the parameters = 1.5 m, r = 0.5 m when b/r = 1, the birefringence and coupling length versus the wavelength with different the elliptical ratio c/b are shown in Figs. 5 and 6, respectively. In Fig. 5, the birefringence
Fig. 4. Coupling length as a function of wavelength with different (r = 0.5 m, b = 0.5 m, c = 0.375 m).
H. He, L. Wang / Optik 124 (2013) 5941–5944
Fig. 5. Birefringence of the fundamental mode as a function of wavelength with different c/b ( = 1.5 m, r = 0.5 m, b/r = 1).
Fig. 6. Coupling length as a function of wavelength with different c/b ( = 1.5 m, r = 0.5 m, b/r = 1).
of the DC-PCFs with c/b = 0.58, 0.86 and 1.00 has nearly invariability, and this birefringence is better than that of the DC-PCFs with c/b = 0.3. The coupling length L for both polarization orientations is illustrated in Fig. 6. It can be clearly seen that the coupling length increases as the value of c/b increases when keeping constant, because of the mode field is compressing along x direction and then makes the model coupling become more difficult. Figs. 7 and 8 show the simulation results of the birefringence and coupling length versus radius (r) of the circular air hole of DC-PCFs at different c/b = 0.6, 0.8 and 1.0 with b/r = 1, = 1.5 m and the operating wavelength = 1.55 m. The birefringence and the coupling length increase as the radius of the circular air hole increases. DC-PCFs with bigger circle air holes result in higher birefringence. This is because the compression degree of the core in
Fig. 7. Birefringence of the fundamental mode as functions of r, when = 1.5, = 1.55 m and b/r = 1.
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Fig. 8. Coupling length as functions of r, when = 1.5, = 1.55 m and b/r = 1.
vertical direction is higher than that in horizontal direction when the air hole radius increase, therefore the asymmetry effect to the light field correspond to the birefringence increase, such as show in Fig. 9. The even mode electric field amplitude for (a) r = 0.4 m and (b) r = 0.7 m of the DC-PCF with = 1.5, = 1.55 m and b/r = 1 are shown in Fig. 9. Moreover, with r increase, the restriction of mode field from the clad air holes is enhanced, therefore, the coupling between two cores becomes difficult and DC-PCFs with smaller circle air holes result in shorter coupling lengths. In Figs. 5–8, the value of the elliptical ratio c/b = 1 means the elliptical air hole becomes circular one, and the structure of our proposed DC-PCFs becomes the previous structure proposed by Fu et al. [25]. Figs. 5 and 7 shows that the birefringence of our DCPCFs is similar to the previous structure, Figs. 6 and 8 show that the coupling lengths of our DC-PCFs are shorter than that of previous structures. The influence of the coefficient b/r and the elliptical ratio c/b on the fiber birefringence and coupling length are illustrated in Figs. 10 and 11, respectively. Fig. 10 shows the simulation results of the birefringence versus the elliptical ratio c/b at different the value of b/r with structure parameters r = 0.5 m, = 1.5 m and = 1.55 m. From Fig. 10, it can be seen that the birefringence increases with the c/b increase for the value of b/r < 1.0, except when the value of c/b is below 0.2. When the c/b is beyond 0.5, the birefringence keeps near stability as c/b increases for the value of c/b = 1.0, which is as same as the result shown in Fig. 5. Fig. 10 shows simultaneity, that the birefringence increases with the value of b/r decreases.
Fig. 9. Even mode field distributions of two circular air hole radius, (a) r = 0.4 m and (b) r = 0.7 m when = 1.5, = 1.55 m and b/r = 1.
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H. He, L. Wang / Optik 124 (2013) 5941–5944
(c/b = 1). With the increase of air-filling fraction, the coupling length as well as the birefringence increases. In conclusion, the presented structure combines high birefringence and short coupling length perfectly, which provide a great application value in high performance optical devices, such as optical fiber coupler or polarization splitters. Acknowledgement This work was supported by the Science and Technology Project of Beijing Municipal Education Commission (Grant Nos. Km200910005019 and Kz201110005010). References Fig. 10. Birefringence of the fundamental mode as functions of elliptical ratio (c/b), when r = 0.5 m, = 1.5 m, = 1.55 m.
