We propose a new method of all-optical buffering with ultra-small core photonic crystal fibers (PCFs) based on stimulated Brillouin scattering. The large refractive index contrast between the near-hollow cladding and the pure-silica core contributes to high nonlinearity of the PCF. Considering the unusual gain spectrum of the ultra-small core PCF, we numerically investigate the influence of these factors to the buffering efficiency. A PCF with a length of 1 meter and a core diameter of 1.06 micrometer is simulated as the storage medium in this paper. It is shown that we can obtain a good buffering efficiency under a very low control power of 2 watt, which promises a significant improvement for the all-optical communication system.
© 2008 Optical Society of America
With the rapid development of the information network, an all-optical memory or buffer has gained a lot of interest recently to realize the storage and buffering of great bits data package, thus to reduce the data loss and enhance the efficiency of the whole communication network. One way is to slow down the light in Moire fiber gratings by greatly reducing its group velocity , but its bandwidth is too narrow to be applicable for high speed system. Another promising method is trapping light in a cooled atomic medium via electromagnetically induced transparency (EIT) [2, 3]. The long storage time (storage time exceeds 1 s for 20-µm-long pulse) and high readout efficiency suggests that it could be useful as a component in long-distance quantum information networks. However, one limitation that the frequency of the data must match precisely with the resonance frequency of the ions or atoms in the storage medium makes it hard for extensive application, and the cryogenic temperature of the operation further increases the production cost. Delightfully, a whole new approach that storing light in an optical fiber through the process of stimulated Brillouin scattering (SBS) was proposed by Z. Zhu et al. recently . They stored light in an optical fiber by converting the optical data pulse into long-lived acoustic excitation via the process of SBS and retrieved them after a controllable time in room temperature. However, a high control-pulse power of ~100 W was needed to ensure the occurrence of SBS in this optical fiber, which may bring risks to the safety and stability of the buffering system.
Photonic crystal fibers (PCFs), which consist of pure silica and a lattice of hollow micro-channels running axially along its length, guide light in the defection of the periodic cladding. Because of the larger refractive index contrast between the netlike cladding and the pure-silica core than the doped fiber, PCFs can have a better ability to confine both acoustic and optical waves to interact in a small region of the core, so as to enhance the nonlinearity of the fiber . However, recent observations have shown that the SBS in PCFs [6, 7] especially in small-core PCFs [8-10] behave unusual. For instance, the Brillouin gain spectrum develops a multi-peaked structure in the 10-GHz range and the threshold power for backward SBS in small-core PCF increases significantly. The multi-peaked structure was attributed to several families of guided acoustic modes, each with different proportions of longitudinal and shear strain, strongly localized to the core . And the increased threshold was rather attributed to the structural variation along the fiber length .
In this paper, we provide a scheme of all-optical buffering in an ultra-small core PCF via the process of SBS. The Brillouin gain spectrum of our PCF is estimated in section 2. In section 3, we mainly introduce our buffering scheme and the numeric method for the following simulation. After considering the unusual characteristic of the gain spectrum of the small-core PCF, we obtain the buffering efficiency under different parameters in section 4. It is shown that a 2-ns-long optical data pulse can be buffered for up to 12 ns with good retrieval efficiency under a very low control-pulse power of 2 W: 20% for 4 ns and 2% for 12 ns. And there are more frequency choices of the control pulse due to the multi-peaked gain spectrum.
2. SBS in ultra-small core PCF
In this paper, we have designed a kind of ultra-small core pure silica PCF as the storage medium for the first time as shown in Fig. 1(a). The pure silica core is surrounded by a triangular array of 0.94-µm-diameter air channels, interhole spacing 1 µm. The core diameter is close to 1.06 µm, and such kind of small-core or smaller-core PCF has been fabricated experimentally . The modal refractive index is calculated to be 1.24, and the effective mode area is 0.5 µm2 for a 1.55-µm-wavelength incident light . In Fig. 1(b) we calculate the modal refractive index of our optical fiber for different-wavelength incident lights with finite element method . The curve shows that the refractive index has a good linear relationship with the wavelength and the fundamental mode of the fiber with 1.55-µm-wavelength incident light is shown in the inset of Fig. 1(b). According to
where nf is the modal refractive index of the fiber and λ is the wavelength of the light, we obtain the group refractive index about 1.523 for our PCF.
