## Abstract

It is known that a decoupled two-core fiber can prevent monochromatic light at a specific wavelength (the decoupling wavelength) launched into one core from coupling to the other core. In this paper, we show that a pulse at the decoupling wavelength launched into one core of such a fiber inevitably splits into two pairs of pulses propagating in the two cores along the fiber. The minimum distance required for pulse splitting to be visible is inversely proportional to the coupling-coefficient dispersion in the fiber and linearly proportional to the pulse width. It would take only several centimeters of a recently demonstrated decoupled two-core photonic-bandgap fiber to observe the pulse-splitting effect with a 100-fs pulse. We also study the effects of self-phase modulation on the pulse propagation dynamics in a decoupled two-core fiber in both the normal and anomalous dispersion regimes.

© 2010 OSA

## 1. Introduction

The mechanism of evanescent-field coupling in a two-core fiber or waveguide forms the basis of numerous modern optical components. A conventional two-core fiber, where two parallel high-index cores are embedded in a uniform low-index cladding, always possesses a finite coupling coefficient, which allows light energy to be transferred between the two cores back and forth periodically along the fiber [1]. The coupling coefficient can be made arbitrarily small by increasing the core separation. It has been shown, however, that in principle the coupling effect in a two-core waveguide can be eliminated completely without increasing the core separation through properly modifying the refractive-index distribution of the waveguide [2], which leads to the concept of a decoupled two-core waveguide. In recent years, several decoupled two-core slab waveguides [3–5] and fibers [6] based on photonic crystal structures have been proposed and a decoupled two-core photonic-bandgap fiber has been demonstrated experimentally [7].

Here we should point out that, in general, the decoupling effect in a decoupled two-core fiber occurs only at a specific wavelength and, therefore, complete decoupling can be observed only for monochromatic light at the decoupling wavelength, i.e., a continuous wave at the decoupling wavelength launched into one of the cores stays in that core over an infinite length of the fiber. In this paper, we study the propagation of a pulse along a decoupled two-core fiber. A pulse, by definition, is not monochromatic, as it is composed of a finite spectrum of light. Therefore, pulse propagation in a decoupled two-core fiber is expected to be different from continuous-wave propagation. The objective of our study is to understand how a pulse evolves along a decoupled two-core fiber.

Pulse propagation in a conventional two-core fiber has been studied extensively by solving the coupled-mode equations, where light coupling between the two cores is characterized by a structure-dependent parameter called the coupling coefficient. In early studies, the wavelength dependence or dispersion property of the coupling coefficient is overlooked and pulse propagation is not much different from continuous-wave propagation (see Ref. 8 and references therein). However, with the coupling-coefficient dispersion taken into account, a pulse can get distorted when it is coupled back and forth between the two cores [9–11]. When the fiber is long enough, two pairs of identical pulses emerge from the two cores and mode coupling stops completely [9–11]. The pulse-breakup effect has been demonstrated experimentally with picosecond pulses and a meters-long conventional two-core fiber [12] and explored for the generation of high-repetition-rate pulse trains [13]. The effects of the coupling-coefficient dispersion on the propagation of short pulses in two-core fibers with dissimilar cores [14] and various nonlinear two-core fibers [15–19] have also been studied. In this paper, we show that a pulse does not couple back and forth between the two cores along a decoupled two-core fiber, but the coupling-coefficient dispersion of the fiber still leads to pulse splitting. In fact, an input pulse, regardless of its width, eventually evolves into two pairs of pulses in the two cores. For the recently demonstrated decoupled two-core photonic-bandgap fiber [7], it should take only several centimeters of the fiber to observe splitting of a 100-fs pulse. In addition, we investigate the effects of self-phase modulation (SPM) on the pulse propagation dynamics in a decoupled two-core fiber. We observe pulse broadening and optical wave breaking in a normal dispersion fiber and soliton formation and pulse compression in an anomalous dispersion fiber.

## 2. Coupled-mode equations

Pulse propagation in a lossless fiber with two identical cores is described by the following pair of coupled-mode equations, which include SPM [10,11]:

*a*

_{1}and

*a*

_{2}are the amplitude envelopes of the pulses in the two cores, respectively;

*z*is the distance along the fiber;

*t*is the time coordinate with reference to the transit time of the pulses;

*β*

_{2}is the group-velocity dispersion (GVD);

*γ*= 2π

*n*

_{2}/(

*λA*

_{eff}) represents SPM with

*n*

_{2},

*λ*, and

*A*

_{eff}being the nonlinear coefficient of the fiber, the free-space optical wavelength, and the effective area of each core, respectively;

*C*is the coupling coefficient at the carrier frequency; and

*C’*= d

*C*/d

*ω*(

*ω*denotes the angular optical frequency) is the coupling-coefficient dispersion at the carrier frequency.

