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Power dependent pulse delay with asymmetric dual-core hybrid photonic crystal fiber coupler
Optics & Laser Technology 55 (2014) 26–36
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Power dependent pulse delay with asymmetric dual-core hybrid photonic crystal fiber coupler Qi Jing n, Xia Zhang, Wei Wei, Yongqing Huang, Xiaomin Ren State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, No. 10, XiTuCheng Road, HaiDian District, Beijing 100876, China
art ic l e i nf o
a b s t r a c t
Article history: Received 4 March 2013 Received in revised form 19 May 2013 Accepted 22 June 2013
We propose a novel asymmetric dual-core hybrid photonic crystal fiber (PCF) coupler composed of a silicon tube as the left core and a silica core as the right core. The control of picosecond pulse delay is achievable by means of power adjusting. The transmission modes, dispersion characteristics and coupling coefficients of the proposed coupler are investigated numerically. The results demonstrate that it is possible to obtain 2.0 ps time delay for soliton pulse with 2.0 ps temporal width within 1 cm length. Further numerical results show that the coupler can generate 10.0 ps undistorted time advance within 5 cm length. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Asymmetric dual-core PCF Slow and fast light Soliton
1. Introduction Slow light phenomenon has attracted a considerable amount of attention since the first experimental demonstration in the 1980s [1]. Many important applications for slow light have also been proposed, such as data synchronization in optical packet switching networks at bit or packet level [2–4], optical buffers and data storage [5–7], and the highly sensitive rotation sensing [8,9]. Generating slow light involves controlling the pulse velocity in optical materials. The corresponding technologies include coherent population oscillations (CPO) [10,11], electromagnetically induced transparency (EIT) [12,13], stimulated Brillouin scattering (SBS) [14,15], coupled-resonator optical waveguide (CROW) [16,17], photonic crystal waveguides [18,19]. However, the limitation for CPO is unsuitable for picosecond and femtosecond pulses, and the EIT needs extreme temperature conditions. The SBS method can be applied at room temperature but its gain bandwidth is usually within tens of megahertz. The CROW could be realized in waveguide with micrometer size but the suitable pulse width is nanosecond. The photonic crystal-based resonance structures could generate both broad bandwidth and large temporal but the regulating process requires auxiliary temperature control. Dual-core fiber couplers have been studied extensively due to its diverse applications such as pulse reshaping and switching, dispersion compensation and multiplexer and demultiplexer [20–22]. However, the conventional dual-core fibers are more complex in production [23]. Due to the development of the fabricating
n
Corresponding author. Tel.: +86 01066820388. E-mail address: [email protected] (Q. Jing).
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technology of microstructure fibers, the emergence of dual-core PCF based couplers provide a new approach for the fiber couplers [24,25] and have obtained various applications due to the short lengths, high effective nonlinear coefficients, strong evanescent field and advantages in integration [26–28]. Recently, dual-core PCF couplers based pulse velocity control has attracted much attention. In 2009, an asymmetric dual-core PCF coupler with six large air holes surrounded left core and one small central air hole constructed right core has been used to generate 46.8 ps delay in 1 m length [29]. Due to the smaller index differences between the two cores, the dual-core coupler drawn from silica based PCF with inconsistent air hole parameters has less design flexibilities and lower delay generation efficiency. Instead of changing the PCF geometric microstructure, integrating crystalline semiconductors and metals into silica capillaries can generate controllable dispersion and nonlinearity characteristics and offer a fundamental platform to exploit the hybrid optoelectronic materials and devices [30]. Depositing germanium [31], indium antimonide [32], zinc selenide [33] and silicon [34,35] into PCF pores have been realized by using molten core technique [31] and high-pressure chemical vapor deposition (HPCVD) technique [30,33]. In 2007, a dual-core coupler with two different deposited materials has generated both the total internal reflection index-guided and the photonic bandgap mechanism [36]. In 2009, a PCF coupler composed of two asymmetric silica cores with lower indices than the background pure silica has been used as wavelength-selective coupler [37]. As the promotion of the hybrid deposition technology, the crystalline silicon tubes can be fabricated in silica capillaries, and its thickness can be precisely controlled [30,38,39]. In this paper, an asymmetric dual-core hybrid PCF coupler is proposed.
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The coupler left core is constructed by depositing silicon tube in one air hole, and its right core is created by omitting seven air holes to serve as a solid silica core. The effective index dissimilarity between the two cores is obtained due to the higher refractive index and nonlinear coefficient of the silicon. The mechanisms of the power dependent pulse delay with the two asymmetric cores are investigated numerically. The results show that the 2.0 ps time delay for Hyperbolic-Secant pulse with 2.0 ps temporal width can be realized within 1 cm length. Additionally, the effects of the slow and fast light phenomena on the pulse temporal profiles are analyzed theoretically. Based on its linear and nonlinear characteristics, this coupler will have practical applications in wavelength-selective narrowband filter [36], high bit rate optical buffer and delay line functionality [29].
2. Device principle and mode coupling characteristics The cross-section of the designed asymmetric dual-core hybrid PCF coupler is illustrated in Fig. 1(a). The air holes are arranged in a triangular lattice in the background of pure silica. The hole-pitch of the two nearest air holes is set as Λ. The diameter of the air holes is d and refractive index is 1.0. The PCF coupler is composed of a large core and a small core. The large core is created by filling seven air holes with silica, and the small core is obtained by depositing silicon tube in an air hole. The center-to-center distance of the two cores is 4Λ. The refractive index of the background silica and the silicon material at 1.550 μm is 1.45 and 3.43, respectively. Fig. 1(b) shows the detailed geometric structure of the silicon tube which is surrounded by six air holes. The silicon tube outer diameter d1 is equal to the air hole diameter d, and the inner diameter d2 satisfies the condition of 0 od2 od. In this asymmetric dual-core coupler, the confined light propagation in the silica core abides by the index-guiding mechanism [40]. However, the mode guiding mechanism in the silicon tube is neither the complete index-guiding mechanism nor the bandgapguiding mechanism [41], since the refractive index of the silicon tube is higher than both of the outside silica and the inside air hole. As a result, the light propagation in the left core is almost constrained in the tube structure. According to the coupled-mode theory, each core in this PCF coupler can be treated as an independent waveguide that is perturbed by the dispersive mode fields propagating in the other core. When the mismatch of the propagation constants between the two cores is large, the light field propagation is independent in each core without power transferring. As the refractive index of the deposited silicon tube can be precisely controlled by varying its structural parameters, the phase matching condition β1(λ0)¼β2(λ0) between the two cores can be obtained at a complete coupling wavelength λ0. As a result, the maximal power transferring between the two cores is constrained within a bandwidth with a central wavelength λ0. The narrow bandwidth for effective coupling between the two cores can be realized by the effective index which is sensitive to the geometric parameters. The asymmetric core arrangement leads to two non-degenerate guided modes with respective dispersion characteristic and mode field distribution. In this paper, the effects of the asymmetric dual-core parameters on the mode effective indices, group velocity dispersion and mode filed distributions are analyzed by finite-element method (FEM) [42] with anisotropic perfectly matched layers (PML). The diameter of air hole is equal to the outer diameter of the silicon tube and is fixed at d ¼d1 ¼0.72 μm. The requirement for endlessly single mode in PCF with one silica-core is the relative hole diameter d/Λo 0.43 [43]. In order to maintain the light propagation in the fundamental mode state, the d/Λ is fixed at 0.3 and the Λ is chosen as 2.0 μm, 2.4 μm, and 3.0 μm. The effective
Fig. 1. Cross-section of the designed asymmetric dual-core hybrid PCF coupler (a) and the silicon tube structure of the left core (b). The background area denotes pure SiO2, the gray color areas represent air holes, and the dark tube denotes the deposited Si.
