Open Access
Add to BibSonomyAbstract
We numerically investigate the polarization-dependent coupling in dual-core photonic crystal fibers (PCFs) selectively filled with gold wires in air holes. It is shown that the even and odd supermodes exhibit significantly different dispersion and loss when one gold wire is filled in between two cores. The enhanced birefringence and polarization-dependent attenuation of the supermodes support the separation of two orthogonally polarized components as well as polarization-dependent transmissions with a high degree of polarization in gold-filled dual core PCFs. Our study suggests that a gold-filled PCF coupler is a new possibility for applications on polarization beam splitters and polarizers.
©2013 Optical Society of America
1. Introduction
Photonic crystal fibers (PCFs) are characterized by guiding light in a central defect of a lattice that consists of air holes parallel to the propagation axis due to the total internal reflection and the existence of a bandgap [1]. Compared with conventional fibers, PCFs have advantages such as endless single-mode operation, ultrahigh nonlinearity, and controllable dispersion [2]. Additionally, the polarization properties of PCFs can be manipulated by filling the cladding holes and hollow core with polymer [3], oil [4], or liquid crystal [5,6]. By manipulating the structure, some PCF devices with specific structures have been fabricated and used to make light couplers [7,8], filters [9], and amplifiers [10]. In recent years, metal nanowire-filled PCFs fabricated by fiber drawing [11] and high-temperature pressure-cell [12] methods have been demonstrated with significant polarization-dependent transmissions, which have been suggested for use as sensors, polarizers, and in-fiber devices [13–17]. Furthermore, a few reports have discussed the optical characteristics of multi-core PCFs integrated with metal wires in which reduction of coupling length and enhancement of energy transfer have been realized by resonance between surface plasmon polariton (SPP) modes with guided-core modes [18]. However, the polarization-dependent coupling and transmission properties in gold-filled dual-core PCF couplers rarely have been reported.
In this paper we numerically study the polarization-dependent coupling in dual-core PCFs selectively filled with gold wires into air holes based on the supermode theory and the finite element method (FEM). First, we compare the dispersion and loss of dual-core PCFs with that of a single-core PCF. The even and odd supermodes show significantly different resonance coupling characteristics. High birefringence is observed in gold-filled dual-core PCFs. Second, we show that linearly polarized components with a high degree of polarization can be separated by propagating in a gold-filled PCF coupler with special length requirements, and high polarization-dependent transmissions can also be observed in a PCF coupler filled with two adjacent gold wires.
2. Dispersion and loss of PCFs filled with gold wires
Figure 1(a) shows a schematic illustration of a dual-core PCF filled with gold microwire in the background silica, into which broadband light is launched and then propagated longitudinally with periodically coupling between two co-parallel cores. Figures 1(b)–1(d) depict the configurations of a single-core PCF filled with a gold wire close to the core as well as dual-core PCFs filled with gold wires between two cores at the intervals of D=2Λ and 3Λ, respectively. Λ is the spacing distance of the air holes. All of the PCF samples share the same parameters such as air hole diameter d=1 μm and spacing distance Λ=2.3 μm, and only the fundamental core-guided modes are discussed here. We perform finite element simulations using the Sellmeier equation for the dispersion of silica [19] and the Drude–Lorentz model for the gold wire [20].
Fig. 1 (a) Schematic illustration of PCF with gold microwire; (b)–(d) configurations of single-core PCF filled with a gold wire close to the core and dual-core PCFs filled with gold wires between two cores at the intervals of D=2Λ and D=3Λ, respectively.
We first discuss the dispersion and loss of a gold-filled single-core PCF shown in Fig. 1(b). Figure 2(a) shows the dependence of effective index and loss on wavelength of x- (the blue curves) and y-polarized (the red curves) core modes. The dashed, dotted, and gray curves in the dispersion diagram correspond to silica, the fundamental core mode in an unfilled PCF, and SPP modes [21] with different orders (m=1, 2, 3) excited on an isolated gold wire embedded in silica surrounded by an air-hole lattice, respectively. The inset on the dispersion diagram is a close-up view of the anti-crossing points of the m=3 SPP mode and the unfilled PCF core mode. Extremely enhanced losses are observed at specific wavelengths where the unfilled PCF core mode and the SPP modes cross. Since phase matching between the core modes and the SPP modes occurs at the anti-crossing points, the core-guided light is able to resonantly couple to the leaky SPP modes on the surface of the gold wire. The effective indices of x- and y-polarized core modes continuously decrease as the wavelength increases, except in the region where resonance coupling with the m=2 SPP mode occurs; there, each mode splits into two branches (the upper and lower branches). This phenomenon can be explained by leaky mode coupling [22] that the two branches perform by completely coupling with the m=2 SPP mode at the phase-matching point, while the core modes perform incomplete coupling with the SPP modes at other phase-matching points [16]. In addition, obvious birefringence between the x- and y-polarized core modes is observed as the wavelength increases.