Fig. 11. Coupling length as functions of elliptical ratio (c/b), when r = 0.5 m, = 1.5 m, = 1.55 m.
Fig. 11 shows the relationship between coupling length and c/b with different b/r = 0.50, 0.75, and 1.00, with fixed = 1.5 m and the wavelength = 1.55 m. The results show that the coupling length increases with the value of c/b or b/r increases. Thus, the central row of elliptical air holes (c/b < 1) having a small size (small b/r) is beneficial to achieve the low coupling length, and the light-propagation coupling between cores becomes easy. The area of air hole is A, and the effective area of air holes is A/2 . With the increasing A/2 , the air filling fraction become large. When decreases in Figs. 3 and 4, the c/b increases in Figs. 5, 6, 10 and 11, or the r increases in Figs. 7 and 8, the A/2 of our proposed DC-PCFs increases. By analyzing Figs. 3–8, 10 and 11, one will find that, with the increasing of the air filling fraction, the coupling length and the birefringence increase simultaneously. The reason is that with the air filling fraction increases, the light field becomes more asymmetric and the mode field restricted from the clad air holes increases, such as shown in Fig. 9. 4. Conclusion A novel dual-core photonic crystal fiber with central row of elliptical air holes is demonstrated in this paper. Relationship among the birefringence, coupling length and various parameters is analyzed numerically. The results show that, when = 1.3 m, r = 0.5 m, b = 0.5 m and c = 0.375 m the birefringence at 1.55 m is almost up to 1 × 10−2 . The coupling length of our proposed structure which has a central row of elliptical air holes (c/b < 1) is shorter than that of structures which have a central row of circular air holes
[1] J.C. Knight, T.A. Birks, P.S.J. Russell, D.M. Atkin, All-silica single-mode optical fiber with photonic crystal cladding, Opt. Lett. 21 (1996) 1547–1549. [2] T.A. Birks, J.C. Knight, P.St.J. Russell, Endlessly single-mode photonic crystal fiber, Opt. Lett. 22 (1997) 961–963. [3] K. Saitoh, Y. Tsuchida, M. Koshiba, N.A. Mortensen, Endlessly single-mode holey fibers: the influence of core design, Opt. Express 13 (2005) 10833–10839. [4] K. Saitoh, M. Koshiba, T. Hasegawa, E. Sasaoka, Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion, Opt. Express 11 (2003) 843–852. [5] J.C. Baggett, T.M. Monro, K. Furusawa, D.J. Richardson, Comparative study of large-mode holey and conventional fibers, Opt. Lett. 26 (2001) 1045–1047. [6] W. Belardi, G. Bouwmans, L. Provino, V. Pureur, M. Douay, A large mode area elliptical hollow photonic crystal fiber, in: Optical Fiber Communication Conference (OFC), Optical Society of America, 2006, p. OFC3. [7] Y.-S. Sun, Y.-F. Chau, H.-H. Yeh, L.-F. Shen, T.-J. Yang, D.P. Tsai, High birefringence photonic crystal fiber with a complex unit cell of asymmetric elliptical air hole cladding, Appl. Opt. 46 (2007) 5276–5281. [8] D. Chen, L. Shen, Highly birefringent elliptical-hole photonic crystal fibers with double defect, J. Lightwave Technol. 25 (2007) 2700–2705. [9] A. Ortigosa-Blanch, J.C. Knight, W.J. Wadsworth, J. Arriaga, B.J. Mangan, T.A. Birks, P.S. Russell, Highly birefringent photonic crystal fibers, Opt. Lett. 25 (2000) 1325–1327. [10] M.J. Steel, J.R.M. Osgood, Elliptical-hole photonic crystal fibers, Opt. Lett. 26 (2001) 229–231. [11] L. Wang, D. Yang, Highly birefringent elliptical-hole rectangular lattice photonic crystal fibers with modified air holes near the core, Opt. Express 15 (2007) 8892–8897. [12] Y. Yue, G.Y. Kai, Z. Wang, T.T. Sun, L. Jin, Y.F. Lu, C.S. Zhang, L.G. Liu, Y. Li, Y.G. Liu, S.Z. Yuan, X.Y. Dong, Highly birefringent elliptical-hole photonic crystal fiber with squeezed hexagonal lattice, Opt. Lett. 32 (2007) 469–471. [13] F. Beltran-Mejia, G. Chesini, E. Silvestre, Ultrahigh-birefringent squeezed lattice photonic crystal fiber with rotated elliptical air holes, Opt. Lett. 32 (2010) 544–546. [14] T.P. Hansen, J. Broeng, S.E.B. Libori, E. Knudsen, A. Bjarklev, J.R. Jensen, H. Simonsen, Highly birefringent