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Abstract
Slow light of stimulated Brillouin scattering in double-clad As2Se3 chalcogenide photonic crystal fibers was investigated theoretically. The influence of air hole size of the inner and outer cladding of the PCFs on Brillouin gain spectrum, Brillouin threshold time delay, and Brillouin gain by taking into account the contribution of the high-order acoustic mode was numerical simulated by full vectorial finite element method. The simulated results indicate that the properties of slow light are affected more obviously by varying the air filling fraction in the inner cladding and less affected by the air hole size of the outer cladding. We found that with the time delay up to 705ns, Brillouin gain up to 40dB can be achieved with pump power of only 10mW in a 1m long chalcogenide PCF when the air filling fraction in the inner cladding is 0.9. The pertinent results can be of great importance for studying and designing the optical devices or optical sensors based on this kind of PCF.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Slow light means the possibility of controlling and reducing the group velocity of optical signal, which has been studied extensively since last decade due to its many potential applications, such as optical delay, all optical buffering, data synchronization, optical memories, opticalsignal processing, microwave photonics, and precise interferometric instruments [1–4]. Slow light can be obtained by various methods, such as Electromagnetically Induced Transparency (EIT) [5], Coherent Population Oscillation (CPO) [6], Stimulated Raman Scattering (SRS) [7], Stimulated Brillouin Scattering (SBS) [8]. Among these methods, SBS has attracted much more attention in generation of slow light in optical fibers. The main advantages of the SBS over other methods are controlling time delay only by tuning the pump power [9], operating at room temperature, compatible with optical telecommunication system and so on . So slow light based on SBS in optical fibers has been recognized as a key technology for designing optical buffers.
Because of low Brillouin gain coefficient, the effect of time delay is very weak in conventional fibers, thus the time delay is small. Researchers try to find some special high nonlinear materials or structures as the medium to achieve large time delay. Chalcogenide [10] and tellurite glass fibers [11] were used to improve delay efficiency of SBS slow light in recent years, which shows potential ability in improving the time delay. Chalcogenide and tellurite glass fibers are both high nonlinearity compound fibers. The As2Se3 chalcogenide glass possesses very high nonlinear refractive index [12] and low material loss which makes it promising candidate for nonlinear applications such as slow light and supercontinuum generation [13–16], the Brillouin gain coefficient of As2Se3 chalcogenide PCF is at least two orders larger than that of the PCF made with silica glass. In 2005, silica glass fibers were presented and used to control the group velocity of optical signal in slow and fast light via SBS [17]. In 2006, a time delay of 19ns in As2S3 single mode fibers via SBS with pump power of 31mW in a fiber of 10m length was presented [18], then a time delay of 37ns over a 5m long As2Se3 fiber was demonstrated by Song et al. [10] and the figure of merit (FOM) is 110 times better than that observed in a conventional single-mode fiber. And then chalcogenide photonic crystal fibers (PCFs) were used to improve the time delay efficiency, realizing 137ns only in a 1m long fiber [12].
PCF guides light by means of a lattice of air holes running along the fiber axis. It is well known that the transmitting properties depend on waveguide structure, PCF has many geometric parameters to be tailored for designing PCF, such as diameter of air hole in different layers, diameter of core, pitch, air hole arrangement, air hole layers and so on. It was demonstrated that PCF has unique and remarkable Brillouin scattering properties [19]. The SBS effect in PCFs can be either suppressed or enhanced by designing a certain PCF structure. A enhanced SBS gain of a PCF that is made of chalcogenide material was reported in Ref. [20], and the SBS gain and tunable slow light of single-mode As2Se3-chalcogenide Photonic Crystal Fiber was reported and showed that the time delay experienced by the pulse can be tuned with the pump power and structural parameters of As2Se3 -chalcogenide photonic crystal fiber, but the proposed PCF has same size air holes [12]. Up to now, the generation of slow light based on SBS in double-clad As2Se3 chalcogenide PCFs has not been reported yet.
