1. Introduction

Recently, slow lights based on optical fiber nonlinearities such as stimulated Brillouin scattering (SBS) [1–7], stimulated Raman scattering (SRS) [8], parametric amplification [9,10], and coherent population oscillations [11], are attracting much attention because of their compatibility with existing telecommunication systems and potential applications such as optical buffering, optical memories, data synchronization, and all-optical signal processing [12]. These slow light systems make use of an optically-controlled narrowband gain process occurring in the optical fiber, in which the group velocity of the signal is strongly altered and consequently a strong slow light effect occurs. In particular, the SBS-based slow light has a number of advantages such as the possibility to create resonance at any wavelength through a simple change of the pump wavelength, the low-power requirement owing to long interaction lengths and small mode areas, and room-temperature operation [3].

The main drawback of this SBS-based slow light has been its narrow bandwidth, normally restricted to 30~45 MHz in conventional single-mode optical fibers, limiting the useful data rates to less than a few tens Mb/s. To broaden this bandwidth, the uses of double and triple Brillouin gain peak configuration have been proposed with respect to the SBS single resonance configuration [4,5]. In addition, direct current modulation of a distributed-feedback (DFB) laser to broaden the pump spectrum has been proposed [6,7]. An operation bandwidth up to 12.6 GHz has been demonstrated with this method, supporting a data rate of over 10 Gb/s.

In applications making use of slow light, real interest lies in demonstrating large relative pulse delay without substantial pulse distortion, the former being defined as the pulse delay normalized to the input pulse width. Unfortunately, these two requirements largely oppose each other. When the resonant gain bandwidth is large, the pulse has small distortion at the cost of small relative pulse delay; while at small resonant gain bandwidth, large relative pulse delay is coming at the cost of great distortion. To obtain large relative pulse delay without substantial distortion, an optimized gain spectrum and the matching of gain bandwidth with pulse bandwidth are needed. In this paper, we present a method to achieve a flat-top gain spectrum by overlapping multiple lines gain spectrum, which ensures the maximum relative pulse delay without substantial distortion. A tunable gain bandwidth can be realized by changing the number of spectral lines and the frequency separation between adjacent spectral lines. In experiment, we make use of SBS-based slow light system. A phase modulator is used to modulate the phase of the pump wave, generating a pump wave with multi-line spectrum, and the maximum Brillouin gain bandwidth can be up to ~330 MHz.

2. Theory

First, we recall an optical pulse propagating through linear optical systems in Ref. [4,5]. The output pulse amplitude A(ω,z) in the frequency domain can be related to the input pulse amplitude A(ω,0) by

where z is the length of the medium and k(ω) is the complex wavenumber as a function of frequency ω. Through Taylor expansion we get k(ω)=∑^∞ _j=0 k_j(ω-ω_c)^j/j!, where ω_c is the carrier frequency of the pulse and k_j≡d^jk(ω)/dω^j. k₀=k(ω₀) is the mean wavenumber magnitude of the optical pulse, k ₁ is the inverse of the group velocity, k ₂ is a measure of the dispersion in the group velocity, and others are higher-order dispersion terms. k ₂ together with higher-order dispersion terms cause pulse distortion of which k ₂ is the dominant source in general. Given the Gaussian-shaped input pulse, k ₂ causes pulse broadening but retaining its Gaussian shape while higher-order dispersion terms cause irregular pulse distortion [13]. According to the analysis of transfer function H(ω)=exp(ik(ω)z) of linear optical system in Ref. [4], an ideal medium has a transfer function |H(ω)|=H ₀ with constant amplitude and a phase that varies linearly with frequency ∠H(ω)=t_pω, where t_p is the total propagation time of the pulse. An optical pulse can propagate through an ideal medium without distortion. For a gain system, an ideal medium means a flat-top gain spectrum and a refractive index varying linearly with frequency. It is difficult to create a custom dispersion profile k(ω). However, one can instead create a custom system by combining multiple spectral lines.