For a decoupled two-core fiber, the coupling coefficient *C* is equal to 0. When the input light is a time-independent continuous wave, Eqs. (1) and (2) are no longer coupled, which implies that the wave stays in the input core over an infinite length of the fiber and there is no power transfer between the two cores. In the case of a pulse, however, the two equations are coupled through the coupling-coefficient dispersion *C’*, i.e., the last terms in the equations.

Unless stated otherwise, we assume that a pulse is launched into one of the cores, namely,

where*a*

_{0}is a real constant. The peak power of the pulse is

*a*

_{0}

^{2}and the total pulse power is 2

*a*

_{0}

^{2}.

## 3. Linear couplers

We first analyze a linear decoupled two-core fiber by solving Eqs. (1) and (2) with the SPM terms removed. To highlight the physics, we further ignore the GVD terms. Keeping only the first and the last terms and assuming that a pulse of an arbitrary shape *a*(0, *t*) is launched into one of the cores, i.e., *a*
_{1}(0, *t*) = *a*(0, *t*) and *a*
_{2}(0, *t*) = 0, we obtain

*z*= 0 and become separated in time as they propagate independently along the fiber. For this pulse-splitting phenomenon to be observable, the time separation between the two sub-pulses should be longer than the width of the input pulse

*T*

_{0}, i.e.,From the above condition, we can define a walk-off distance at which the two sub-pulses become separated:For

*C′*= − 1 ps/m, we have, for example,

*z*= 5 cm and 500 m for

_{w}*T*

_{0}= 100 fs and 1 ns, respectively. An input pulse, regardless of its width, evolves into two pairs of identical pulses in the two cores along the fiber, in spite of the fact the two cores are completely decoupled at the carrier frequency of the pulse. While it is not obvious why a dispersive coupling coefficient can cause pulse splitting, the phenomenon can be understood intuitively with the concept of intermodal dispersion [9]. A fiber with two single-mode cores is actually a bimodal waveguide that supports a symmetric mode and an antisymmetric mode, which are the supermodes or normal modes of the composite two-core structure. The pulse launched into one of the cores excites the two normal modes equally [1,9]. In a decoupled two-core fiber, these two modes have identical propagation constants or phase velocities, but their group velocities are in general different. Therefore, the pulses carried by these two modes propagate along the fiber at different velocities and eventually walk off. The two sub-pulses in Eqs. (4) and (5) can be interpreted as the pulses carried by the two normal modes and the pulse-splitting effect is a manifestation of the intermodal dispersion in the bimodal structure.

We now consider a practical decoupled two-core photonic-bandgap fiber [7]. From the coupling characteristics and the dispersion curves given in Ref. 7, we obtain *C′* = − 1 ps/m and *β*
_{2} = 50 ps^{2}/km (estimated values only). Assuming the input condition Eq. (3), we solve Eqs. (1) and (2) numerically with a Fourier series analysis method [15]. The pulse propagation dynamics are shown in Fig. 1(a)
for *T*
_{0} = 100 fs and in Fig. 1(b) for *T*
_{0} = 1 ps, where *U* = |*a*
_{1}|^{2}/*a*
_{0}
^{2} and *V* = |*a*
_{2}|^{2}/*a*
_{0}
^{2} are normalized power envelopes in the two cores, respectively.

As shown in Fig. 1, the input pulse splits into two pairs of pulses in the two cores along the fiber. It takes a distance of ~5 cm for the 100-fs pulse to split or ~50 cm for the 1-ps pulse to split, as suggested by Eq. (7). We can see more significant pulse broadening for the 100-fs pulse, because of the large GVD. As the pulses travel down the fiber, the two pulses in each core become more and more separated. There is no periodic power transfer between the two cores, yet the cross core keeps taking power from the input core till all the input power is equally distributed among the two pairs of pulses in the two cores. The propagation dynamics does not change when we change the sign of GVD. The pulse propagation dynamics in a decoupled two-core fiber is characteristically different from that in a conventional two-core fiber.