In this paper, we extended our previous work [21] and propose a double-clad As2Se3 chalcogenide PCF to enhance the efficiency of SBS-based slow light generation. We systematically investigated the slow light induced by SBS in double-clad As2Se3 chalcogenide PCFs with different structures, the Brillouin gain of 40dB and time delay of 705ns are achieved with pump power of only 10mW in a 1m long chalcogenide PCF. This implementation shows that the chalcogenide PCF can be one of the best candidate medium for practical applications of SBS slow light.
2. Design of the proposed PCF
The cross-sectional profile of the proposed PCF is shown in Fig. 1, which consists of an array of air holes arranged in hexagonal lattice pattern in As2Se3-based chalcogenide glass. Five layers of air hole are designed with three layers in the inner cladding and two layers in the outer cladding. The pitch denotes the centre to centre distance of air holes which is taken as $\Lambda $. In our simulation, the pitch of the PCF is kept as a constant of 2.6µm. d1 and d2 represent the diameter of air hole in the inner and outer cladding, respectively. ${\textrm{d}_1}/\Lambda $ and ${\textrm{d}_2}/\Lambda $ denote the air filling fraction (AFF) in the inner and outer, respectively. Then the structures of the PCF can be tailored by varying the AFFs in the inner cladding and outer cladding and the properties of slow light are simulated by full vector finite element method (FEM).
3. Principle of SBS base slow light
SBS is a nonlinear phenomenon induced by the interaction between a high-intensity pump wave and a low-intensity signal wave with the help of an acoustic wave. The two waves have unequal Brillouin frequencies, the difference of which is responsible for the generation of acoustic wave via the process of electrostriction. The acoustic waves produce a periodic modulation of the refractive index and generate a moving grating along the fiber. Then Brillouin scattering occurs when light is diffracted backward on this moving grating, giving rise to frequency shifted called Stokes wave and anti-Stokes components.
Suppose ${\omega _p}$ and ${\omega _s}$ are the frequencies of the pump wave and signal wave, their intensities are ${I_p}$ and ${I_s}$, respectively. ${\alpha _p}$ and ${\alpha _s}$ are the fiber loss of pump and single waves respectively. Then the coupled nonlinear differential equations for the pump and signal wave are expressed as [22]
Assuming that the pump wave is undepleted and $({\alpha _p} \approx {\alpha _s}) = \alpha$, then Eqs. (1) and (2) can be defined by the following equation [23] where ${L_{eff}}$ denotes the effective length of the fiber and can be taken form as where L represents the fiber length, $\alpha$ is the material loss of the fiber.The acoustic displacement distribution can be calculated using the relation [24]
where $u$ is the acoustic field in the fiber, ${\omega _a}$ denotes the angular frequency of the acoustic wave, $v_l^{}$ is the longitudinal acoustic velocity, ${v_l} = 2250m/s$. The backscattered SBS only happens when ${\beta _a} = 2{\beta _o}$, where ${\beta _o} = 2\pi {n_{eff}}/{\lambda _p}$ is the optical propagation constant, ${n_{eff}}$ is the effective index of optical fundamental mode and ${\lambda _p}$ is the pump wavelength.The Brillouin frequency shift (BFS) of the $i$ th-order longitudinal acoustic mode is given by [25]
The Brillouin gain spectrum (BGS) between each acoustic mode is expressed as [18]${I_i}$ is the overlap integral between the optical and the $i$ th-order acoustic modes and expressed as [26]
In order to avoid the pulse getting distorted, the value of pump power must be smaller than ${P_{th}}$. That maximum allowable pump power be calculated by using following equation [15]
where ${A_{eff}}$ is the effective mode area, K is the polarization factor.The time delay is a key property of slow light which describe the arrival time difference of output pulse with and without SBS. It can be defined as [27]
where ${P_p}$ is the input pump power.The Brillouin gain (G) can be expressed as [28]
The pulse broadening factor B is taken the following form4. Results and discussions
4.1 Optical and acoustic mode contribution
Figures 2(a) and 2(b) show the simulated plots of the optical fundamental mode and the acoustic displacement distribution of the acoustic fundamental mode with ${\textrm{d}_\textrm{1}}/\Lambda $ = 0.4, ${\textrm{d}_2}/\Lambda $=0.6, respectively. It can be seen that not only the optical field distribution of optical fundamental mode but also that of the acoustic fundamental mode is confined tightly in the core region because of the guided wave structure, this leads to strong optical and acoustic interaction so that the highest peak of Brillouin gain appears here.
