Open Access
Add to BibSonomyAbstract
We report on a numerical analysis of the nonlinear coupling characteristics in triangular triple-core photonic crystal fibers (TTC-PCFs) by using coupled mode theory. The results show that the coupling of the TTC-PCFs exhibit more excellent power selectivity than that of the dual-core PCF and sharper optical switching and coupling-band with lower critical power are implemented in asymmetric TTC-PCF. By adjusting the parameters of the TTC-PCF structure and length, a coupling-band power controlled with better flatness will be obtained, in which more than 90% input power can be transferred. These results maybe offer a new possibility for application fields including optical switching, pulse shaping and pulse compressing.
©2010 Optical Society of America
1. Introduction
The transmittance of the optical fiber nonlinear directional coupler will appear dramatic reversal when the input power increases to a critical value, resulting in coupling switching [1–3], which would have a great prospect in the applications of all-optical logical operation, ultrafast all-optical switching, multiplexer and demultiplexer, etc. Whereas the nonlinear coefficient of a traditional optical fiber is usually very small, this greatly restricts the development of the optical fiber nonlinear directional coupler. Compared with the traditional optical fiber, the nonlinear coefficient of a photonic crystal fiber (PCF) is usually greater for several orders of magnitude and could be conveniently adjusted by changing the fiber structure [4]. The nonlinear directional coupler based on dual-core PCF is very convenient to achieve optical switching, which has been demonstrated in the past years [5–7]. The linear and nonlinear coupling properties of the dual-core PCF have been investigated theoretically and experimentally [8–14]. Furthermore, the soliton switching in dual-core PCF has been also realized by carefully designing the coupler geometry [15]. However, so far the studies on coupling properties of the PCF nonlinear directional coupler are mainly concentrated on the dual-core PCF. Only few reports deal with the nonlinear coupling in a triple-core PCF directional coupler [16]. In contrast with dual-core PCF, the triple-core PCF reveals many novel characteristics because of more coupling between cores.
In this paper, we construct a TTC-PCF by filling the air-holes at triangular positions, and numerically simulate the nonlinear coupling behaviors based on coupled-mode theory (CMT) under different incidence conditions of the power pulse [17]. Then we analyze the transmittance of the symmetric and asymmetric TTC-PCFs with different lengths, respectively. The results show that such a TTC-PCF could offer a new possibility for application fields including optical switching, pulse shaping and pulse compressing.
2. Numerical simulation model
The CMT is considered as one of the most effective method to analyze the coupling propagation of the light wave in optical waveguides. According to Helmholtz equation, the electric field in triple-core PCF can be approximately expressed as
Where, Fm(x, y), Αm(z, t), and β 0m are the modal spatial distribution, the slowly varying envelope and the propagation constant of the mth core, respectively. Taking Eq. (1) into the Helmholtz equation, considering the case of pumping pulse with a broad width T 0, then, the dispersion length L D = T 0 2/|β 2| is much greater than the PCF length L and the nonlinear length L NL, so that the effect of the group velocity dispersion could be neglected. Here β 2 is the group velocity dispersion coefficient. The coupled-mode equations for the triple-core PCF will be written asWhere, γm = (n 2 ω 0)/(cA eff) (m = 1,2,3) depicts the nonlinear coefficient of the mth core, n 2 is the nonlinear refractive index, c is the light speed in vacuum, A eff is the effective mode area; ηlm and κlm describes the ratio of the cross-phase (XPM) to the self-phase (SPM) coefficient and the coupling coefficient between the lth and mth cores, respectively. κlm dependences on the index profile n(x, y) and the modal distribution F(x, y), and it can be expressed as [18]Where, β is the propagation constant, k 0 is the wave number, Ω is the transverse section of the PCF, nl(x, y) is the index profile except the lth core. Equation (2) illustrates that, besides the coupling coefficient κ, the nonlinear effect of the SPM plays an important role in the coupling between those three cores.Figure 1 shows the cross geometries of the TTC-PCFs used in our analysis, which is formed by filling the air-holes in the triangular positions 1, 2 and 3 with background material. Where, x and y are the transverse directions and the numbers 1, 2 and 3 represent the three cores, respectively. Figure 1(a) has a symmetric structure, where d 1 is the diameter of the air-hole, Λ is the distance of the adjacent two air-holes and D = 3Λ is the distance of the adjacent two cores, n = 1.45 is the refractive index of the medium. Figure 1(b) shows an asymmetric TTC-PCF, which has distinct coupling coefficients and identity propagation constants, and can be obtained by changing the diameter of the first layer air-holes around the cores, where d 2 and d 3 are the diameters of the two kinds of the air-holes.