## 4. Nonlinear couplers

It is known that SPM can lead to many interesting pulse dynamics in conventional two-core fibers [15–19]. Here we investigate the effects of SPM in a decoupled two-core fiber. To facilitate discussion, we use the soliton power of a single-core fiber as the unit to measure the input power:

The parameters used in our simulation are*C*= 0,

*C′*= − 1 ps/m,

*A*

_{eff}= 100 μm

^{2},

*λ =*1.5 μm,

*n*= 3.2 × 10

_{2}^{−20}m

^{2}/W (for silica glass), and

*T*

_{0}= 100 fs. Since it is possible to tailor the value of GVD over a wide range with a photonic crystal structure, we consider both a normal dispersion fiber with

*β*

_{2}= 50 ps

^{2}/km and an anomalous dispersion fiber with

*β*

_{2}= −50 ps

^{2}/km. For both fibers, the single-core soliton power

*P*

_{s}is equal to 3.7 kW.

For the normal dispersion fiber, the propagation dynamics for three input power levels *a*
_{0}
^{2} = *P*
_{s}, 4*P*
_{s}, and 9*P*
_{s} are shown in Fig. 2
and the pulse envelopes at different distances along the fiber are shown more clearly in Fig. 3
. In all the three cases, the input pulse is split into two pairs of pulses in the two cores before SPM produces any significant effects. At a low input power, the propagation dynamics is similar to the linear case, where pulse broadening is mainly caused by the large GVD. At a high input power, however, SPM reinforces the pulse broadening effect on the split pulses. As shown in Fig. 2(b) and (c) and also in Fig. 3(b) and (c), over a sufficiently long distance, each pulse in the two cores is broadened into a parabolic or nearly rectangular shape, which is consistent with the observation with a single-core fiber [20]. The small ripples across the pulses, as shown in Fig. 3(c), can be attributed to the interaction of the pulses and the effect of optical wave breaking [21].

For the anomalous dispersion fiber, the propagation dynamics for three input power levels *a*
_{0}
^{2} = *P*
_{s}, 4*P*
_{s}, and 9*P*
_{s} are shown in Fig. 4
and the pulse envelopes at different distances along the fiber are shown in Fig. 5
. In the first two cases, the input pulse splits before SPM produces any significant effects. When SPM acts on the split pulses, it compensates for the GVD as in the case of a single-core fiber. In particular, at *a*
_{0}
^{2} = 4*P*
_{s}, each split pulse evolves into a soliton-like pulse, whose width does not change significantly with the distance, as shown in Fig. 4(b) and Fig. 5(b). This is consistent with the fact that each split pulse carries only a quarter of the input power, so the input power needed to turn the four split pulses into soliton-like pulses must be four times of the single-core soliton power. The small ripples across the pulses are likely to be caused by pulse interaction. At *a*
_{0}
^{2} = 9*P*
_{s}, SPM overtakes GVD and leads to pulse compression even at a short distance, and most of the input power is trapped in the input core at a sufficiently long distance, as shown in Fig. 4(c) and Fig. 5(c).

## 5. Conclusion

A decoupled two-core fiber does not in general provide complete decoupling for pulse transmission. In the presence of coupling-coefficient dispersion, a pulse of any width launched into one of the cores inevitably splits into two pairs of pulses in the two cores. The pulse-splitting distance is inversely proportional to the coupling-coefficient dispersion and increases linearly with the pulse width. It should take only a few centimeters of the recently demonstrated decoupled two-core photonic-bandgap fiber to observe pulse splitting with a 100-fs pulse. When the pulse-splitting effect is strong, SPM acts on the split pulses, which can lead to various nonlinear propagation dynamics, depending on the sign of GVD and the input power. The fiber could find applications in pulse shaping, pulse generation, and pulse delay control. In an ordinary single-mode fiber, pulses that have the same carrier frequencies propagate at the same group velocity. In a decoupled two-core fiber, however, the two split pulses in each core, which have the same carrier frequency, propagate at different group velocities. This characteristic of the fiber can lead to new dynamics of nonlinear pulse interaction. Finally, it is yet to be proved whether it is possible to design a decoupled two-core fiber for complete decoupling operation with pulses, which demands a coupling coefficient that vanishes over the bandwidth of the input pulse.

## Acknowledgement

The research was supported by the Fundamental Research Funds for the Central Universities, Project No. CDJZR10 16 00 06.