However, there are higher acoustic modes existing in these structures of the PCFs. we demonstrated the simulated plots of high-order acoustic modes in Fig. 3 and Fig. 4 with the AFF is 0.4, 0.6 and 0.9 in the inner cladding and outer cladding respectively. It can be seen that the high-order acoustic modes are not tightly confined in the core region as the acoustic fundamental mode is, they penetrate in the inner cladding more or less according to the different structures. In Fig. 3, it obviously indicates that the confinement ability of high-order acoustic mode enhances in fiber core with the AFF in the inner cladding increasing, Thus the mode is more confined in the core area as shown in Fig. 3(a) to (c).
Fig. 3. High order acoustic modes with ${\textrm{d}_2}/\Lambda = 0.6$ and (a) ${\textrm{d}_1}/\Lambda = 0.4$ (b) ${\textrm{d}_1}/\Lambda = 0.6$ (c) ${\textrm{d}_1}/\Lambda = 0.9$
Fig. 4. Higher-order acoustic modes with ${\textrm{d}_1}/\Lambda = 0.4$ and (a) ${\textrm{d}_2}/\Lambda = 0.4$ (b) ${\textrm{d}_2}/\Lambda = 0.6$ (c) ${\textrm{d}_2}/\Lambda = 0.9$.
Figure 4 depicts the distribution of high-order acoustic mode of proposed photonic crystal fibers with different outer claddings. We see that the acoustic field is distributed in the inner cladding, and it changes slightly with increasing of AFF in the outer cladding comparing with that of Fig. 3. In Fig. 4(a), both d1 and d2 are same and small, which can be treated as one cladding with 5 air hole rings, the high-order acoustic mode penetrates in all air hole rings. In Fig. 4(b) and (c), the high-order acoustic mode is confined in the inner cladding but the distribution changes slightly although the AFF in the outer cladding changes obviously. That's to say, the distribution of high acoustic modes is slightly affected by varying AFF in the outer cladding.
Figure 5 depicts the overlap integral as a function of the resonance frequency of the corresponding acoustic mode, which denotes the overlap efficiency between optical mode and acoustic modes. By calculating the spatial overlap between the optical fundamental mode and the acoustic modes, we are able to highlight the relevance of each acoustic mode in the BGS. The overlap efficiency between optical mode and fundamental acoustic mode is the highest because their modes are tightly confined in the core, and it decreases when optical mode overlap with the high-order acoustic modes because the high-order acoustic mode is distributed in the inner or outer cladding.
4.2 Properties of stimulated Brillouin scattering
The variation of Brillouin threshold in the proposed PCFs with the different structures in the inner cladding and outer cladding is shown in Fig. 6. It can be seen from Fig. 6(a) that the Brillouin thresholds of both main peak and second peak decrease when the AFF in the inner cladding increases, but the second peak decreases sharply when the AFFs in the both inner and outer cladding are less than 0.5. The Brillouin threshold is affected more obviously by the AFF in the inner cladding than that by the AFF in the outer cladding, because the high-order acoustic mode is affected more easily by AFF in the inner cladding, as we mentioned in 4.1 section. It also can be seen from Fig. 6(b) that the threshold of second peak decreases obviously when the ${\textrm{d}_2}/\Lambda $ increases from 0.4 to 0.5, then the threshold of main peak keeps constant and that of the second peak decrease slowly with AFF in the inner cladding increasing. The maximum pump power of 10 mW is applied in the simulation in order to avoid the output pulse getting distorted.