Fig. 2 Dispersion and loss spectra of x- (blue lines) and y-polarized (red lines) core modes in (a) single-core and (b) dual-core PCFs (short dashed lines: even supermodes; solid lines: odd supermodes) filled with a gold wire. The dashed, dotted, and thin (gray) lines represent the silica, the fundamental guided modes in the unfilled PCFs, and the SPP modes (m=1, 2, 3) on an isolated gold wire embedded in silica surrounded by air-hole lattice, respectively. The insets are the close-up view of m=3 anti-crossing points.
Next we study the dispersion and loss of a gold-filled dual core PCF with D=2Λ [Fig. 1(c)], and we then compare them with that of the single-core PCF. Figure 2(b) shows the dispersion and loss spectra of x- (the blue curves) and y-polarized (the red curves) guided modes. Here, the dashed and solid curves represent the even and odd supermodes, and the dotted curves in the dispersion diagram represent the core-guided even and odd supermodes in an unfilled dual-core PCF, respectively. We can see that the even and odd supermodes exhibit quite dissimilar dispersion and loss. In the vicinity of the anti-crossing point (~766 nm) for the core-guided supermodes and the m=3 SPP mode, the losses of even supermodes are 10 times greater than that of odd supermodes, and one can observe the dispersion curves of even supermodes depict obvious perturbation [inset in Fig. 2(b)]. This is because only the core-guided even supermodes meet phase matching with the m=3 SPP mode. Figures 3(a) and 3(b) show the axial Poynting vector distributions of the even and odd supermodes as well as the SPP modes on an isolated gold wire embedded in silica surrounded by an air-holes lattice at λ=766 nm and 1076 nm, respectively. The arrows indicate the polarization orientation of the electric fields. The m=3 SPP mode has two degenerate states, and the opposite lobes of each state are in-phase, while the core-guided even supermodes possess a similar property, and the core-guided odd supermodes have phase difference π between the fields in two cores. Therefore, only the core-guided even supermodes meet the phase-matching condition at this anti-crossing point and extend onto the surface of the gold wire, resulting in great enhancements on the loss curves of even supermodes. We can also find that the opposite lobes of the m=2 SPP mode are out-of-phase and only the core-guided odd supermodes meet phase-matching conditions (λ=1076 nm for y-polarization; λ=1082 nm for x-polarization), which give rise to the appearance of extremely enhanced loss and splitting of dispersion curves of the odd supermodes. Additionally, the birefringence between the x- and y-polarized supermodes is obviously enhanced for wavelengths beyond 1000 nm, which provides an effective way to achieve polarization-dependent coupling in dual-core PCF couplers.
Fig. 3 Axial Poynting vector distributions of the even and odd supermodes as well as the SPP modes at (a) 766 nm and (b) 1076 nm. The SPP modes are excited on an isolated gold wire embedded in silica surrounded by air-hole lattice. The axial Poynting vectors are displayed as log10(Sz). The arrows indicate the instantaneous electric-field orientation.
To gain further insight into the optical properties of a gold-filled dual-core PCF, we discuss the dual-core PCF filled with two gold wires between two cores shown in Fig. 1(d) (D=3Λ). Figures 4(a) and 4(b) show the dispersion and loss spectra, respectively. It is noted that both the even and odd supermodes of x- and y- polarized components perform resonance coupling with the m=2, and 3 SPP modes, and four peaks appear on the loss curves in the vicinity of 1076 nm [inset in Fig. 4(b)]. That means the two degenerate states of the m=2 SPP mode will split into four nondegenerate hybridized SPP modes (bonding and anti-bonding modes) when two adjacent gold wires are integrated into the PCF as in Ref [17], and their dispersion curves cross with the core-guided supermodes at different wavelengths. However, each hybridized SPP mode can only produce resonance coupling with one core-guided supermode due to the relationship between their polarization states, resulting in the separation of loss peaks. The x- and y-polarized core-guided even supermodes produce resonance coupling with the bonding SPP modes at λ=1080 nm and 1100 nm [C and D points in Fig. 4 (b)], and the core-guided odd supermodes produce resonance coupling with the anti-bonding SPP modes at λ=1040 nm and 1075 nm [A and B points in Fig. 4 (b)], respectively. In addition, two orthogonally polarized components exhibit more distinct birefringence when at wavelengths beyond 1100 nm, where the difference [|Re(neeff)-Re(noeff)|] of the x-polarized component is much greater than that of the y-polarized one. The marked difference of dispersion and loss between x- and y-polarized supermodes in dual-core PCFs filled with two gold wires will greatly affect the energy coupling between two cores, suggesting a new way to enrich the applications of special PCF couplers.