To investigate the Brillouin scattering in our PCF, we must first obtain the Brillouin gain spectrum. In general large-core optical fiber, the gain spectrum is a single-peak structure with a Brillouin frequency shift of ~11.2 GHz for the silica fiber according to G. P. Agrawal , and we can simply obtain the Brillouin gain by 
where νB is Brillouin frequency shift, and ΔνB is the Brillouin linewidth which is related to the phonon lifetime. The peak value of the gain occurring at ν=νB is given by
where n is refractive index of the fiber, p 12 is the longitudinal elasto-optic coefficient, c is light velocity in vacuum, λp is the pump wavelength, ρ0 is the material density and νA is acoustic velocity in fiber.
However, in small-core PCF, the Brillouin spectrum is significantly different from that in general fiber. When the core diameter of the fiber is reduced to the scale of the pump light, the coupling between the shear wave and longitudinal wave becomes strong near the silica/air interfaces, and the acoustic modes contain proportions of both shear and dilatational strain, each localized to the core region . Because of the different traveling velocities of the shear and longitudinal waves (vS=3,764.8 m/s and vL=5,970.7 m/s in silica), the coupling acoustic modes result to families of hybrid modes with diverse dispersion relations, each with its own Brillouin frequency shift and linewidth, which causes the multi-peaked structure in the Brillouin gain spectrum. Also, the threshold power for SBS increases greatly as the core diameter of the fiber reduces. This can be explained by the nanostructure of the fiber core, which enables acoustic-optical interactions in a small region of the defects of the periodical nanostructure [17, 18]. According to P. Dainese et al. , we believe that our PCF also has a multi-peaked structure and reduced Brillouin shift in the gain spectrum since our PCF is approaching to the smallest-core PCF in structure and parameters in Ref. 10. We roughly estimate the frequency shift and linewidth of our PCF for different peaks, and then plot the Brillouin gain spectrum shown in Fig. 2. It is shown that there are three gain peaks in the Brillouin spectrum of our ultra-small core PCF, each with a frequency shift 9.76, 9.95 and 10.22 GHz, respectively. The peak gain coefficients are calculated to be 2.97×10-11, 1.84×10-11 and 2.72×10-11 m/W according to Eq. (3). Here, we only qualitatively demonstrate the characteristics of Brillouin gain spectrum in our PCF, which has little influence for the result of our method. A complete understanding of the spectrum relies on precisely measuring the Brillouin gain spectrum through experiment or carefully solving the full acoustic wave equation to obtain the actual acoustic modes.
3. Buffering scheme and numeric method
The light buffering in our scheme mainly consists of two optical-acoustic converting processes based on SBS in optical fiber [19, 20]. After entering the PCF, the data package pulses modulate the refractive index of the medium periodically because of electrostriction, in which a time-varying electric field creates a time-varying change in density of the material. By interacting with an additional ‘write’ pulse propagating from another end of the PCF, the data pulses are converted into a long-lived acoustic wave, which moves on in the fiber with a velocity of sound. In this process, a large part of the data pulse’s energy has been depleted into the form of Stokes wave which is absorbed by the control pulse synchronously. Another part transfers to the acoustic wave and realizes the information storage. After a controllable buffering time, another control ‘read’ pulse propagates with the same direction of the ‘write’ pulse and converts the acoustic wave back into the data pulses. In this process, part of control ‘read’ pulse’s energy is transferred to the data pulse again. Thus we complete the light buffering. The whole process is based on SBS in PCF. The data pulse, control pulse and acoustic wave satisfy the SBS conditions. That is to say, the carrier frequency of the control pulse is lower than the center frequency of the incident data pulse by a Brillouin frequency shift ΩB, which is proportional to the speed of sound in material. The optical pulses in our PCF propagate in fundamental modes with a group velocity c/ng, where c is the light velocity in vacuum. While the acoustic wave maybe exists with different hybrid modes because of the coupling of the shear wave and longitudinal wave. In this case, we can choose different Brillouin frequency shifts corresponding to the gain peak to enhance the particular acoustic excitation and improve the retrieval efficiency at a wavelength near 1.55 µm .