indices neff of the two individual modes are calculated after setting those parameters and then the supermodes neff are obtained. During the simulation, the silicon tube inner diameter d2 is chosen as 0.5587 μm, 0.6596 μm and 0.8147 μm for the purpose of satisfying λ0 ¼1.550 μm when the Λ changes. The results show that the deviation of d2 for 2.0 μm oΛo 3.0 μm is less than 0.26 μm. The neff of the corresponding modes are shown in Fig. 2. As shown in Fig. 2(a), the real parts of the fundamental mode neff in silicon tube and silica core decreases as the wavelength increases, and the neff curves intersect at the complete coupling wavelength λ0 ¼ 1.550 μm under the linear condition. In the wavelength region λo1.550 μm, the neff of the silicon tube is larger than that of the neff in silica core. The neff in silicon tube decreases fast as the wavelength
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two individual modes increases as the Λ increases but the phase matching wavelength is maintaining at 1.550 μm. As the Λ increases from 2.0 μm to 3.0 μm, the neff curves of supermodes and individual modes become closer, which means the supermodes neff has larger slope changing at 1.550 μm as Λ becomes larger. According to Vg ¼ (2πc/λ2)(dλ/dβ) [44], the group index ng of the individual modes and supermodes can be expressed as ng_Si=SiO2 ¼ ng_E=O ¼
dβSi=SiO 2 dω
2
λ ¼ 2π
dβSi=SiO 2 dλ
c V g_Si=SiO2
¼c
c
dβE=O λ2 dβE=O ¼ dω 2π dλ
V g_E=O
¼c
ð1Þ
where Vg, β and c denotes to group velocity, propagation constant and the speed of light in vacuum, respectively. The subscripts Si/SiO2 and E/O correspond to the individual modes (silicon/silica) and the supermodes (even/odd), respectively. Results in Fig. 3(a) show the wavelength dependent ng of the two supermodes for three different Λ, and the differences of ng between the two supermodes and the two individual modes are presented in Fig. 3(b). As shown in Fig. 3(a), the ng curves of the two supermodes intersect at complete coupling wavelength 1.550 μm and have steeper slope within 1.545–1.555 μm as the Λ increases. It is shown in Fig. 3(b) that the ng of the supermodes is higher than that of the silica core and is lower than that of the silicon tube. Furthermore, the maximal changing rate of the ng differences lies in 1.545– 1.555 μm.
Fig. 2. Effective indices neff of the two individual guiding modes (in the SiO2 core and the Si tube) and the supermodes (Even mode and Odd mode) as a function of wavelength under linear condition. The hole-pitch Λ is set as 2.0 μm (a), 2.4 μm (b), and 3.0 μm (c), respectively.
increases, and its value becomes smaller than that of the silica core as λ41.550 μm. In both shorter and larger wavelength regions around λ0, the neff differences between individual modes and supermodes are not obvious. However, either of the supermodes neff curve obtains a turning point at λ0 and leaves the original trend for the other individual mode. It is clearly shown in Fig. 2(a)–(c) that the neff of
Fig. 3. Wavelength dependent ng of the two supermodes (a) and the ng differences between the two supermodes and the two individual modes for Λ ¼ 2.0 μm, 2.4 μm, and 3.0 μm (b).
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The GVD (group velocity dispersion) of the supermodes is defined as [44] DE=O ¼
d 1 ð Þ dλ V g_E=O
ð2Þ
The effects of the Λ variations on the supermodes GVD within 1.540–1.560 μm are depicted in Fig. 4. It is shown that the supermodes GVD has peak values at 1.550 μm and increases rapidly as the Λ increases. The maximal value is 727500 ps/km/nm, 7 32500 ps/km/nm and 742500 ps/ km/nm for Λ ¼2.0 μm, 2.4 μm and 3.0 μm, respectively. The Λ¼ 2.4 μm and d2 ¼0.6596 μm are chosen as the main structural parameters in calculations for concisely. The x-polarization electric field distribution mode (Ex) at the 1.545 μm, 1.550 μm and 1.555 μm are evaluated by the FEM and are shown in Fig. 5. As the y-polarized state has the similar field distribution as the x-polarized state and it is omitted for simplicity. As shown in Fig. 5(c) and (d), the odd mode and even mode have symmetric field distribution at 1.550 μm, which matches the
Fig. 4. Wavelength dependent effects of the Λ on the supermodes GVD.
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complete linear coupling wavelength λ0 in Fig. 2. Due to the large neff differences between the two cores at λo1.550 μm or λ41.550 μm, the light propagation in each core can hardly transfers into the other one. The symmetric mode field distribution can only be excited at 1.550 μm. These inferences can be verified in Fig. 5(a), (b), (e) and (f) where the linear coupling is restrained and the supermodes fields have obvious asymmetric distribution. According to the supermodes ng shown in Fig. 3 and the mode field distributions in Fig. 5, the full-width at half-maximum (FWHM) bandwidth of the linear coupling is less than 5 nm. Considering the sensitive effects of the geometric parameters on the neff, a stronger coupling can be achieved at an expected wavelength by changing d, d1, d2 and Λ. When short (i.e. picosecond) pulses with higher intensity are injected into one core of the PCF coupler, the neff of silicon tube and silica core will be changed as [44] ~ nðω; jEj2 Þ ¼ nðωÞ þ n2 jEj2
ð3Þ
where n(ω) is the linear part, n2 is the nonlinear-index coefficient, |E|2 is the light intensity. Additionally, the power induced neff change can be expressed as Δn ¼ n2 jEj2 ¼ n2 P 0 =Aeff where P0 is the pulse peak power and Aeff ¼ πr2 denotes the effective mode field area. The Aeff of the fundamental modes in silicon tube and silica core are calculated as 10.01 μm2 and 40.19 μm2. We choose the peak power as P0 ¼12950 W and the Δn of the silicon tube and silica core is 5.596 10 3 and 7.087 10 6 1.095 10 5. It is evident that the neff change of silicon tube is much larger than that of silica core, so that the power dependent changes of neff in silica core can be ignored. The neff and ng of the individual modes and supermodes under 12950 W are shown in Fig. 6. As shown in Fig. 6, the complete coupling characteristics at 1.550 μm are destroyed and the coupling wavelength is shifted to 1.555 μm. The corresponding GVD values are shown in Fig. 7. It is shown that the supermodes GVD peak value varies to 1.555 μm and increases from 732500 ps/km/nm to 734100 ps/ km/nm. Due to power dependent changes of the neff, ng and GVD, the injected pulse can exhibit tunable propagation delay and advance phenomena.
Fig. 5. The x-polarization electric field distribution mode (Ex) of the supermodes at 1545 nm (a/b), 1550 nm (c/d) and 1555 nm (e/f). The symmetric (c/d) and asymmetric (a/b/e/f) characteristics of the mode distribution are depicted.