Fig. 4 (a) Dispersion and (b) loss spectra of x- (blue lines) and y-polarized (red lines) supermodes in a D=3Λ dual-core PCF filled with two gold wires into the air holes between two cores (dashed lines: even supermodes; solid lines: odd supermodes). Inset: close-up view of loss spectrum. (A, B, C, D correspond to the peaks at λ=1040 nm, 1075 nm, 1080 nm, and 1110 nm, respectively.)
3. Polarization-dependent coupling in gold-filled PCF couplers
The supermode theory can be used to introduce the periodic coupling between two co-parallel cores in a dual-core PCF coupler. The light fields in two cores can be represented in the even [ue(z) = u1(z) + u2(z)] and odd [uo(z) = u1(z)-u2(z)] supermodes with independent evolution equation as
where, u1(z) and u2(z) are the light fields in separated cores, and where βe and βo are the propagation constant of the even and odd supermodes that can be obtained by finite element simulation. The evolution of light fields in two cores can be solved numerically as u1,2(z)=(ue(z) ± uo(z))/2. For initial light fields u1(0)=1, u2(0)=0, and PCF coupler length L, the output light fields in the two cores are u1,2(z)=[exp(iβeL)±exp(iβoL)]/2. The coupling length is relate to βe and βo, and can be expressed as Lc=π/|βe-βo|. The Lc values of x- (Lcx) and y-polarized (Lcy) components are very close to each other in an unfilled dual-core PCF coupler. Filling gold wire in the air hole between two cores can significantly enhance the birefringence and polarized-dependent loss of guided supermodes, which thus leads to obvious differences of coupling length between the two orthogonal components.Table 1 gives the coupling lengths Lcx and Lcy in gold-filled PCF couplers [Figs. 1(c) and 1(d)] at λ=1330 nm and 1550 nm. It is clearly noticed that in addition to the enhancement of coupling efficiency as shown in Ref [18], more importantly integrating gold wire between two cores can also increase the coupling length difference between x- and y- polarized components. Particularly, the ratio of coupling lengths of the y- and x-polarized components (Lcy/Lcx) increases from 1.50 to 4.03 at λ=1330 nm when increasing with one more gold wire. Distinct coupling of the lengths of two orthogonally polarized components has potential applications for polarization-dependent splitters and polarizers.
Table 1. Coupling Lengths of Gold Wire-filled Dual-core PCF Couplers at 1330 nm and 1550 nm
Figure 5(a) shows the coupling lengths Lcx and Lcy (the dotted curves) and their ratio Lcy/Lcx (the solid curve) vs. the wavelength of the gold-filled PCF coupler with D=2Λ. The circle and square dotted curves represent the x- and y-polarized components, respectively. Notice that the value of Lcy/Lcx almost maintains the ratio coupling length of around 1.5 in the range 1100 nm to 1600 nm. The enhancement of birefringence introduced by filling a gold wire gives rise to the polarization-dependent coupling between two co-parallel cores, which supports separating two orthogonally polarized components into two cores before dramatic attenuation. Figure 5(b) shows the transmissions of x- (the thin curves) and y-polarized (the thick curves) components vs. propagation length at λ=1330 nm for linear polarized light launched into core 1 (the left core), with an angle of 45° between the polarized direction and the x-axis. The solid and dashed curves represent the transmissions of cores 1 and 2 (the right core), respectively. The x-polarized component has shorter coupling length and faster dissipation than the y-polarized one. After propagating length L0 = 2Lcy, the light in core 1 is just y-polarized and in core 2 only x-polarized. The inset in Fig. 5(b) gives the transmission ratios between the minimum and the maximum transmissions of two orthogonally polarized components in each core vs. wavelength after propagating L0 in such a gold-filled PCF coupler, where the circle and square dots represent the transmission ratio Tx1/Ty1 in core 1 and Ty2/Tx2 in core 2, respectively. Note that the transmission ratios in two cores both reach the maximum around 1310 nm and exist in a broadband of about 100 nm in which the light keeps a high degree of polarization. The results show that we can realize the separation of two orthogonally polarized components by means of polarization-dependent coupling in a gold-filled dual-core PCF coupler, which is different from the polarization-dependent loss in the reported gold-filled PCFs.