We simulate the process of the light buffering in our PCF by solving the one-dimensional coupled wave equations, which involve a forward data field (+z direction), a backward control field (-z direction), and a forward acoustic field. Under the slowly varying envelop approximation (SVEA), the three-wave coupled wave equations are given by :
where Ad and Ac are slow-varying electric field envelops of the data and control pulses, respectively, and Q is related to the slow-varying envelop of the acoustic wave ρ by Q=2ν 2 a ρ/iγeε 0ΩB τB, νa is acoustic velocity, γe is the electrostriction coefficient of the fiber, ε 0 is the vacuum permittivity, ΩB is the Brillouin frequency shift, Γ=1/2τB is Brillouin linewidth, τB is the acoustic lifetime, νg is the group velocity of the optical pulses in fiber, η=cε0nf/2, c is light velocity in vacuum, nf is the modal refractive index of the fiber, and gB is the SBS gain coefficient. We numerically solve Eqs. (4) by using an implicit finite differencing in time and a down-winding or backward differencing scheme in space . All field amplitudes are known initially at t=0. The incident data pulses are known for all t at z=0 and the control pulses are known for all t at z=L, where L is the length of our PCF.
In our simulation, a 1-m-long PCF is used as the storage medium. A data pulse is approximated by a super-Gaussian envelop of form exp(-t2r) with r=10 for rectangular pulse and r=2 for smooth pulse. Considering the enhanced threshold power of SBS in our ultra-small core PCF, we increase our pump power by 10 times compared with Ref. 4. Therefore, a 100-mW-power data pulse with a center wavelength around 1.55 µm and a full width at half maximum (FWHM) of 2 ns is used in our simulation. The control write/read pulses are also approximated by super-Gaussian pulse with r=6 and a FWHM of 1.5 ns. The data pulse and control pulses are shown in Fig. 3. The data pulse is converted to an acoustic wave after interacting with the control write pulse. After a controllable buffering time Ts, the control read pulse retrieves the acoustic wave back to the incident data pulse. This is the process of light buffering.
4. Numeric results and discussion
Since the Brillouin gain spectrum shows a multi-peaked structure, each peak with a different gain coefficient and Brillouin frequency shift, we choose the control pulse with different power and frequency to satisfy the SBS condition. It shows the retrieval efficiencies for different control power from 0 W to 100 W and different buffering time of Ts=4, 6, 8, 10 and 12 ns in Fig. 4. The retrieval efficiency is defined as the energy ratio between the retrieved data pulse and the incident data pulse. For different gain peak, we calculate the retrieval efficiencies for different control power and buffering time, respectively. We can find that the efficiency curves vary periodically with the control power and the peak values augment as the power increases. Besides, for the same control-pulse power, the retrieval efficiency drops rapidly with the increasing buffering time. This can be explained as a result of the attenuation of the acoustic wave. Since the optical phonon decays exponentially in the medium, when the buffering time exceeds the acoustic lifetime τB, the majority of the converted acoustic wave has depleted, so the readout efficiency is reduced greatly.
The periodical structure of the efficiency curve can be explained by introducing the control pulse area Θ=(gBνgnfε0c/16τB)1/2∫ Ac(t)dt where the integration is carried out over the duration of the control pulse . The retrieved data pulse depends on Θ by~sin(Θ), so when Θ equals to(m+1/2)π, where m =0,1,2,…, the retrieval efficiencies will achieve their peak values as seen in Fig. 4. The peak positions of the retrieval efficiencies for different gain peak are listed in Table 1. By contrast, we both provide the theoretical and stimulant results of the control power for the first three peak efficiencies. It is shown that the results between the theory and simulation are precisely consistent, which verifies that our simulation is reliable. Also, for a higher gain coefficient, the efficiency curve will shift down, while the corresponding efficiency peaks remain the same regardless the different control power. For a large control power, the interaction among the data pulse, acoustic wave and the control pulse experiences resonate enhancement, thus the energy of the retrieval data pulse may exceed that of the original pulse, which may probably explain the efficiency >100% in Fig. 4. However, this process is kind of complicated and the high control power can also bring damage to the optical system.