index-guiding photonic crystal fibers, IEEE Photon. Technol. Lett. 13 (2001) 588–590. [15] G.B. Ren, Z. Wang, S.Q. Lou, S.S. Jian, Modal interference in dual-core photonic crystal fibers, Acta Phys. Sinica 53 (2004) 2600–2607. [16] K. Saitoh, S. Yuichiro, K. Massanori, Coupling characteristics of dual-core photonic crystal fiber couplers, Opt. Express 11 (2003) 3188–3195. [17] R. Salgueiro Jose, S. Kivshar Yuri, Nonlinear dual-core photonic crystal fiber couplers, Opt. Lett. 30 (2005) 1858–1860. [18] B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, A.H. Greenaway, Experimental study of dualcore photonic crystal fibre, Electron. Lett. 36 (2000) 1358–1359. [19] F. Fogli, L. Saccomandi, P. Bassi, G. Bellanca, S. Trillo, Full vectorial BPM modeling of indexguiding photonic crystal fibers and couplers, Opt. Express 10 (2002) 54–59. [20] B.H. Lee, J.B. Eom, J. Kim, D.S. Moon, U.-C. Paek, G.-H. Yang, Photonic crystal fiber coupler, Opt. Lett. 27 (2002) 812–814. [21] L. Zhang, C.X. Yang, Polarization splitter based on photonic crystal fibers, Opt. Express 11 (2003) 1015–1019. [22] K. Wen, J.Y. Wang, R. Wang, Polarization splitter based on two-core rectangularlattice photonic crystal fibers, Chin. J. Quantum Electron. 25 (2008) 505–508. [23] M.Y. Chen, B. Sun, Y.K. Zhang, X.X. Fu, Design of broadband polarization splitter based on partial coupling in square-lattice photonic-crystal fiber, Appl. Opt. 49 (2010) 3042–3048. [24] F. Shi, Y. Wu, M. Li, Y. Zhao, L. Zhao, Highly birefringent two-mode photonic crystal fibers with near-zero flattened dispersion, IEEE Photon. J. 3 (2011) 1181–1188. [25] B. Fu, S.-G. Li, Y.-Y. Yao, L. Zhang, M.-Y. Zhang, Design of two kinds of dualcore high birefringence and high coupling degree photonic crystal fibers, Opt. Commun. 283 (2010) 4064–4068.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Numerical analysis of birefringence and coupling length on dual-core photonics crystal fiber with complex air holes Huimei He a,b,∗ , Li Wang a a b
College of Applied Sciences, Beijing University of Technology, Beijing 100124, China College of Science, Beijing University of Chemical Technology, Beijing 100029, China
a r t i c l e
i n f o
Article history: Received 21 November 2012 Accepted 18 April 2013
Keywords: Dual-core photonic crystal fibers Birefringence Coupling length
a b s t r a c t A type of high birefringence dual-core photonic crystal fibers (DC-PCFs) with a central row of elliptical air holes have been proposed. The transverse electric field vector distributions of the two modes are evaluated, the birefringence or coupling length with the different parameters is numerically analyzed based on finite-element method. The numerical results show values for the birefringence of 8.247 × 10−3 (for wavelength, = 1.5 m and lattice length, = 1.3 m), and for the coupling lengths about 3.1 mm and 2.6 mm ( = 1.5 m and = 1.5 m) to modes of x and y polarized, respectively. With the increasing of the air-filling fraction in proposed DC-PCF, the coupling length becomes longer and the birefringence becomes higher. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Since photonic crystal fibers (PCFs) which consist of a central defect region in a periodic array of air holes were fabricated [1], it has been extensively researched due to their unique properties, such as endlessly single-mode propagation capability [2,3], dispersion management [4], large mode area [5,6], high birefringence [7,8], and the remarkable flexibility in the design of PCF structure. One of these special characteristics of PCFs is their high birefringence. These highly birefringent PCFs have a great deal potential of practical use in high performance optical devices, such as fiber optical gyroscope, couplers, and polarization maintaining photonic devices. Many methods have been recommended to induce high birefringence in index-guiding PCFs, such as, using different air hole sizes or elliptical air holes in the cladding instead of circular ones [9–11], squeezing the air holes lattice [12,13], or using asymmetric core [14]. Destroying the symmetry of the PCFs structure is the key point of all methods to obtain higher birefringence in PCFs. Recently, dual-core photonic crystal fibers (DC-PCFs) which introduced two defects in the central region of PCFs have been widely researched and applied in optical fiber coupler [15–20]. For example, model interference in DC-PCFs has been studied by Ren [15]. The coupling characteristics of DC-PCFs and its application to a multiplexer–demultiplexer have been investigated by Saitoh [16]. The nonlinear phenomenon in DC-PCF couplers has been discussed