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dispersion and nonlinear effects. The full vectorial beam propagation method (BPM) [45] has been widely used to investigate the propagating distance dependent light field. For the first time, a full vectorial finite-element beam propagation method (FEM-BPM) [46] is presented for efficiently analyzing 3-D anisotropic optical waveguides, diffuse waveguides and magnetooptic devices. In order to focus on revealing the time dependent and power controlled pulse switching dynamics and time delay in the two asymmetric cores, the coupledmode theories in the vectorial form are used. The nonlinear Schrödinger equations (NLSE) for coupler with two asymmetric cores can be derived as [47,48] ∂A1 ∂z
þ ∑
m¼1
im1 βm1 ∂m A1 m! ∂t m
¼ ðiκ 12 κ′12 ∂t∂ iκ″212 ∂t∂ 2 ÞA2 þ iδa A1 α21 A1 2
þiðγ 1 jA1 j2 þ C 12 jA2 j2 ÞA1
∂A2 ∂z
im1 βm2 ∂m A2 ∂ iκ″21 ∂2 Þ m ¼ ðiκ 21 κ′21 ∂t m! ∂t 2 ∂t 2 m¼1
þ ∑
A1 iδa A2
α2 A2 þ iðγ 2 jA2 j2 þ C 21 jA1 j2 ÞA2 2 ð4Þ
Fig. 6. The neff (a) and ng (b) of the individual modes and supermodes under 12,590 W.
where the subscript n (1, 2) denoting one of the two cores. An ðz; tÞ represents the varying envelopes of the pulses carried by the elemental modes of the two waveguides in isolation, αn is the loss coefficient, βmn is the m-order dispersion coefficient at the center frequency ω0 , γn is the nonlinear coefficient, and C12(C21) is the crossphase modulation (XPM) coefficient between the two cores. The parameters κ 12 and κ 21 are the linear coupling coefficients between the two cores, κ′12 and κ′21 are the first-order coupling coefficient dispersion, which accounts for the wavelength dependence of the coupling coefficient. The κ″12 and κ ″21 describe the second-order dispersive coupling coefficient. δa ¼ 12ðβ01 β02 Þ is used to describe the asymmetry between the dual cores. When the pulse temporal width is within 10 fs 10 ps, the κ″ in silica based coupler is about 10 27 s2/m [48] which is much smaller than the value of κ and κ′, so that the κ″ is ignored in calculation. We choose the retarded frame as T ¼ tβ12 z (Z ¼ z), and neglect the higher-order dispersion coefficients and the XPM coefficient C12(C21), and the simplified equations can be defined as ∂A1 ∂Z
iβ21 ∂ A1 β31 ∂ A1 1 þ ðβ11 β12 Þ ∂A ∂T þ 2! ∂T 2 3! ∂T 3 2
3
∂ α1 ∂A2 ÞA2 A1 þ iδa A1 þ iγ 1 jA1 j2 A1 ∂T 2 ∂Z iβ22 ∂2 A2 β32 ∂3 A2 ∂ ¼ ðiκ 21 κ′21 ÞA1 þ ∂T 2! ∂T 2 3! ∂T 3
¼ ðiκ 12 κ′12
Fig. 7. Wavelength dependent effects of the Λ on the GVD of the supermodes when the injected power is 12,590 W.
3. Equivalent solid core model for calculating coupling coefficients Optical switches and wavelength-selectors can be constructed by dual-core couplers. The distorted soliton pulses can be used to realize power switching which is supported by the balance between
Fig. 8. Cross-section of the equivalent solid dual-core hybrid coupler. The small core is Si and the large core is SiO2 and the cladding is solid material.
Q. Jing et al. / Optics & Laser Technology 55 (2014) 26–36
Fig. 9. Wavelength dependent neff of the model (a) and the neff differences between the model and original coupler (b) under linear and nonlinear conditions.
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Fig. 11. Wavelength dependent coupling coefficients of equivalent model under linear (a) and nonlinear (b) conditions.
According to the conventional coupled-mode theory [49], the analytical expression of coupling coefficients based on the HE11 modes is used in calculation: ðρp ρq Þ1=2 κ pq ¼
V 2p u2p
þ ðwq ρp =ρq Þ
2
Bp1=4 Bq1=4 Ap
1=2 1=2 ðzÞ GðtÞ Bq Gpq pq 2Bp
ðF p F q Þ1=2 ð6Þ
Fig. 10. The propagation delay deviations between the model and original coupler on the scale of ps/cm.
2 α2 A2 iδa A2 þ iγ 2 A2 A2 2
ð5Þ
As shown in Eq. (5), the βmn and δa can be derived from the neff, and the γn can be obtained from the γ n ¼ n2 ω=cAeff [44]. The coupling coefficients κ and κ′ are crucial for depicting pulse dynamics and can be numerically calculated, but the process is complex and tedious.
where p, q¼1, 2 (or 2, 1), ρp ¼ dp =2, V p ¼ k0 ρp np ð1n2cl =n2p Þ1=2 , 1=2 2 up ¼ ρp ðk0 n2p β2p Þ , wp ¼ ðV 2p u2p Þ1=2 , and k0 ¼ 2π=λ. The parameters A, B, F and G are defined in reference [49]. The first-order coupling coefficient dispersion can be obtained as: κ′pq ¼ ∂κ pq =∂ω. It should be clearly noted that the pulse propagation in the asymmetric hybrid PCF coupler cannot be directly solved by Eqs. (5) and (6) which are unique for the directional coupler with two parallel solid cores. In order to analytically calculate the coupling coefficient, an equivalent solid dualcore fiber model is proposed in this paper and its cross-section is illustrated in Fig. 8. As shown in Fig. 8, the silicon core, silica core and cladding have solid geometric constructure, and the refractive indices are nSi and nSiO2 and ncl, respectively. The diameters of two cores are dSi and dSiO2 , and the center-to-center distance d0 is 4Λ (9.6 μm). In order to make the model has similar refractive index as original coupler, the equivalent diameter of solid silicon core should meet the condition of 0odSi od1, and the refractive index ncl of the solid background should bepffiffiffiwithin nair oncl onSiO2 . According to the effective core radius Λ= 3 for the triangular PCF whose core is created by missing
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Fig. 12. Switching dynamics of 2.0 ps Sech pulse with 0.0789 W peak power propagating 4.2796 cm.