For the simulation below, we achieve a control-pulse area of ~π/2(m=0) for both the write and read pulses for the first gain peak in the Brillouin spectrum, i.e. the control power is ~2 W. Fig. 5(a), (b) shows one rectangular and smooth data pulse before and after buffering. The blue and green lines to the left side of the dashed line show the incident data pulse in the absence and presence of the control pulse, respectively. The curves to the right side of the dashed line are the retrieved data pulses from the acoustic wave for different buffering time, where we have scaled them by a factor of 2 for clarity. In this case, we achieve a readout efficiency of approximate 20% for Ts=4ns, which is equal to 2 pulse widths of the data. And for 6-pulse-width buffering time Ts=12ns, the readout efficiency turns out to be 2%. It suggests that our method is efficient for high-speed, all-optical data buffering and related applications. The readout efficiency drops with the increasing of the buffering time because of the decay of the acoustic wave. For the smooth data pulse in Fig.5 (b), we show that the readout efficiency is slightly higher than the rectangular pulse in Fig.5 (a). The steep edge of the rectangular data pulse in the time domain contributes to its high-frequency content in the frequency domain. The spectrum of control pulse is slightly wider than the spectrum of data pulse for a little narrower pulse width, so the control pulse is not broad enough to cover the whole data-pulse spectrum. That is why the retrieval efficiency of the smooth pulse is slightly higher than that of rectangular pulse.
To verify that our scheme is also efficient for a sequence of data pulses, we simulate the light buffering for two and three data pulses with rectangular and smooth shape. The control power is also 2 W. It shows the result of two and three data pulses with rectangular and smooth envelops in Fig. 6 and Fig. 7, respectively. Considering the concision of the figure, we haven’t shown the depleted data pulses. The incident data pulses are displayed to the left of the dashed line and the retrieved pulse to the right of it. For the two data pulses, we scale the retrieved data pulses by a factor of 5. The readout efficiency is about 6% for Ts=8ns and 2% for Ts=12ns.
For three data pulses, we scale the retrieved data pulses by a factor of 10 for clarity. The readout efficiency is about 1.6% for Ts=12ns and 1% for Ts=14ns. Similarly, the retrieval efficiencies are slightly higher for smooth pulses than for rectangular pulses. And the retrieved data pulses have a good repetition with the incident pulses, which enables the fidelity of our light buffering and shows the validity of our buffering scheme.
In conclusion, we demonstrate that a kind of ultra-small core PCF can be used as storage medium for all-optical buffering. The PCF with large refractive index contrast between the micro-structured cladding and the pure-silica core can perfectly confine the light and acoustic wave in a small region, thus enhances the acoustic-optical interaction. The buffering process is based on SBS, and the acoustic wave acts as the interim carrier of the optical-pulse information. We can store a sequence of pulse in our PCF for a controllable time limited by the acoustic lifetime. The unusual characteristic of the Brillouin gain spectrum enable us to have more choice of the control pulse to retrieve different acoustic mode, each with different retrieval efficiency. With an ultra-small core PCF instead of general SMF as the storage medium, we obtain comparable buffering efficiencies with that in Ref. 4 while greatly reducing the control power of ~100 W to 2 W. We believe that such buffering based on PCF will bring advantages to the all-optical communication system, not only on the efficiency but also on the safety and stability of the network system. However, there are still some technical difficulties for such buffering, such as, the coupling problems between the ultra-small core PCF and other optical devices, and the difficulty for the fabrication of such kind of small core PCF.
Compared with the general optical fiber, PCFs have many potential advantages in all-optical storage and buffering. First, we can greatly decrease the control-pulse power by reducing the effective mode area as described above. Second, to achieve a longer buffering time, we can choose the fiber material with longer acoustic lifetimes or fill the fiber core with high-pressure gas. Of course, other approach can also be taken to increase the buffering efficiency, such as using fiber material with a larger SBS gain coefficient.
This work was supported by the National Natural Science Foundation of China under Grants No.10574050, 10774054 and 60706013, the National Key Basic Research Special Foundation under Grant No. 2006CB806006 and State Key Laboratory of Precision Spectroscopy.
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