∗ Corresponding author at: College of Applied Sciences, Beijing University of Technology, Beijing 100124, China. E-mail address: [email protected] (H. He). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.04.124
by Salgueiro Jose and Kivshar Yuri [17]. Highly birefringent DC-PCFs can also be used as polarization splitters. For example, a DCPCFs polarization splitter is demonstrated by Zhang [21], and highly birefringent DC-PCFs polarization splitters with diverse structures are reported by Wen [22] and Chen [23]. In this paper, a new design of high birefringence DC-PCFs combining circular and elliptical air holes is proposed. The guided modes, birefringence and coupling length are analyzed numerically using the perpendicular wave module of the commercially available COMSOL MULTIPHYSICS software. 2. Structure and theory Fig. 1 shows the structure of our proposed DC-PCFs. In the cladding of DC-PCFs, there are triangular lattice of circular and elliptical air holes, and the central row is formed by the elliptical air holes. DC-PCFs cores which are separated by one air hole are constituted by the omission of three neighboring air holes. The structure of our proposed DC-PCFs is combining both asymmetries of the core and the cladding. The DC-PCFs is characterized by lattice length (), the circular air hole’s radius (r), the elliptical air hole’s half length along x direction (c) and along y direction (b). The elliptical ratio is c/b. The relationship between the elliptical and circular air holes is determined by b/r. The refractive index of the silica cladding and the air is 1.444 (at 1550 nm) and 1.0, respectively. In DC-PCFs, there are four modes, which include the even y x , E y . Their propagation x modes Eeven , Eeven and the odd modes Eodd odd y y constants and effective refractive indices are ˇex , ˇe , ˇox , ˇo , and nxe , y y x ne , no , no , respectively.
5942
H. He, L. Wang / Optik 124 (2013) 5941–5944
Fig. 1. Cross-section of the proposed DC-PCF.
According to the theory of mode coupling, the coupling of a DCPCF can be described by the use of the odd and even modes. The coupling length L is the distance along the fiber in which the optical power in the PCF will be entirely transferred from one fiber core to the other. In DC-PCF the coupling length is defined as [20] Li =
ˇei − ˇoi
=
2(nie − nio )
,
i = x, y
(1)
where is the operation wavelength, ˇei and ˇoi are the propagation constants, nie and nio are the effective indices, corresponding to even mode and odd mode in i-polarized state, respectively. In general, optical fiber birefringence is mostly described by the model birefringence, and its defined formula is y
B = neff − nxeff
(2) y
where nxeff and neff are the fundamental mode effective indices of two orthogonal polarization state. In this paper, the guided modes and the effective refractive indices of the proposed DC-PCFs were calculated using the perpendicular wave module of commercially available COMSOL MULTIPHYSICS software, and then the coupling length was obtained by Eq. (1). The even and odd transmission modes of the proposed DC-PCFs are evaluated, their x-polarized electric field distributions and vectors are shown in Fig. 2, where the = 1.5 m, r = 0.5 m, b = 0.5 m and c = 0.375 m. 3. Results The influence of the lattice length () on the fiber birefringence and coupling length, with fixed r = 0.5 m, b = 0.5 m and c = 0.375 m, is illustrated in Figs. 3 and 4, respectively.