## References and links

**1. **A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. **62**(11), 1267–1277 (1972). [CrossRef]

**2. **C. G. Someda, “Antiresonant decoupling of parallel dielectric waveguides,” Opt. Lett. **16**(16), 1240–1242 (1991). [CrossRef] [PubMed]

**3. **S. Boscolo, M. Midrio, and C. G. Someda, “Coupling and decoupling of electromagnetic waves in parallel 2D photonic crystal waveguides,” IEEE J. Quantum Electron. **38**(1), 47–53 (2002). [CrossRef]

**4. **F. S.-S. Chien, Y.-J. Hsu, W.-F. Hsieh, and S.-C. Cheng, “Dual wavelength demultiplexing by coupling and decoupling of photonic crystal waveguides,” Opt. Express **12**(6), 1119–1125 (2004). [CrossRef] [PubMed]

**5. **Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE J. Quantum Electron. **41**(1), 76–84 (2005). [CrossRef]

**6. **Z. Wang, G. Kai, Y. Liu, J. Liu, C. Zhang, T. Sun, C. Wang, W. Zhang, S. Yuan, and X. Dong, “Coupling and decoupling of dual-core photonic bandgap fibers,” Opt. Lett. **30**(19), 2542–2544 (2005). [CrossRef] [PubMed]

**7. **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]

**8. **K. S. Chiang, “Theory of pulse propagation in optical directional couplers,” J. Nonlinear Opt. Phys. Mater. **14**(2), 133–147 (2005). [CrossRef]

**9. **K. S. Chiang, “Intermodal dispersion in two-core optical fibers,” Opt. Lett. **20**(9), 997–999 (1995). [CrossRef] [PubMed]

**10. **K. S. Chiang, “Coupled-mode equations for pulse switching in parallel waveguides,” IEEE J. Quantum Electron. **33**(6), 950–954 (1997). [CrossRef]

**11. **K. S. Chiang, “Propagation of short optical pulses in directional couplers with Kerr nonlinearity,” J. Opt. Soc. Am. B **14**(6), 1437–1443 (1997). [CrossRef]

**12. **K. S. Chiang, Y. T. Chow, D. J. Richardson, D. Traverner, L. Dong, L. Reekie, and K. M. Lo, “Experimental demonstration of intermodal dispersion in a two-core optical fiber,” Opt. Commun. **143**(4-6), 189–192 (1997). [CrossRef]

**13. **P. Peterka, P. Honzatko, J. Kanka, V. Matejec, and I. Kasik, “Generation of high-repetition rate pulse trains in a fiber laser through a twin-core fiber,” Proc. SPIE **5036**, 376–381 (2003). [CrossRef]

**14. **M. Liu, K. S. Chiang, and P. Shum, “Evaluation of intermodal dispersion in a two-core fiber with non-identical cores,” Opt. Commun. **219**(1-6), 171–176 (2003). [CrossRef]

**15. **P. Shum, K. S. Chiang, and W. A. Gambling, “Switching dynamics of short optical pulses in a nonlinear directional coupler,” IEEE J. Quantum Electron. **35**(1), 79–83 (1999). [CrossRef]

**16. **P. M. Ramos and C. R. Paiva, “All-optical pulse switching in twin-core fiber couplers with intermodal dispersion,” IEEE J. Quantum Electron. **35**(6), 983–989 (1999). [CrossRef]

**17. **S. C. Tsang, K. S. Chiang, and K. W. Chow, “Soliton interaction in a two-core optical fiber,” Opt. Commun. **229**(1-6), 431–439 (2004). [CrossRef]

**18. **M. Liu, K. S. Chiang, and P. Shum, “Propagation of short pulses in an active nonlinear two-core optical fiber,” IEEE J. Quantum Electron. **40**(11), 1597–1602 (2004). [CrossRef]

**19. **M. Liu and K. S. Chiang, “Propagation of ultrashort pulses in a nonlinear two-core photonic crystal fiber,” Appl. Phys. B **98**(4), 815–820 (2010). [CrossRef]

**20. **H. Nakatsuka, D. Grischkowsky, and A. C. Balant, “Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion,” Phys. Rev. Lett. **47**(13), 910–913 (1981). [CrossRef]

**21. **W. J. Tomlinson, R. H. Stolen, and A. M. Johnson, “Optical wave breaking of pulses in nonlinear optical fibers,” Opt. Lett. **10**(9), 457–459 (1985). [CrossRef] [PubMed]