one air hole [43],pthe ffiffiffi equivalent diameter of the silica core should be suitable for 2Λ= 3 odSiO2 o 4Λ. In the simulation, we optimize the diameters and the refractive index of the equivalent model, and choose the parameters as: ncl ¼ 1.428, dSi ¼ 0.6810 μm and dSiO2 ¼ 4.2967 μm. The used temporal span is 40 ps with resolution 4.9 fs, and the spatial step is chosen as 10 μm. The calculated equivalent neff are shown in Fig. 9(a) under linear (low power) and nonlinear (12, 950 W) conditions and the index differences between the virtual model and the original coupler are shown in Fig. 9(b). As shown in Fig. 9(a), the equivalent model has the same coupling wavelengths 1550 nm and 1555 nm as the original coupler. It is shown in Fig. 9(b) that the maximal neff difference between the model and original device under low power condition is 2.2 10 3 (silicon core) and 2.3 10 4 (silica core). Due to the large input power 12,950 W, the maximal neff difference of silicon core is enlarged to 3.5 10 3. The neff differences induced light propagation delay deviations have been depicted in Fig. 10. As shown in Fig. 10, the delay deviation of the silicon core is more sensitive to the light intensity than that of the silica core, and
the maximal value is 1.14 10 1 ps/cm at 1540 nm under 12,950 W. As the desired coupler size is centimeter magnitude, the delay deviation between the model and original device is small enough to be ignored. The corresponding coupling coefficients obtained from Eq. (6) are shown in Fig. 11. As shown in Fig. 11, coupling coefficient κ_Si_SiO2 represents the coupling level from silica core to silicon core, and κ_SiO2 _Si pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi shows the contrary case, and κ ¼ κ_Si_SiO2 κ_SiO2 _Si. It is shown that the two cores have the same coupling coefficients at 1550 nm and 1555 nm, which means complete light field coupling under linear and nonlinear conditions. 4. Power dependent pulse advance and delay The time delay (advance) ΔT for a light pulse passing through a coupler with length L is defined as [50] ΔT ¼
L Δng ðλ0 Þ c
ð7Þ
Q. Jing et al. / Optics & Laser Technology 55 (2014) 26–36
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Fig. 13. Switching dynamics of 2.0 ps Sech pulse with 12950 W peak power propagating 1 cm.
where Δng(λ0) is the group index change at the wavelength λ0. Based on the changes of the power controlled neff and ng of the supermodes, the tunable time delay and advance can be obtained by adjusting the pulse intensity. The Hyperbolic-Secant shaped (Sech) pulse with temporal width T0 ¼2.0 ps (FWHM) is used in the simulation. The calculated linear coupling length is Lc ¼2.1398 cm, and the dispersion coefficients at 1550 nm are 1 1 β1 ¼ 4:9097 106 ps km , β2 ¼ 1:7792 102 ps2 km , β3 ¼ 3:8 1 1 2 271 10 ps km for silicon tube, and β1 ¼ 4:8904 106 1 1 ps km , β2 ¼ 3:2283 101 ps2 km , β3 ¼ 1:2896 101 ps3 1 km for silica core. The nonlinear coefficients of silicon tube 1 and silica core are γ Si ¼ 1:752 103 W1 km and γ SiO2 ¼ 2:9 1 870 W1 km , respectively. The switching dynamics of 2.0 ps Sech pulse with 0.0789 W peak power (i.e. satisfying the fundamental soliton condition N ¼ ðγP 0 T 20 =jβ2 jÞ1=2 ¼ 1 in silicon tube) and 1550 nm central wavelength propagating 4.2796 cm (i.e. twice of the Lc ¼2.1398 cm) is presented in Fig. 12.
As shown in Fig. 12, the pulse has complete power transferring at linear coupling length with no distortion. The numerical results have verified that the coupler has the same switching dynamics for pulse injected into either silicon tube or silica core. When the peak power is increased to 12,950 W, the total power transferring at 2.1398 cm is cut off and the coupling length decreases to 3.8107 mm. The corresponding dispersion coefficients for silicon 1 tube at 1550 nm are changed to: β1 ¼ 5:1081 106 ps km , 1 1 3 2 2 3 β2 ¼ 1:2074 10 ps km and β3 ¼ 4:6281 10 ps km , while the dispersion coefficient changes of silica core can be ignored. Fig. 13 depicts the switching dynamics of the pulse injected into silicon tube within 1 cm. It is shown in Fig. 13(a) that due to the increased dispersion coefficient β2, the pulse has been obviously broadened after propagating 1 cm. Additionally, the positive β2 together with the self-phase modulation (SPM) and self-steepening (SS) have enhanced oscillations and steepening on the pulse edges [44].
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Fig. 14. Comparison of the pulse temporal profiles between peak power 0.0789 W and 12,950 W.
Fig. 16. Comparison of the pulse temporal profiles between 1 cm and 4.2796 cm under 12,950 W.
Fig. 15. Switching dynamics of 2.0 ps Sech pulse with 12,950 W peak power propagating 4.2796 cm.
Q. Jing et al. / Optics & Laser Technology 55 (2014) 26–36
As shown in Fig. 13(b), the normalized amplitude in silica core at 1 cm is less than 0.05 which is much lower than the 0.4 under the linear condition. The pulse temporal profiles at 1 cm generated from 0.0789 W and 12,950 W are compared in Fig. 14. As presented in Fig. 14, the time delay phenomenon has been generated in silicon tube. The corresponding time delay for 0.0789 W and 12950 W is 0.2 ps and 2.0 ps, respectively. It is shown that the broadened temporal width (4.2 ps) and steepened edges are the costs of generating 2.0 ps delay within 1 cm. In the common on–off keying (OOK) modulation system, the doubled pulse width is the limit of generating code errors. As a result, the shortest length and largest power for generating one pulse width delay is 1 cm and 12,950 W, respectively. If the same pulse is injected into silica core with 12,950 W, the time advance phenomenon is excited. Fig. 15 shows the switching dynamics of the pulse injected into silica core within 4.2796 cm. As shown in Fig. 15(a), the pulse profile has not been obviously deteriorated after propagating 4.2796 cm. This result can be ascribed to the much longer dispersion length LD ¼39.9 m and the much longer nonlinear length LNL ¼2.6 cm in silica core, while the two parameters in silicon tube are LD ¼ 1.1 m and LNL ¼44.1 μm under the same power. It is shown in Fig. 15(b) that due to the differences of β, κ and κ′, the pulse has been split since 7.6214 mm (i.e. twice of the Lc ¼3.8107 mm), and the normalized amplitude within the whole propagating length is less than 0.04. The pulse temporal profiles at 1 cm and 4.2796 cm under 12,950 W are compared in Fig. 16. As shown in Fig. 16, the silica core excided pulse switching dynamics has generated 2.0 ps and 9.3 ps time advance at 1 cm and 4.2796 cm, respectively. It is obvious that the pulse after propagation has almost the same temporal profile with slight variations on the pedestal. We can conclude that the proposed coupler could generate five times of the pulse width (i.e. 10 ps) advance within 5 cm length without obvious distortion.
5. Conclusion We have proposed an asymmetric dual-core hybrid PCF coupler for generating power dependent time delay and advance. The left core of this coupler is constructed by a silicon tube in one air hole, and the right core is created by omitting seven air holes. In order to analytically calculate the coupling coefficient, we propose an equivalent solid dual-core model which has the similar coupling characteristics as the PCF coupler. As a result of the unbalanced chirps generated from the SPM and normal dispersion, the pulse profile will be broadened and steepened after time delay propagation. In order to avoid pulse interferences caused by broadened temporal width, the shortest device length for generating 2.0 ps delay is 1 cm. Further numerical results show that the coupler can generate more than 10.0 ps advance within 5 cm length without significant pulse distortion. This coupler will have practical applications in wavelength-selective narrowband filter, high bit rate optical buffer and delay line functionality. This work was supported by National Basic Research Program of China (2010CB327605), National Natural Science Foundation of China (61077049), Program for New Century Excellent Talents in University of China (NCET-08-0736) and the 111 Project of China (B07005). References [1] Chu S, Wong S. Linear pulse propagation in a absorbing medium. Physical Review Letters 1982;48:738–41. [2] Boyd RW, Gauthier DJ. ‘Slow’ and ‘Fast’ light. In: Wolf E, editor. Progress in Optic, 43. Amsterdam: Elsevier; 2002. p. 497–530.