Fig. 2. Amplitude of the electric field for (a) even mode and (b) odd mode of the DC-PCF. Arrow diagrams show corresponding electric field vectors.
Fig. 3. Birefringence of the fundamental mode as a function of wavelength with different (r = 0.5 m, b = 0.5 m, c = 0.375 m).
Fig. 3 shows the relationship between the birefringence and operating wavelength with different lattice length = 1.3, 1.5, 1.6, 1.9 and 2.2 m. For certain lattice length , the birefringence increases as the wavelength increases. When keeping constant, the values of the birefringence increases as decrease, it is because that a larger will make the DC-PCFs structure more asymmetric, and then it results the effective refractive index in x-polarized reduced less than in y-polarized. When = 1.3 m, the birefringence is 8.247 × 10−3 at 1.55 m operating wavelength, which is higher than that of previous structure [24]. In Fig. 4, we have simulated the coupling length L for both polarization orientations with different lattice length = 1.5, 1.6, 1.9 and 2.2 m. The coupling length increases as the wavelength decreases, because the mode field extends a little to the cladding region and the model coupling from one core to another becomes difficult with the wavelength decreased. Moreover, the coupling length increases when is squeezed, it is because that the core compressed degree in horizontal direction is higher than vertical compression with the decreases of , so the mode field is extending in vertical direction, and the model coupling between two cores becomes difficult. When keeping and constant, the coupling lengths of xpolarized and y-polarized are different, because there is different of the effective refractive index between the even and odd mode. When = 1.5 m, the coupling lengths are 3.1 mm and 2.63 mm at wavelength = 1.55 m for x-polarized and y-polarized state, respectively. To illustrate the relative of birefringence and coupling length to the central row air holes of our proposed fiber with the parameters = 1.5 m, r = 0.5 m when b/r = 1, the birefringence and coupling length versus the wavelength with different the elliptical ratio c/b are shown in Figs. 5 and 6, respectively. In Fig. 5, the birefringence
Fig. 4. Coupling length as a function of wavelength with different (r = 0.5 m, b = 0.5 m, c = 0.375 m).
H. He, L. Wang / Optik 124 (2013) 5941–5944
Fig. 5. Birefringence of the fundamental mode as a function of wavelength with different c/b ( = 1.5 m, r = 0.5 m, b/r = 1).
Fig. 6. Coupling length as a function of wavelength with different c/b ( = 1.5 m, r = 0.5 m, b/r = 1).
of the DC-PCFs with c/b = 0.58, 0.86 and 1.00 has nearly invariability, and this birefringence is better than that of the DC-PCFs with c/b = 0.3. The coupling length L for both polarization orientations is illustrated in Fig. 6. It can be clearly seen that the coupling length increases as the value of c/b increases when keeping constant, because of the mode field is compressing along x direction and then makes the model coupling become more difficult. Figs. 7 and 8 show the simulation results of the birefringence and coupling length versus radius (r) of the circular air hole of DC-PCFs at different c/b = 0.6, 0.8 and 1.0 with b/r = 1, = 1.5 m and the operating wavelength = 1.55 m. The birefringence and the coupling length increase as the radius of the circular air hole increases. DC-PCFs with bigger circle air holes result in higher birefringence. This is because the compression degree of the core in
Fig. 7. Birefringence of the fundamental mode as functions of r, when = 1.5, = 1.55 m and b/r = 1.