In the SBS-based slow light system, we apply phase modulation to the pump wave, and obtain a pump with the carrier frequency and side frequencies. The frequency separation between adjacent spectral lines is equal to the modulation frequency. Every Brillouin gain spectrum induced by spectral lines of the pump wave overlaps, generating a broadband gain spectrum, and the equal-amplitude spectral lines may result in a flat-top gain spectrum. k(ω) has a good linearity in the region of the flat-top. We consider here a system composed of multiple Brillouin gain lines. The wavenumber of such a system is given by

where n₀ is the background refractive index, g₀/z is the line-center amplitude gain coefficient for each line, γ is the bandwidth, δ is the frequency separation between adjacent spectral lines, n is the order of side frequency, m is the maximum order of side frequency, and υ is the detuning from the line center. In all, the total spectral lines number N is 2m+1 in the above system. Total gain G and summation refractive index n_s can be expressed as

The total gain spectrum and summation refractive index with different frequency separation and spectral lines are shown in Fig. 1. Figure 1 (a) and (b) are the evolution of the total gain and summation refractive index with δ/γ=1 when N is 3,5,11 and 41, respectively. It can be seen that the gain bandwidth increases with N, and a broadband and flat-top gain can be obtained through overlapping. As for summation refractive index, the region of linear variation with frequency is extended with N. Under the condition of δ/γ=2 , the simulation results are shown in Fig. 1 (c) and (d), respectively. At a large separation, there is fluctuation on both total gain and summation refractive index in the line center. The fluctuation increases and the linearity of summation refractive index in line center decreases with the frequency separation, which result in great pulse distortion. Therefore, there is a maximum frequency separation for a constrained pulse distortion. The gain bandwidth broadening is realized mainly by increasing the number of spectral lines, while the frequency separation limits the bandwidth tuning range.

 

Fig. 1. The total gain spectrum with δ/γ=1(a) and δ/γ=2 (c), and their corresponding summation refractive index (b) and (d) by overlapping different spectral lines.

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3. Experiment and results

The experimental setup is shown in Fig. 2. Light from a 100 mW, 40 kHz linewidth, 1550 nm wavelength DFB laser is divided into two parts with a 3dB coupler. One part is injected into Brillouin optical fiber ring laser, generating a Brillouin down shift Stokes wave as probe wave; the other part is modulated by a phase modulator, then amplified by an Erbium doped fiber amplifier (EDFA) and routed via circulator OC2 to pump a 500-m conventional single-mode optical fiber (used as slow light medium). The Stokes wave is modulated by an intensity modulator to obtain approximately Gaussian-shaped probe pulse and counter-propagate with respect to the pump wave in the fiber, in which the probe pulse is amplified and the Brillouin gain is determined by pump power. One percent of the pump wave is extracted with a 1:99 coupler, then detected by a confocal scanning F-P optical spectrum analyzer (OSA) with a free spectrum range of 2 GHz (the resolution is 7 MHz).

 

Fig. 2. Experimental setup. P: polarizer; PM: phase modulator; EDFA: Erbium-doped fiber amplifier; VOA: variable optical attenuator; OC: optical circulator; PC: polarizer controller; IM: intensity modulator; OI: optical isolator; D: detector.