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Contents lists available at SciVerse ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Power dependent pulse delay with asymmetric dual-core hybrid photonic crystal fiber coupler Qi Jing n, Xia Zhang, Wei Wei, Yongqing Huang, Xiaomin Ren State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, No. 10, XiTuCheng Road, HaiDian District, Beijing 100876, China
art ic l e i nf o
a b s t r a c t
Article history: Received 4 March 2013 Received in revised form 19 May 2013 Accepted 22 June 2013
We propose a novel asymmetric dual-core hybrid photonic crystal fiber (PCF) coupler composed of a silicon tube as the left core and a silica core as the right core. The control of picosecond pulse delay is achievable by means of power adjusting. The transmission modes, dispersion characteristics and coupling coefficients of the proposed coupler are investigated numerically. The results demonstrate that it is possible to obtain 2.0 ps time delay for soliton pulse with 2.0 ps temporal width within 1 cm length. Further numerical results show that the coupler can generate 10.0 ps undistorted time advance within 5 cm length. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Asymmetric dual-core PCF Slow and fast light Soliton
1. Introduction Slow light phenomenon has attracted a considerable amount of attention since the first experimental demonstration in the 1980s [1]. Many important applications for slow light have also been proposed, such as data synchronization in optical packet switching networks at bit or packet level [2–4], optical buffers and data storage [5–7], and the highly sensitive rotation sensing [8,9]. Generating slow light involves controlling the pulse velocity in optical materials. The corresponding technologies include coherent population oscillations (CPO) [10,11], electromagnetically induced transparency (EIT) [12,13], stimulated Brillouin scattering (SBS) [14,15], coupled-resonator optical waveguide (CROW) [16,17], photonic crystal waveguides [18,19]. However, the limitation for CPO is unsuitable for picosecond and femtosecond pulses, and the EIT needs extreme temperature conditions. The SBS method can be applied at room temperature but its gain bandwidth is usually within tens of megahertz. The CROW could be realized in waveguide with micrometer size but the suitable pulse width is nanosecond. The photonic crystal-based resonance structures could generate both broad bandwidth and large temporal but the regulating process requires auxiliary temperature control. Dual-core fiber couplers have been studied extensively due to its diverse applications such as pulse reshaping and switching, dispersion compensation and multiplexer and demultiplexer [20–22]. However, the conventional dual-core fibers are more complex in production [23]. Due to the development of the fabricating
n
Corresponding author. Tel.: +86 01066820388. E-mail address: [email protected] (Q. Jing).
0030-3992/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlastec.2013.06.029
technology of microstructure fibers, the emergence of dual-core PCF based couplers provide a new approach for the fiber couplers [24,25] and have obtained various applications due to the short lengths, high effective nonlinear coefficients, strong evanescent field and advantages in integration [26–28]. Recently, dual-core PCF couplers based pulse velocity control has attracted much attention. In 2009, an asymmetric dual-core PCF coupler with six large air holes surrounded left core and one small central air hole constructed right core has been used to generate 46.8 ps delay in 1 m length [29]. Due to the smaller index differences between the two cores, the dual-core coupler drawn from silica based PCF with inconsistent air hole parameters has less design flexibilities and lower delay generation efficiency. Instead of changing the PCF geometric microstructure, integrating crystalline semiconductors and metals into silica capillaries can generate controllable dispersion and nonlinearity characteristics and offer a fundamental platform to exploit the hybrid optoelectronic materials and devices [30]. Depositing germanium [31], indium antimonide [32], zinc selenide [33] and silicon [34,35] into PCF pores have been realized by using molten core technique [31] and high-pressure chemical vapor deposition (HPCVD) technique [30,33]. In 2007, a dual-core coupler with two different deposited materials has generated both the total internal reflection index-guided and the photonic bandgap mechanism [36]. In 2009, a PCF coupler composed of two asymmetric silica cores with lower indices than the background pure silica has been used as wavelength-selective coupler [37]. As the promotion of the hybrid deposition technology, the crystalline silicon tubes can be fabricated in silica capillaries, and its thickness can be precisely controlled [30,38,39]. In this paper, an asymmetric dual-core hybrid PCF coupler is proposed.
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27
The coupler left core is constructed by depositing silicon tube in one air hole, and its right core is created by omitting seven air holes to serve as a solid silica core. The effective index dissimilarity between the two cores is obtained due to the higher refractive index and nonlinear coefficient of the silicon. The mechanisms of the power dependent pulse delay with the two asymmetric cores are investigated numerically. The results show that the 2.0 ps time delay for Hyperbolic-Secant pulse with 2.0 ps temporal width can be realized within 1 cm length. Additionally, the effects of the slow and fast light phenomena on the pulse temporal profiles are analyzed theoretically. Based on its linear and nonlinear characteristics, this coupler will have practical applications in wavelength-selective narrowband filter [36], high bit rate optical buffer and delay line functionality [29].
2. Device principle and mode coupling characteristics The cross-section of the designed asymmetric dual-core hybrid PCF coupler is illustrated in Fig. 1(a). The air holes are arranged in a triangular lattice in the background of pure silica. The hole-pitch of the two nearest air holes is set as Λ. The diameter of the air holes is d and refractive index is 1.0. The PCF coupler is composed of a large core and a small core. The large core is created by filling seven air holes with silica, and the small core is obtained by depositing silicon tube in an air hole. The center-to-center distance of the two cores is 4Λ. The refractive index of the background silica and the silicon material at 1.550 μm is 1.45 and 3.43, respectively. Fig. 1(b) shows the detailed geometric structure of the silicon tube which is surrounded by six air holes. The silicon tube outer diameter d1 is equal to the air hole diameter d, and the inner diameter d2 satisfies the condition of 0 od2 od. In this asymmetric dual-core coupler, the confined light propagation in the silica core abides by the index-guiding mechanism [40]. However, the mode guiding mechanism in the silicon tube is neither the complete index-guiding mechanism nor the bandgapguiding mechanism [41], since the refractive index of the silicon tube is higher than both of the outside silica and the inside air hole. As a result, the light propagation in the left core is almost constrained in the tube structure. According to the coupled-mode theory, each core in this PCF coupler can be treated as an independent waveguide that is perturbed by the dispersive mode fields propagating in the other core. When the mismatch of the propagation constants between the two cores is large, the light field propagation is independent in each core without power transferring. As the refractive index of the deposited silicon tube can be precisely controlled by varying its structural parameters, the phase matching condition β1(λ0)¼β2(λ0) between the two cores can be obtained at a complete coupling wavelength λ0. As a result, the maximal power transferring between the two cores is constrained within a bandwidth with a central wavelength λ0. The narrow bandwidth for effective coupling between the two cores can be realized by the effective index which is sensitive to the geometric parameters. The asymmetric core arrangement leads to two non-degenerate guided modes with respective dispersion characteristic and mode field distribution. In this paper, the effects of the asymmetric dual-core parameters on the mode effective indices, group velocity dispersion and mode filed distributions are analyzed by finite-element method (FEM) [42] with anisotropic perfectly matched layers (PML). The diameter of air hole is equal to the outer diameter of the silicon tube and is fixed at d ¼d1 ¼0.72 μm. The requirement for endlessly single mode in PCF with one silica-core is the relative hole diameter d/Λo 0.43 [43]. In order to maintain the light propagation in the fundamental mode state, the d/Λ is fixed at 0.3 and the Λ is chosen as 2.0 μm, 2.4 μm, and 3.0 μm. The effective
Fig. 1. Cross-section of the designed asymmetric dual-core hybrid PCF coupler (a) and the silicon tube structure of the left core (b). The background area denotes pure SiO2, the gray color areas represent air holes, and the dark tube denotes the deposited Si.