Fig. 5 (a) Coupling length (dotted lines) of x- (circle) and y-polarized (square) components, and their ratio (solid line) vs. wavelength; (b) transmissions of x- (thin lines) and y-polarized (thick lines) components in two cores vs. propagation length. Inset: Ratio between the minimum and the maximum transmissions of two orthogonally polarized components after propagating L0.
Figure 6(a) shows the transmissions of x- (the thin curves) and y-polarized (the thick curves) components in a PCF coupler as shown in Fig. 1(d) vs. propagation length at λ=1330 nm. The x-polarized component decays dramatically and its loss closes to 17 dB after propagating 1.8 mm, while the loss of y-polarized one is about 6 dB. Then the output light just consists of the y-polarized component, and an optical polarizer is achieved. However, high overall SPP-induced loss limits the coupler in applications as a polarizer. In order to lower the loss of linear polarized light output from such a polarizer, we analyze the situation of two adjacent gold wires integrated into cladding away from two cores as shown in the inset of Fig. 6(b). Figure 6(b) shows the loss spectrum of the even (dashed curves) and odd (solid curves) supermodes nearby the m=2 anti-crossing points where the thick and thin curves represent the y- and x-polarized supermodes, respectively. We can also find that the loss peaks happen by splitting (A and D points for the y-polarized component, B and C points for the x-polarized component) as a result of mode hybridization, and the separation of loss peaks provides excellent polarization-dependent loss spectra. Figure 6(c) shows the transmissions of x- (the thin curves) and y-polarized (the thick curves) components in two cores vs. PCF length at λ=1030 nm. Notice that with increasing propagation distance, the y-polarized components sharply reduce, and its loss is close to 15 dB after propagating 1.5 cm, yet the loss of the x-polarized transmission is only about 2.6 dB. In addition, the polarization-dependent loss spectra can be modulated by filling two adjacent gold wires with different diameters [17] so that the operating wavelengths of the optical filter and polarizer based on the gold-filled PCF coupler can also be modulated by adjusting the wire size.
Fig. 6 (a) Transmissions of x- (thin lines) and y-polarized (thick lines) components in a PCF filled with two gold wires between two cores vs. propagation length at wavelength 1330nm; (b) loss spectra of the two orthogonally even (dashed lines) and odd (solid lines) supermodes. (A, 1032nm; B, 1068nm; C, 1069nm; D, 1093nm). Inset: schematic transverse structures of PCF; (c) transmissions of x- (thin lines) and y-polarized (thick lines) components in PCF as shown in the inset of (b) vs. propagation length at λ = 1030nm.
4. Conclusions
We numerically investigate the dispersion and loss of single- and dual-core PCFs selectively filled with gold wires. Obvious enhancement of fiber birefringence and distinct resonance of the even and odd supermodes are obtained by filling a gold wire between two cores in a dual-core PCF. High polarization-dependent coupling and transmission are observed, and two orthogonally polarized components can be separated in such a gold-filled PCF coupler. The polarization properties of output beams can be improved by introducing hybridization of surface plasmon resonance (SPR). Our study can be potentially used to design polarizers and polarization filters, as well as polarization beam splitters, based on gold-filled dual-core PCFs.
Acknowledgments
This work was supported by the National Basic Research Program (973 Program) of China under Grant No. 2012CB921900, the Doctorate Foundation (CX201119), the Technology Innovation Foundation (2011KJ01011) of Northwestern Polytechnical University, and the Ministry of Education Fund for Doctoral Students Newcomer Awards.
References and links
1. P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006). [CrossRef]
2. J. C. Knight, “Photonic crystal fibres,” Nature 424(6950), 847–851 (2003). [CrossRef] [PubMed]
3. B. Eggleton, C. Kerbage, P. Westbrook, R. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9(13), 698–713 (2001). [CrossRef] [PubMed]
4. R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” in OSA Trends in Optics and Photonics (TOPS) Vol. 70, of 2002 OSA Technical Digest Series, Optical Fiber Communication Conference, Postconference Edition (Optical Society of America, 2002), pp. 466–468.