3. Results and analysis
We first analyze the nonlinear coupling behavior when broad pulse launched into the symmetric TTC-PCF shown in Fig. 1(a). Where, d 1 = 1.2μm, Λ = 2.3μm, at λ = 800nm, the coupling coefficient between the three identical cores is κ = 8m−1. The calculation results are as following: the nonlinear coefficient γ = 27.5W−1km−1 when n 2 = 3.0 × 10−20m2/W, the group velocity dispersion coefficient β 2≈20ps2/km at the pumping wavelength, and the dispersion length L D = 50m for a 1ps input pulse. Considering the ration of the XPM η<<1, the nonlinear effect of the XPM can be neglected in the simulation processes.
We define the needed lengths for maximum power exchange and maximum power coupled back as coupling length and beat length, respectively. Figure 2(a) shows the coupling between the three cores in a symmetric TTC-PCF for a pulse with a peak power of 1W launched into core 1. The red and blue curves represent the light power in the input core and the other cores, respectively. Figures 2(b)-(d) show the normalized mode field distributions at different lengths. It is seen that, about 90% input power can be transferred from the input core 1 to the adjacent cores 2 and 3 after propagating in one-coupling-length of 0.139m, and completely transferred back after propagating in one-beat-length of 0.278m. In such case the two cores 2 and 3 can be considered as an effective single core, so the coupling between the three identical cores could be equivalent to that between two asymmetric cores.
Fig. 2 (a) Normalized powers of the three cores in a symmetric TTC-PCF vs. the propagation distance when P 0 = 1W; (b)-(d) normalized mode field distributions at 0m, 0.05m and 0.139m.
To examine the nonlinear coupling properties of the symmetric TTC-PCF, Fig. 3 display the power transmittances Ti = |Ai(z)|2/|A 0|2 of the three cores in a fixed length L c vs. the input power P 0. L c is the coupling length under the initial condition P 0 = 1W. Where, the red curve represents the input core 1 and the blue curves represent the other cores 2 and 3, respectively. It is shown that, the input power is almost coupled into the core 2 and 3 after propagating L c. When P 0<1kW, the coupled power slowly increases as P 0 increases. However, when P 0>1kW, the coupled power dramatically decreases. Especially, the coupling will be cut-off when P 0>2.2kW, and then the input power will be confined in the initial core. These results clearly show that the nonlinear effect of the SPM starts to play a highly significant role in power coupling when the input power is beyond the critical power. Thus, the phase shift induced by SPM increases with the input power, the coupling between the adjacent cores decreases correspondingly. So we obtain a remarkable rising edge nearby 2kW. The results demonstrate that the symmetric TTC-PCF has similar linear and nonlinear coupling properties to the asymmetric TTC-PCF reported in Ref [11].
Fig. 3 Transmittances of the three cores vs. the input power P 0 in a symmetric TTC-PCF with length L c.
The asymmetric TTC-PCF shown in Fig. 1(b) can be simply obtained by adjusting the diameter of the first layer air-holes around the core 1. The diameters of the three kinds of air-holes are d 1 = 1.2μm, d 2 = 0.96μm, and d 3 = 1.38μm, respectively. In such case, the asymmetric coupling coefficient can be obtained on the premise of keeping the propagation constants identical, at λ = 800nm, κ 12 = 6m−1, κ 23 = 8m−1, κ 13 = 6m−1, and the nonlinear coefficients of the three cores are γ 1 = 30W−1km−1, γ 2 = 27.5W−1km−1, and γ 3 = 27.5W−1km−1, respectively. The dispersion coefficients are still around 20ps2/km at the pumping wavelength in three cores.
Figure 4 gives the transmittances of the input cores vs. the input power in the TTC-PCFs with fixed length L c. Where, the dashed and solid lines depict the symmetric and asymmetric TTC-PCFs, and the colors blue and red stand for the cases of the power launched into core 1 and 2, respectively. It is shown that, the power-controlled transmittance switching will be obtained with increasing the input power, and the critical powers of these three curves are all about 1kW. However, the transmittance curve of the asymmetric TTC-PCF is markedly sharper than that of the symmetric one. Besides, the transmittance curves for input core 1 and 2 in the asymmetric TTC-PCF are also distinct. The curve for core 2 in the asymmetric TTC-PCF has an improved sharper switching in a relatively short power range.