indices neff of the two individual modes are calculated after setting those parameters and then the supermodes neff are obtained. During the simulation, the silicon tube inner diameter d2 is chosen as 0.5587 μm, 0.6596 μm and 0.8147 μm for the purpose of satisfying λ0 ¼1.550 μm when the Λ changes. The results show that the deviation of d2 for 2.0 μm oΛo 3.0 μm is less than 0.26 μm. The neff of the corresponding modes are shown in Fig. 2. As shown in Fig. 2(a), the real parts of the fundamental mode neff in silicon tube and silica core decreases as the wavelength increases, and the neff curves intersect at the complete coupling wavelength λ0 ¼ 1.550 μm under the linear condition. In the wavelength region λo1.550 μm, the neff of the silicon tube is larger than that of the neff in silica core. The neff in silicon tube decreases fast as the wavelength
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two individual modes increases as the Λ increases but the phase matching wavelength is maintaining at 1.550 μm. As the Λ increases from 2.0 μm to 3.0 μm, the neff curves of supermodes and individual modes become closer, which means the supermodes neff has larger slope changing at 1.550 μm as Λ becomes larger. According to Vg ¼ (2πc/λ2)(dλ/dβ) [44], the group index ng of the individual modes and supermodes can be expressed as ng_Si=SiO2 ¼ ng_E=O ¼
dβSi=SiO 2 dω
2
λ ¼ 2π
dβSi=SiO 2 dλ
c V g_Si=SiO2
¼c
c
dβE=O λ2 dβE=O ¼ dω 2π dλ
V g_E=O
¼c
ð1Þ
where Vg, β and c denotes to group velocity, propagation constant and the speed of light in vacuum, respectively. The subscripts Si/SiO2 and E/O correspond to the individual modes (silicon/silica) and the supermodes (even/odd), respectively. Results in Fig. 3(a) show the wavelength dependent ng of the two supermodes for three different Λ, and the differences of ng between the two supermodes and the two individual modes are presented in Fig. 3(b). As shown in Fig. 3(a), the ng curves of the two supermodes intersect at complete coupling wavelength 1.550 μm and have steeper slope within 1.545–1.555 μm as the Λ increases. It is shown in Fig. 3(b) that the ng of the supermodes is higher than that of the silica core and is lower than that of the silicon tube. Furthermore, the maximal changing rate of the ng differences lies in 1.545– 1.555 μm.
Fig. 2. Effective indices neff of the two individual guiding modes (in the SiO2 core and the Si tube) and the supermodes (Even mode and Odd mode) as a function of wavelength under linear condition. The hole-pitch Λ is set as 2.0 μm (a), 2.4 μm (b), and 3.0 μm (c), respectively.
increases, and its value becomes smaller than that of the silica core as λ41.550 μm. In both shorter and larger wavelength regions around λ0, the neff differences between individual modes and supermodes are not obvious. However, either of the supermodes neff curve obtains a turning point at λ0 and leaves the original trend for the other individual mode. It is clearly shown in Fig. 2(a)–(c) that the neff of
Fig. 3. Wavelength dependent ng of the two supermodes (a) and the ng differences between the two supermodes and the two individual modes for Λ ¼ 2.0 μm, 2.4 μm, and 3.0 μm (b).
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The GVD (group velocity dispersion) of the supermodes is defined as [44] DE=O ¼
d 1 ð Þ dλ V g_E=O
ð2Þ
The effects of the Λ variations on the supermodes GVD within 1.540–1.560 μm are depicted in Fig. 4. It is shown that the supermodes GVD has peak values at 1.550 μm and increases rapidly as the Λ increases. The maximal value is 727500 ps/km/nm, 7 32500 ps/km/nm and 742500 ps/ km/nm for Λ ¼2.0 μm, 2.4 μm and 3.0 μm, respectively. The Λ¼ 2.4 μm and d2 ¼0.6596 μm are chosen as the main structural parameters in calculations for concisely. The x-polarization electric field distribution mode (Ex) at the 1.545 μm, 1.550 μm and 1.555 μm are evaluated by the FEM and are shown in Fig. 5. As the y-polarized state has the similar field distribution as the x-polarized state and it is omitted for simplicity. As shown in Fig. 5(c) and (d), the odd mode and even mode have symmetric field distribution at 1.550 μm, which matches the
Fig. 4. Wavelength dependent effects of the Λ on the supermodes GVD.
29
complete linear coupling wavelength λ0 in Fig. 2. Due to the large neff differences between the two cores at λo1.550 μm or λ41.550 μm, the light propagation in each core can hardly transfers into the other one. The symmetric mode field distribution can only be excited at 1.550 μm. These inferences can be verified in Fig. 5(a), (b), (e) and (f) where the linear coupling is restrained and the supermodes fields have obvious asymmetric distribution. According to the supermodes ng shown in Fig. 3 and the mode field distributions in Fig. 5, the full-width at half-maximum (FWHM) bandwidth of the linear coupling is less than 5 nm. Considering the sensitive effects of the geometric parameters on the neff, a stronger coupling can be achieved at an expected wavelength by changing d, d1, d2 and Λ. When short (i.e. picosecond) pulses with higher intensity are injected into one core of the PCF coupler, the neff of silicon tube and silica core will be changed as [44] ~ nðω; jEj2 Þ ¼ nðωÞ þ n2 jEj2
ð3Þ
where n(ω) is the linear part, n2 is the nonlinear-index coefficient, |E|2 is the light intensity. Additionally, the power induced neff change can be expressed as Δn ¼ n2 jEj2 ¼ n2 P 0 =Aeff where P0 is the pulse peak power and Aeff ¼ πr2 denotes the effective mode field area. The Aeff of the fundamental modes in silicon tube and silica core are calculated as 10.01 μm2 and 40.19 μm2. We choose the peak power as P0 ¼12950 W and the Δn of the silicon tube and silica core is 5.596 10 3 and 7.087 10 6 1.095 10 5. It is evident that the neff change of silicon tube is much larger than that of silica core, so that the power dependent changes of neff in silica core can be ignored. The neff and ng of the individual modes and supermodes under 12950 W are shown in Fig. 6. As shown in Fig. 6, the complete coupling characteristics at 1.550 μm are destroyed and the coupling wavelength is shifted to 1.555 μm. The corresponding GVD values are shown in Fig. 7. It is shown that the supermodes GVD peak value varies to 1.555 μm and increases from 732500 ps/km/nm to 734100 ps/ km/nm. Due to power dependent changes of the neff, ng and GVD, the injected pulse can exhibit tunable propagation delay and advance phenomena.
Fig. 5. The x-polarization electric field distribution mode (Ex) of the supermodes at 1545 nm (a/b), 1550 nm (c/d) and 1555 nm (e/f). The symmetric (c/d) and asymmetric (a/b/e/f) characteristics of the mode distribution are depicted.