5. T. Alkeskjold, J. Lægsgaard, A. Bjarklev, D. Hermann, A. Anawati, J. Broeng, J. Li, and S. T. Wu, “All-optical modulation in dye-doped nematic liquid crystal photonic bandgap fibers,” Opt. Express 12(24), 5857–5871 (2004). [CrossRef] [PubMed]
6. G. Ren, P. Shum, X. Yu, J. Hu, G. Wang, and Y. Gong, “Polarization dependent guiding in liquid crystal filled photonic crystal fibers,” Opt. Commun. 281(6), 1598–1606 (2008). [CrossRef]
7. B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, and A. H. Greenaway, “Experimental study of dual-core photonic crystal fiber,” Electron. Lett. 36(16), 1358–1359 (2000). [CrossRef]
8. P. Li, J. Zhao, and X. Zhang, “Nonlinear coupling in triangular triple-core photonic crystal fibers,” Opt. Express 18(26), 26828–26833 (2010). [CrossRef] [PubMed]
9. Y. Zhu, P. Shum, H. W. Bay, X. Chen, C. H. Tan, and C. Lu, “Wide-passband, temperature-insensitive, and compact π-phase-shifted long-period gratings in endlessly single-mode photonic crystal fiber,” Opt. Lett. 29(22), 2608–2610 (2004). [CrossRef] [PubMed]
10. X. Fang, M. Hu, C. Xie, Y. Song, L. Chai, and C. Wang, “High pulse energy mode-locked multicore photonic crystal fiber laser,” Opt. Lett. 36(6), 1005–1007 (2011). [CrossRef] [PubMed]
11. J. Hou, D. Bird, A. George, S. Maier, B. T. Kuhlmey, and J. C. Knight, “Metallic mode confinement in microstructured fibres,” Opt. Express 16(9), 5983–5990 (2008). [CrossRef] [PubMed]
12. M. A. Schmidt, L. N. Prill Sempere, H. K. Tyagi, C. G. Poulton, and P. St. J. Russell, “Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires,” Phys. Rev. B 77(3), 033417 (2008). [CrossRef]
13. X. Zhang, R. Wang, F. M. Cox, B. T. Kuhlmey, and M. C. J. Large, “Selective coating of holes in microstructured optical fiber and its application to in-fiber absorptive polarizers,” Opt. Express 15(24), 16270–16278 (2007). [CrossRef] [PubMed]
14. H. W. Lee, M. A. Schmidt, H. K. Tyagi, L. Prill Sempere, and P. St. J. Russell, “Polarization-dependent coupling to plasmon modes on submicron gold wire in photonic crystal fiber,” Appl. Phys. Lett. 93(11), 111102 (2008). [CrossRef]
15. H. W. Lee, M. A. Schmidt, R. F. Russell, N. Y. Joly, H. K. Tyagi, P. Uebel, and P. St. J. Russell, “Pressure-assisted melt-filling and optical characterization of Au nanowires in microstructured fibers,” Opt. Express 19(13), 12180–12189 (2011). [CrossRef] [PubMed]
16. A. Nagasaki, K. Saitoh, and M. Koshiba, “Polarization characteristics of photonic crystal fibers selectively filled with metal wires into cladding air holes,” Opt. Express 19(4), 3799–3808 (2011). [CrossRef] [PubMed]
17. H. W. Lee, M. A. Schmidt, and P. St. J. Russell, “Excitation of a nanowire ‘molecule’ in gold-filled photonic crystal fiber,” Opt. Lett. 37(14), 2946–2948 (2012). [CrossRef] [PubMed]
18. S. Zhang, X. Yu, Y. Zhang, P. Shum, Y. Zhang, L. Xia, and D. Liu, “Theoretical study of dual-core photonic crystal fibers with metal wire,” IEEE Photon. J. 4(4), 1178–1187 (2012). [CrossRef]
19. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1989).
20. P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006). [CrossRef] [PubMed]
21. M. A. Schmidt and P. St. J. Russell, “Long-range spiraling surface plasmon modes on metallic nanowires,” Opt. Express 16(18), 13617–13623 (2008). [CrossRef] [PubMed]
22. Z. Zhang, Y. Shi, B. Bian, and J. Lu, “Dependence of leaky mode coupling on loss in photonic crystal fiber with hybrid cladding,” Opt. Express 16(3), 1915–1922 (2008). [CrossRef] [PubMed]