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Different side frequencies can be obtained by changing the amplitude and waveform of modulation signal applied in the phase modulator. In experiment, the half wave voltage of the phase modulator is 8.3 V, and the modulation frequency is 42 MHz which is about the Brillouin gain bandwidth in our optical fiber. When modulation signal is single frequency, the amplitude of carrier wave and side frequencies follow Bessel function of the first kind, i.e., the amplitude of the nth order side frequency is $J_n\left(\beta \right)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\exp \left[j\left(\beta \sin x-\mathrm{nx}\right)\right]\bullet \text{d}x,$ , where β is modulation index. When the modulation index is 1.435, the amplitude of carrier wave equals to that of the first order side frequency, and then three equal-amplitude spectral lines can be obtained. Figure 3(a) is the modulation signal of a sine wave, and the corresponding output pump spectrum is shown in Fig. 3(b). It comprises carrier wave, the first and second order side frequencies. The amplitudes of three central spectral lines are equal and large. In order to obtain more equal-amplitude spectral lines, one should apply the modulation signal with multiple frequencies. As for the number of equal-amplitude spectral lines more than three, the solution is not single. In this experiment, we obtain five equal-amplitude spectral lines by use of the modulation signal in Fig. 3(c). The spectrum of this distorted sine wave is shown in Fig. 4, in which Fig. 4 (a) and (b) are amplitude and phase of basic frequency and harmonic waves, respectively. It can be seen that the spectrum mainly include basic frequency, second and third harmonic waves. The corresponding output pump spectrum has eleven spectral lines including the first to fifth order side frequencies and carrier wave, and the five central spectral lines have approximately equal amplitude, as is shown in Fig. 3(d). The frequency separation between adjacent two spectral lines is equal to the modulation frequency. To obtain more spectral lines, we can change the amplitude and waveform of the modulation signal. Another simple method is to use cascaded phase modulators, the amplitude and waveform of modulation signal applied in every phase modulator are the same, but the modulation frequency increases by M times. M denotes the obtained number of the equal-amplitude spectral lines with the first phase modulator, and thus the total number of spectral lines increases exponentially with the number of phase modulators.

 

Fig. 3. The modulation signal of a sine wave (a) and the corresponding output pump spectrum (b); the modulation signal of a distorted sine wave (c) and the corresponding output pump spectrum (d).

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Fig. 4. The spectrum of the modulation signal shown in Fig. 3(c). (a) and (b) are amplitude and phase of basic frequency and harmonic waves, respectively.

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Gaussian-shaped 6 ns and 4.2 ns (FWHM) Stokes pulses as probe pulse are obtained by an intensity modulator. The pump power is controlled to get Brillouin gain value of 20 dB in the case of both three and five equal-amplitude spectral lines of the pump wave. Figure 5 shows the temporal evolution of the Stokes pulse at different input pulse widths. Under the condition of three equal-amplitude spectral lines, for the input Stokes pulses of 6 ns and 4.2 ns, the pump power are both 240 mW, the slow light delay are both 4.7 ns (relative pulse delay are 0.78 and 1.12, respectively), and the corresponding Brillouin gain bandwidth is ~150 MHz. The output Stokes pulses widths vary from 6 ns to 7.6 ns and from 4.2 ns to 7.7 ns, and the broadening factor are 1.27 and 1.83, respectively. Under the condition of five equal-amplitude spectral lines, the pump power are both 450 mW, the slow light delay are both 2.2 ns (relative pulse delay are 0.37 and 0.52, respectively), and the corresponding Brillouin gain bandwidth is ~330 MHz. The Stokes pulse of 6 ns has very slight distortion in waveform and the width is still 6 ns. The Stokes pulse of 4.2 ns is broadened to 4.7 ns with a broadening factor of 1.12. It can be seen that the Brillouin gain bandwidth increases as the number of the spectral lines increases. For a fixed Brillouin gain bandwidth, as the input pulse width increases, pulse distortion decreases at the cost of decreasing relative pulse delay.

 

Fig. 5. The input pulse width is 6 ns (a) and 4.2 ns (b) and different curves represent temporal evolution of input Stokes pulse (solid lines), output Stokes pulse in three equal-amplitude spectral lines (dot lines) and output Stokes pulse in five equal-amplitude spectral lines (dashed lines).

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4. Conclusions

In this paper, we present a method to achieve broadband and flat-top gain spectrum through overlapping multiple gain lines, which can be used to increase slow light relative pulse delay and reduce pulse distortion. The increase of gain bandwidth can be achieved through increasing the number of spectral lines. The key technique is how to obtain multiple equal-amplitude spectral lines. In our experiment, we use a phase modulator to modulate the phase of the pump wave, generating a pump wave with multi-line spectrum. Two different modulation signals are used, generating three and five equal-amplitude lines and achieving Brillouin gain bandwidth of ~150 MHz and ~330 MHz, respectively. More spectral lines can be obtained by changing the amplitude and waveform of the modulation signal. In addition, we propose another simple method using cascaded phase modulators, which makes the spectral lines increase exponentially with the number of phase modulators.