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Q. Jing et al. / Optics & Laser Technology 55 (2014) 26–36
dispersion and nonlinear effects. The full vectorial beam propagation method (BPM) [45] has been widely used to investigate the propagating distance dependent light field. For the first time, a full vectorial finite-element beam propagation method (FEM-BPM) [46] is presented for efficiently analyzing 3-D anisotropic optical waveguides, diffuse waveguides and magnetooptic devices. In order to focus on revealing the time dependent and power controlled pulse switching dynamics and time delay in the two asymmetric cores, the coupledmode theories in the vectorial form are used. The nonlinear Schrödinger equations (NLSE) for coupler with two asymmetric cores can be derived as [47,48] ∂A1 ∂z
þ ∑
m¼1
im1 βm1 ∂m A1 m! ∂t m
¼ ðiκ 12 κ′12 ∂t∂ iκ″212 ∂t∂ 2 ÞA2 þ iδa A1 α21 A1 2
þiðγ 1 jA1 j2 þ C 12 jA2 j2 ÞA1
∂A2 ∂z
im1 βm2 ∂m A2 ∂ iκ″21 ∂2 Þ m ¼ ðiκ 21 κ′21 ∂t m! ∂t 2 ∂t 2 m¼1
þ ∑
A1 iδa A2
α2 A2 þ iðγ 2 jA2 j2 þ C 21 jA1 j2 ÞA2 2 ð4Þ
Fig. 6. The neff (a) and ng (b) of the individual modes and supermodes under 12,590 W.
where the subscript n (1, 2) denoting one of the two cores. An ðz; tÞ represents the varying envelopes of the pulses carried by the elemental modes of the two waveguides in isolation, αn is the loss coefficient, βmn is the m-order dispersion coefficient at the center frequency ω0 , γn is the nonlinear coefficient, and C12(C21) is the crossphase modulation (XPM) coefficient between the two cores. The parameters κ 12 and κ 21 are the linear coupling coefficients between the two cores, κ′12 and κ′21 are the first-order coupling coefficient dispersion, which accounts for the wavelength dependence of the coupling coefficient. The κ″12 and κ ″21 describe the second-order dispersive coupling coefficient. δa ¼ 12ðβ01 β02 Þ is used to describe the asymmetry between the dual cores. When the pulse temporal width is within 10 fs 10 ps, the κ″ in silica based coupler is about 10 27 s2/m [48] which is much smaller than the value of κ and κ′, so that the κ″ is ignored in calculation. We choose the retarded frame as T ¼ tβ12 z (Z ¼ z), and neglect the higher-order dispersion coefficients and the XPM coefficient C12(C21), and the simplified equations can be defined as ∂A1 ∂Z
iβ21 ∂ A1 β31 ∂ A1 1 þ ðβ11 β12 Þ ∂A ∂T þ 2! ∂T 2 3! ∂T 3 2
3
∂ α1 ∂A2 ÞA2 A1 þ iδa A1 þ iγ 1 jA1 j2 A1 ∂T 2 ∂Z iβ22 ∂2 A2 β32 ∂3 A2 ∂ ¼ ðiκ 21 κ′21 ÞA1 þ ∂T 2! ∂T 2 3! ∂T 3
¼ ðiκ 12 κ′12
Fig. 7. Wavelength dependent effects of the Λ on the GVD of the supermodes when the injected power is 12,590 W.
3. Equivalent solid core model for calculating coupling coefficients Optical switches and wavelength-selectors can be constructed by dual-core couplers. The distorted soliton pulses can be used to realize power switching which is supported by the balance between
Fig. 8. Cross-section of the equivalent solid dual-core hybrid coupler. The small core is Si and the large core is SiO2 and the cladding is solid material.
Q. Jing et al. / Optics & Laser Technology 55 (2014) 26–36
Fig. 9. Wavelength dependent neff of the model (a) and the neff differences between the model and original coupler (b) under linear and nonlinear conditions.
31
Fig. 11. Wavelength dependent coupling coefficients of equivalent model under linear (a) and nonlinear (b) conditions.
According to the conventional coupled-mode theory [49], the analytical expression of coupling coefficients based on the HE11 modes is used in calculation: ðρp ρq Þ1=2 κ pq ¼
V 2p u2p
þ ðwq ρp =ρq Þ
2
Bp1=4 Bq1=4 Ap
1=2 1=2 ðzÞ GðtÞ Bq Gpq pq 2Bp
ðF p F q Þ1=2 ð6Þ
Fig. 10. The propagation delay deviations between the model and original coupler on the scale of ps/cm.
2 α2 A2 iδa A2 þ iγ 2 A2 A2 2
ð5Þ
As shown in Eq. (5), the βmn and δa can be derived from the neff, and the γn can be obtained from the γ n ¼ n2 ω=cAeff [44]. The coupling coefficients κ and κ′ are crucial for depicting pulse dynamics and can be numerically calculated, but the process is complex and tedious.
where p, q¼1, 2 (or 2, 1), ρp ¼ dp =2, V p ¼ k0 ρp np ð1n2cl =n2p Þ1=2 , 1=2 2 up ¼ ρp ðk0 n2p β2p Þ , wp ¼ ðV 2p u2p Þ1=2 , and k0 ¼ 2π=λ. The parameters A, B, F and G are defined in reference [49]. The first-order coupling coefficient dispersion can be obtained as: κ′pq ¼ ∂κ pq =∂ω. It should be clearly noted that the pulse propagation in the asymmetric hybrid PCF coupler cannot be directly solved by Eqs. (5) and (6) which are unique for the directional coupler with two parallel solid cores. In order to analytically calculate the coupling coefficient, an equivalent solid dualcore fiber model is proposed in this paper and its cross-section is illustrated in Fig. 8. As shown in Fig. 8, the silicon core, silica core and cladding have solid geometric constructure, and the refractive indices are nSi and nSiO2 and ncl, respectively. The diameters of two cores are dSi and dSiO2 , and the center-to-center distance d0 is 4Λ (9.6 μm). In order to make the model has similar refractive index as original coupler, the equivalent diameter of solid silicon core should meet the condition of 0odSi od1, and the refractive index ncl of the solid background should bepffiffiffiwithin nair oncl onSiO2 . According to the effective core radius Λ= 3 for the triangular PCF whose core is created by missing
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Fig. 12. Switching dynamics of 2.0 ps Sech pulse with 0.0789 W peak power propagating 4.2796 cm.