Fig. 4 Transmittances of the input cores vs. the input power P 0 in the TTC-PCFs with length L c. Dashed and solid lines correspond to the symmetric and asymmetric TTC-PCFs, respectively.
Figure 5 shows the transmittances of the input cores vs. the input power in the TTC-PCFs with fixed length L b. Where, L b equals the beat length under the initial condition P 0 = 1W, the dashed line represents the symmetric structure, and the blue and red solid lines represent the asymmetric structures corresponding to the input core 1 and core 2, respectively. It is shown that, the input power is completely transferred into the input core again after propagating L b. However, as the input power increases, the SPM effect makes the PCF length no longer equal to the beat length, resulting in more power be transferred into the adjacent cores and the appearance of the coupling-band power controlled. As the input power increases to the critical power, the de-phasing induced by SPM between three cores becomes large enough. In consequence, the input pulse remains confined in the initial core. As a result, a coupling-band can be obtained with increasing the input power. Comparing the three curves, we would like to note that, the transmittance curves of the three cases are similar. However, the critical power of the asymmetric TTC-PCF is 300W lower than that of the symmetric one, and the coupling-band obtained in the case of core 2 incidences in the asymmetric TTC-PCF is broader than that of the others.
Fig. 5 Transmittances of the input cores vs. the input power P 0 in the TTC-PCF with length L b. Dashed and solid lines correspond to symmetric and asymmetric TTC-PCFs, respectively.
Figures 6(a) and 6(b) show the transmittances of the core 2 in the symmetric and asymmetric TTC-PCFs with different lengths, respectively. Where, the blue and red curves correspond to the fiber lengths L b and 2L b, respectively, and the black curve in Fig. 6(b) represents a fiber length of 0.56m which is just 0.04m shorter than 2L b. From Fig. 6(a), it can be seen that, the transmittance curve of the input core with length 2L b exhibits two coupling-bands in the region of 1.5kW and 1.7kW, respectively. The first one is broader than that of the TTC-PCF with length L b, but the first critical power is lower. The second one is extraordinary sharp and the cut-off power is equal to that of the TTC-PCF with length L b. For the results presented in Fig. 6(b), similar features can be also observed. The critical power of the first band is 1kW and the cut-off power is 1.25kW, which are lower than those of the coupling-band in the TTC-PCF with length L b. Remarkably, almost 80% input power can be transferred into the adjacent cores in this coupling-band. The second coupling-band is also very sharp, and the cut-off power is 1.5kW, which is in accord with the TTC-PCF with length L b. In order to improve the flatness of the first coupling-band, we chose a 0.56m long TTC-PCF, which is just 0.04m shorter than 2L b. The result is shown in Fig. 6(b) as black curve. Note that, more than 90% input power can be transferred into the adjacent cores in a range of 250W, and the second coupling-band is similar to the case of the TTC-PCF with length 2L b.
Fig. 6 Transmittances of the input core vs. the input power P 0 in (a) symmetric and (b) asymmetric TTC-PCFs with different lengths.
The results shown in Fig. 6 also demonstrate that the nonlinear effect of the SPM increases with the input power, which leads to the changes of the coupling/beat length. The PCF length is no longer equal to double beat length gradually, hence the power transferred from the input core increases, and the first coupling-band appears. With further increasing the incident power, the beat length gradually approach the PCF length, therefore the first coupling-band disappears. The second coupling-band appears similarly when the effective length of the TTC-PCF is equal to the coupling length. So as the incident power increases to the fixed cut-off power, it will be confined in the input core. The broad coupling-band obtained here is considerable to generate “gate” pulse and separate pulses with different powers, while the sharp one is available to create an extremely narrow pulse.