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Fig. 8. Coupling length as functions of r, when = 1.5, = 1.55 m and b/r = 1.
vertical direction is higher than that in horizontal direction when the air hole radius increase, therefore the asymmetry effect to the light field correspond to the birefringence increase, such as show in Fig. 9. The even mode electric field amplitude for (a) r = 0.4 m and (b) r = 0.7 m of the DC-PCF with = 1.5, = 1.55 m and b/r = 1 are shown in Fig. 9. Moreover, with r increase, the restriction of mode field from the clad air holes is enhanced, therefore, the coupling between two cores becomes difficult and DC-PCFs with smaller circle air holes result in shorter coupling lengths. In Figs. 5–8, the value of the elliptical ratio c/b = 1 means the elliptical air hole becomes circular one, and the structure of our proposed DC-PCFs becomes the previous structure proposed by Fu et al. [25]. Figs. 5 and 7 shows that the birefringence of our DCPCFs is similar to the previous structure, Figs. 6 and 8 show that the coupling lengths of our DC-PCFs are shorter than that of previous structures. The influence of the coefficient b/r and the elliptical ratio c/b on the fiber birefringence and coupling length are illustrated in Figs. 10 and 11, respectively. Fig. 10 shows the simulation results of the birefringence versus the elliptical ratio c/b at different the value of b/r with structure parameters r = 0.5 m, = 1.5 m and = 1.55 m. From Fig. 10, it can be seen that the birefringence increases with the c/b increase for the value of b/r < 1.0, except when the value of c/b is below 0.2. When the c/b is beyond 0.5, the birefringence keeps near stability as c/b increases for the value of c/b = 1.0, which is as same as the result shown in Fig. 5. Fig. 10 shows simultaneity, that the birefringence increases with the value of b/r decreases.
Fig. 9. Even mode field distributions of two circular air hole radius, (a) r = 0.4 m and (b) r = 0.7 m when = 1.5, = 1.55 m and b/r = 1.
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(c/b = 1). With the increase of air-filling fraction, the coupling length as well as the birefringence increases. In conclusion, the presented structure combines high birefringence and short coupling length perfectly, which provide a great application value in high performance optical devices, such as optical fiber coupler or polarization splitters. Acknowledgement This work was supported by the Science and Technology Project of Beijing Municipal Education Commission (Grant Nos. Km200910005019 and Kz201110005010). References Fig. 10. Birefringence of the fundamental mode as functions of elliptical ratio (c/b), when r = 0.5 m, = 1.5 m, = 1.55 m.
Fig. 11. Coupling length as functions of elliptical ratio (c/b), when r = 0.5 m, = 1.5 m, = 1.55 m.
Fig. 11 shows the relationship between coupling length and c/b with different b/r = 0.50, 0.75, and 1.00, with fixed = 1.5 m and the wavelength = 1.55 m. The results show that the coupling length increases with the value of c/b or b/r increases. Thus, the central row of elliptical air holes (c/b < 1) having a small size (small b/r) is beneficial to achieve the low coupling length, and the light-propagation coupling between cores becomes easy. The area of air hole is A, and the effective area of air holes is A/2 . With the increasing A/2 , the air filling fraction become large. When decreases in Figs. 3 and 4, the c/b increases in Figs. 5, 6, 10 and 11, or the r increases in Figs. 7 and 8, the A/2 of our proposed DC-PCFs increases. By analyzing Figs. 3–8, 10 and 11, one will find that, with the increasing of the air filling fraction, the coupling length and the birefringence increase simultaneously. The reason is that with the air filling fraction increases, the light field becomes more asymmetric and the mode field restricted from the clad air holes increases, such as shown in Fig. 9. 4. Conclusion A novel dual-core photonic crystal fiber with central row of elliptical air holes is demonstrated in this paper. Relationship among the birefringence, coupling length and various parameters is analyzed numerically. The results show that, when = 1.3 m, r = 0.5 m, b = 0.5 m and c = 0.375 m the birefringence at 1.55 m is almost up to 1 × 10−2 . The coupling length of our proposed structure which has a central row of elliptical air holes (c/b < 1) is shorter than that of structures which have a central row of circular air holes
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