Fig. 6. Variation of Brillouin threshold with the AFF (a) in the inner cladding (b) in the outer cladding.
The influence of structural parameters on BGS is shown in Fig. 7 and Fig. 8. Figure 7 depicts the BGS for AFFs in the inner cladding of from 0.4 to 0.9. We can see that the BGS shows two peaks behavior. The fundamental acoustic mode that overlaps with the fundamental optical mode leads to the first peak (higher peak) of the BGS, and the high-order acoustic mode overlaps with the fundamental optical mode to induce the second peak which becomes more and more pronounced with the AFF in the inner cladding increasing. This is because when ${d_1}$ increases, the ${A_{eff}}$ decreases, and the confinement of both optical mode and acoustic modes enhances and focuses to the center of the core, thus the overlap integral between optical mode and high-order acoustic increases to induce higher second peak. In addition, we note that the BGS of the main peak decreases, this is due to the strong index contrast between the fiber core and inner cladding. The PCF illustrates a BFS of the main peak around 8.110 GHz with a second peak around 8.130 GHz as shown in Table 1. When ${\textrm{d}_1}/\Lambda \textrm{ = }0.4$, the PCF illustrates the BFS of the main peak around 8.1301 GHz and the second peak at 8.1407 GHz. the BFS of both main peaks and second peaks decrease with ${\textrm{d}_1}/\Lambda $ increasing.
Table 1. Variation of BFS (GHz) with ${\textrm{d}_1}/\Lambda $
Figure 8 illustrates the BGS varying with AFF in the outer cladding. It can be seen that the second peak is very weak at ${\textrm{d}_2}/\Lambda \textrm{ = }0.4$, this is because both d1 and d2 are so small that the ${A_{eff}}$ is large. The high-order acoustic mode is leaked in the both inner and outer cladding area, although the optical fundamental mode is confined in the fiber core. The second peaks increase obviously with increase of ${\textrm{d}_2}/\Lambda \ge 0.5$,. After that it increases continually. We confirm that the air holes in the outer cladding can confine the high-order acoustic modes when the confinement ability of inner cladding is weak. Besides, the confinement ability of outer cladding becomes stronger with increase of d2. These conclusions are agree on well with those mentioned in section 4.1. In these structures, BFSs of the main peaks and the second peaks keep constants of 8.1301 GHz and 8.1407 GHz respectively. This means that the structure of outer cladding don't affect the BFS. These results agree on well with the result of Ref. [13].
According to Eq. (12), we calculated the Brillouin gain G varies with the AFFs for different pump power as illustrated in Fig. 9. Figures 9(a) and (b) demonstrate AFF in the inner cladding obviously affect the Brillouin gain G of the both main peak and second peak due to dependence of ${g_B}$ and ${A_{eff}}$ on AFF in the inner cladding. It also can be seen that the Brillouin gain G is affected more obviously with the pump power increasing, and the maximum Brillouin gain of 40dB can be obtained at ${\textrm{d}_1}/\Lambda $=0.9 and the pump power of 10 mW. Figure 9(c) and (d) depict the Brillouin gain G of the both main peak and the second peak as a function of the AFF in the outer cladding for different pump power. We found that the AFF in the outer cladding does not affect G of the main peak, but slightly affect G of the second peak. The Brillouin gain of the second peak increases with increase of ${\textrm{d}_\textrm{2}}/\Lambda $ when ${\textrm{d}_\textrm{2}}/\Lambda $ < 0.5, and then keep almost unchanged. The Brillouin gains of the both main and second peak increase with increase of the pump power.
Fig. 9. Brillouin gain varying with AFF, (a) the main peak in the inner cladding (b) the second peak in the inner cladding (c) the main peak in the outer cladding (d) the second peak in the outer cladding.