one air hole [43],pthe ffiffiffi equivalent diameter of the silica core should be suitable for 2Λ= 3 odSiO2 o 4Λ. In the simulation, we optimize the diameters and the refractive index of the equivalent model, and choose the parameters as: ncl ¼ 1.428, dSi ¼ 0.6810 μm and dSiO2 ¼ 4.2967 μm. The used temporal span is 40 ps with resolution 4.9 fs, and the spatial step is chosen as 10 μm. The calculated equivalent neff are shown in Fig. 9(a) under linear (low power) and nonlinear (12, 950 W) conditions and the index differences between the virtual model and the original coupler are shown in Fig. 9(b). As shown in Fig. 9(a), the equivalent model has the same coupling wavelengths 1550 nm and 1555 nm as the original coupler. It is shown in Fig. 9(b) that the maximal neff difference between the model and original device under low power condition is 2.2 10 3 (silicon core) and 2.3 10 4 (silica core). Due to the large input power 12,950 W, the maximal neff difference of silicon core is enlarged to 3.5 10 3. The neff differences induced light propagation delay deviations have been depicted in Fig. 10. As shown in Fig. 10, the delay deviation of the silicon core is more sensitive to the light intensity than that of the silica core, and
the maximal value is 1.14 10 1 ps/cm at 1540 nm under 12,950 W. As the desired coupler size is centimeter magnitude, the delay deviation between the model and original device is small enough to be ignored. The corresponding coupling coefficients obtained from Eq. (6) are shown in Fig. 11. As shown in Fig. 11, coupling coefficient κ_Si_SiO2 represents the coupling level from silica core to silicon core, and κ_SiO2 _Si pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi shows the contrary case, and κ ¼ κ_Si_SiO2 κ_SiO2 _Si. It is shown that the two cores have the same coupling coefficients at 1550 nm and 1555 nm, which means complete light field coupling under linear and nonlinear conditions. 4. Power dependent pulse advance and delay The time delay (advance) ΔT for a light pulse passing through a coupler with length L is defined as [50] ΔT ¼
L Δng ðλ0 Þ c
ð7Þ
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Fig. 13. Switching dynamics of 2.0 ps Sech pulse with 12950 W peak power propagating 1 cm.
where Δng(λ0) is the group index change at the wavelength λ0. Based on the changes of the power controlled neff and ng of the supermodes, the tunable time delay and advance can be obtained by adjusting the pulse intensity. The Hyperbolic-Secant shaped (Sech) pulse with temporal width T0 ¼2.0 ps (FWHM) is used in the simulation. The calculated linear coupling length is Lc ¼2.1398 cm, and the dispersion coefficients at 1550 nm are 1 1 β1 ¼ 4:9097 106 ps km , β2 ¼ 1:7792 102 ps2 km , β3 ¼ 3:8 1 1 2 271 10 ps km for silicon tube, and β1 ¼ 4:8904 106 1 1 ps km , β2 ¼ 3:2283 101 ps2 km , β3 ¼ 1:2896 101 ps3 1 km for silica core. The nonlinear coefficients of silicon tube 1 and silica core are γ Si ¼ 1:752 103 W1 km and γ SiO2 ¼ 2:9 1 870 W1 km , respectively. The switching dynamics of 2.0 ps Sech pulse with 0.0789 W peak power (i.e. satisfying the fundamental soliton condition N ¼ ðγP 0 T 20 =jβ2 jÞ1=2 ¼ 1 in silicon tube) and 1550 nm central wavelength propagating 4.2796 cm (i.e. twice of the Lc ¼2.1398 cm) is presented in Fig. 12.
As shown in Fig. 12, the pulse has complete power transferring at linear coupling length with no distortion. The numerical results have verified that the coupler has the same switching dynamics for pulse injected into either silicon tube or silica core. When the peak power is increased to 12,950 W, the total power transferring at 2.1398 cm is cut off and the coupling length decreases to 3.8107 mm. The corresponding dispersion coefficients for silicon 1 tube at 1550 nm are changed to: β1 ¼ 5:1081 106 ps km , 1 1 3 2 2 3 β2 ¼ 1:2074 10 ps km and β3 ¼ 4:6281 10 ps km , while the dispersion coefficient changes of silica core can be ignored. Fig. 13 depicts the switching dynamics of the pulse injected into silicon tube within 1 cm. It is shown in Fig. 13(a) that due to the increased dispersion coefficient β2, the pulse has been obviously broadened after propagating 1 cm. Additionally, the positive β2 together with the self-phase modulation (SPM) and self-steepening (SS) have enhanced oscillations and steepening on the pulse edges [44].
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Fig. 14. Comparison of the pulse temporal profiles between peak power 0.0789 W and 12,950 W.
Fig. 16. Comparison of the pulse temporal profiles between 1 cm and 4.2796 cm under 12,950 W.
Fig. 15. Switching dynamics of 2.0 ps Sech pulse with 12,950 W peak power propagating 4.2796 cm.
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As shown in Fig. 13(b), the normalized amplitude in silica core at 1 cm is less than 0.05 which is much lower than the 0.4 under the linear condition. The pulse temporal profiles at 1 cm generated from 0.0789 W and 12,950 W are compared in Fig. 14. As presented in Fig. 14, the time delay phenomenon has been generated in silicon tube. The corresponding time delay for 0.0789 W and 12950 W is 0.2 ps and 2.0 ps, respectively. It is shown that the broadened temporal width (4.2 ps) and steepened edges are the costs of generating 2.0 ps delay within 1 cm. In the common on–off keying (OOK) modulation system, the doubled pulse width is the limit of generating code errors. As a result, the shortest length and largest power for generating one pulse width delay is 1 cm and 12,950 W, respectively. If the same pulse is injected into silica core with 12,950 W, the time advance phenomenon is excited. Fig. 15 shows the switching dynamics of the pulse injected into silica core within 4.2796 cm. As shown in Fig. 15(a), the pulse profile has not been obviously deteriorated after propagating 4.2796 cm. This result can be ascribed to the much longer dispersion length LD ¼39.9 m and the much longer nonlinear length LNL ¼2.6 cm in silica core, while the two parameters in silicon tube are LD ¼ 1.1 m and LNL ¼44.1 μm under the same power. It is shown in Fig. 15(b) that due to the differences of β, κ and κ′, the pulse has been split since 7.6214 mm (i.e. twice of the Lc ¼3.8107 mm), and the normalized amplitude within the whole propagating length is less than 0.04. The pulse temporal profiles at 1 cm and 4.2796 cm under 12,950 W are compared in Fig. 16. As shown in Fig. 16, the silica core excided pulse switching dynamics has generated 2.0 ps and 9.3 ps time advance at 1 cm and 4.2796 cm, respectively. It is obvious that the pulse after propagation has almost the same temporal profile with slight variations on the pedestal. We can conclude that the proposed coupler could generate five times of the pulse width (i.e. 10 ps) advance within 5 cm length without obvious distortion.
5. Conclusion We have proposed an asymmetric dual-core hybrid PCF coupler for generating power dependent time delay and advance. The left core of this coupler is constructed by a silicon tube in one air hole, and the right core is created by omitting seven air holes. In order to analytically calculate the coupling coefficient, we propose an equivalent solid dual-core model which has the similar coupling characteristics as the PCF coupler. As a result of the unbalanced chirps generated from the SPM and normal dispersion, the pulse profile will be broadened and steepened after time delay propagation. In order to avoid pulse interferences caused by broadened temporal width, the shortest device length for generating 2.0 ps delay is 1 cm. Further numerical results show that the coupler can generate more than 10.0 ps advance within 5 cm length without significant pulse distortion. This coupler will have practical applications in wavelength-selective narrowband filter, high bit rate optical buffer and delay line functionality. This work was supported by National Basic Research Program of China (2010CB327605), National Natural Science Foundation of China (61077049), Program for New Century Excellent Talents in University of China (NCET-08-0736) and the 111 Project of China (B07005). References [1] Chu S, Wong S. Linear pulse propagation in a absorbing medium. Physical Review Letters 1982;48:738–41. [2] Boyd RW, Gauthier DJ. ‘Slow’ and ‘Fast’ light. In: Wolf E, editor. Progress in Optic, 43. Amsterdam: Elsevier; 2002. p. 497–530.
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