4. Conclusions
The nonlinear couplings of the TTC-PCFs have been numerically analyzed by calculating the transmittance based on the CMT. It is shown that the coupling between cores will be changed significantly with increasing the input power, resulting in the appearances of optical switching and coupling-band. The coupling performance in the symmetric TTC-PCF is similar to that in an asymmetric dual-core PCF. However, the asymmetric TTC-PCF has obvious advantages to achieve sharper switching and coupling-band with lower critical power, compared with the dual-core PCF. A relatively flat coupling-band can be achieved in the asymmetric TTC-PCF with approximate two-beat length. More than 90% input power can be transferred into the adjacent cores. The results show that the couplers based on the TTC-PCF may permit the achievement of optical switching, pulse shaping, pulse compressing.
Acknowledgements
This work was supported by the fundamental research foundation of northwestern polytechnical university under No. JC200950.
References and links
1. S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51(15), 1135–1137 (1987). [CrossRef]
2. S. R. Friberg, A. M. Weiner, Y. Silberberg, B. G. Sfez, and P. S. Smith, “Femotosecond switching in a dual-core-fiber nonlinear coupler,” Opt. Lett. 13(10), 904–906 (1988). [CrossRef] [PubMed]
3. C. Schmidt-Hattenberger, U. Trutschel, and F. Lederer, “Nonlinear switching in multiple-core couplers,” Opt. Lett. 16(5), 294–296 (1991). [CrossRef] [PubMed]
4. P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003). [CrossRef] [PubMed]
5. A. Betlej, S. Suntsov, K. G. Makris, L. Jankovic, D. N. Christodoulides, G. I. Stegeman, J. Fini, R. T. Bise, and D. J. Digiovanni, “All-optical switching and multifrequency generation in a dual-core photonic crystal fiber,” Opt. Lett. 31(10), 1480–1482 (2006). [CrossRef] [PubMed]
6. A. Tonello, M. Szpulak, J. Olszewski, S. Wabnitz, A. B. Aceves, and W. Urbanczyk, “Nonlinear control of soliton pulse delay with asymmetric dual-core photonic crystal fibers,” Opt. Lett. 34(7), 920–922 (2009). [CrossRef] [PubMed]
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 fibre,” Electron. Lett. 36(16), 1358–1359 (2000). [CrossRef]
8. Z. Wang, T. Taru, T. A. Birks, J. C. Knight, Y. Liu, and J. Du, “Coupling in dual-core photonic bandgap fibers: theory and experiment,” Opt. Express 15(8), 4795–4803 (2007). [CrossRef] [PubMed]
9. J. Laegsgaard, O. Bang, and A. Bjarklev, “Photonic crystal fiber design for broadband directional coupling,” Opt. Lett. 29(21), 2473–2475 (2004). [CrossRef] [PubMed]
10. K. Saitoh, Y. Sato, and M. Koshiba, “Coupling characteristics of dual-core photonic crystal fiber couplers,” Opt. Express 11(24), 3188–3195 (2003). [CrossRef] [PubMed]
11. J. R. Salgueiro and Y. S. Kivshar, “Nonlinear dual-core photonic crystal fiber couplers,” Opt. Lett. 30(14), 1858–1860 (2005). [CrossRef] [PubMed]
12. K. L. Reichenbach and C. Xu, “Numerical analysis of light propagation in image fibers or coherent fiber bundles,” Opt. Express 15(5), 2151–2165 (2007). [CrossRef] [PubMed]
13. X. Sun, “Wavelength-selective coupling of dual-core photonic crystal fiber with a hybrid light-guiding mechanism,” Opt. Lett. 32(17), 2484–2486 (2007). [CrossRef] [PubMed]
14. M. Liu and P. Shum, “Generalized coupled nonlinear equations for the analysis of asymmetric two-core fiber coupler,” Opt. Express 11(2), 116–119 (2003). [CrossRef] [PubMed]
15. K. R. Khan, T. X. Wu, D. N. Christodoulides, and G. I. Stegeman, “Soliton switching and multi-frequency generation in a nonlinear photonic crystal fiber coupler,” Opt. Express 16(13), 9417–9428 (2008). [CrossRef] [PubMed]
16. Y. Yan and J. Toulouse, “Nonlinear inter-core coupling in triple-core photonic crystal fibers,” Opt. Express 17(22), 20272–20281 (2009). [CrossRef] [PubMed]
17. W. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11(3), 963–983 (1994). [CrossRef]
18. P. Di Bin and N. Mothe, “Numerical analysis of directional coupling in dual-core microstructured optical fibers,” Opt. Express 17(18), 15778–15789 (2009). [CrossRef] [PubMed]