4.3 Slow light
As mentioned above, the Brillouin gain of the proposed double-clad As2Se3 chalcogenide photonic crystal fiber is so high to make it a potential candidate for slow light generation. The dependence of time delay on structures of the fiber with different input pump power is simulated from Eq. (11) as illustrated in Fig. 10. It can be seen from Figs. 10(a) and (b) that the time delay of both main peak and second peak increases with increasing ${\textrm{d}_1}/\Lambda $ and it also can be tuned by varying the pump power. When ${\textrm{d}_1}/\Lambda = 0.9$, ${P_p}$=10 mW, the maximum time delay can be obtained up to 705 ns. We also can see from Figs. 10(c) and (d) that the time delays keep constant with the increasing of AFF in the outer cladding, that means the AFF in the outer cladding does not affect the delay time. However, when ${\textrm{d}_2}/\Lambda $ < 0.5, the delay time of the second peak increases with increase of ${\textrm{d}_2}/\Lambda $, then ${\textrm{d}_2}/\Lambda $ slightly affect the slow light of the second peak. For either the main peak or the second peak, the delay times increase with the increase of the input pump power as shown in Fig. 11.
Fig. 10. Time delay varying with AFF (a) the main peak in the inner cladding (b) the second peak in the inner cladding (c)the main peak in the outer cladding (d) the second peak in the outer cladding
Fig. 11. Time delay varying with pump power when ${\textrm{d}_1}/\Lambda = 0.9$, ${\textrm{d}_2}/\Lambda = 0.6$
We optimize the structure of PCF by setting ${\textrm{d}_1}/\Lambda = 0.9$ and ${\textrm{d}_2}/\Lambda = 0.6$, and analyze the time delay varying from pump power of both main peak and second peak. Figure 11 indicates that time delay increases linearly with increasing pump power for both main peak and second peak, but the delay time of the main peak increases more sharply than that of the second peak, which indicates that the pump power affect the delay time of the main peak more seriously.
Figure 12 depicts the output pulse waves at the pump power of 1 mW for different AFF (a) in the inner cladding and (b) in the outer cladding. It can be seen from Fig. 12(a) that the delay time of the output wave increases with increasing AFF in the inner cladding, but almost keeps constant by varying AFF in the outer cladding. This conclusion is consistent with that mentioned above. However, the output waves are broadened for both ${\textrm{d}_1}/\Lambda $ and ${\textrm{d}_\textrm{2}}/\Lambda $, the AFF in the inner cladding influences the broadening factor more obviously than the AFF in the outer cladding does. The trailing edge of the output wave becomes flattening compared to the leading edge, and when ${\textrm{d}_1}/\Lambda $ increases, the pulse broadening factor increases, the output pulse waves are broadened namely. In Fig. 12(b), the output waves are almost overlapping, which means the output pulse waves are less affected by the AFF in the outer cladding.
In order to further study the pulse broadening, we demonstrate the pulse broadening factor varying from pump power with the different structures in the inner cladding, as illustrated in Fig. 13. The pulse broadening factor increases with the increase of the pump power, and the larger the pump power is, the more obviously the AFF in the inner cladding affects the pulse broadening factor. Thus, we should choose a proper pump power and the structure of the fiber to avoid the pulse getting distorted.
5. Summary
In conclusion, a highly nonlinear double-clad As2Se3 chalcogenide PCF is proposed and the influence of structures of this kind of fiber on BGS, Brillouin threshold, Brillouin gain as well as tunable slow light generation based on SBS is investigated. We found that the properties of slow light are more affect by varying the air filling fraction in the inner cladding, but less affect by varying the air filling fraction in the outer cladding. Though simulation and optimizing, the Brillouin gain of 40 dB and time delay of 705 ns are achieved with pump power of only 10 mW in a 1 m long double-clad As2Se3 chalcogenide PCF. This implementation shows that the proposed PCF can be one of the best candidate medium for designing optical components based on SBS slow light.
Funding
National Natural Science Foundation of China (61665005); Natural Science Foundation of Gansu Province (17JR5RA132).
Disclosures
The authors declare no conflicts of interest.
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