1. Introduction

During the past few years, the group-index control of optical pulses, i.e. slow and fast light, in optical fibers have been among the hot issues and widely studied in photonics societies. A number of experimental demonstrations have been reported based on nonlinear optical processes such as stimulated Brillouin or Raman scattering, optical parametric amplification, coherent population oscillation, and soliton collision [1–11]. In most cases, the group index change is measured by the variation of the arrival time of the peak position of a pulse [2–8] or the time delay of the intensity modulated patterns [9, 10]. While the former method is established on the clear basis that the peak of the pulse moves at the group velocity, the latter has not been deeply analyzed for gain-assisted slow light configurations, although the optical intensity-modulation is commonly used in several applications [12–14].

In this paper, we present the theoretical analysis and the experimental confirmation of the effects of the Brillouin slow light on sinusoidally intensity-modulated waveforms. We show that the DC and the AC parts of the waveform experience different Brillouin gain according to the modulation frequency and that the time delay of the intensity-modulated signal are directly related to the phase-index change of the slow light configuration when the gain shape is symmetric. Although the slow light-induced signal degradation has long been pointed out by experimental and numerical investigations [15–17], our analytic approach can also provide better understanding of the effect by the intuition that the random data streams are basically composed of sinusoidally modulated waves with different frequencies.

2. Principle

We consider two types of loss-less intensity modulators for analysis; an ideal intensity modulator (IM) and a conventional electro-optic modulator based on a Mach-Zehnder interferometer (MZI) as illustrated in Fig. 1. E_in (I_in) and E_out (I_out) denote the input and the output amplitudes (intensities) of the electric field of the propagating optical waves, respectively.

 

Fig. 1. Two different types of intensity modulators considered for analysis; (a) an ideal intensity modulator, and (b) an intensity modulator based on a Mach-Zehnder interferometer.

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It is supposed that the output intensity is sinusoidally modulated at the frequency ω_m in the IM, and the phase of the electric field in one of the arms is sinusoidally modulated at ω_m in the MZI. The modulation parameters a, b are positive constants. For convenience, an AC and a DC part of a waveform are defined as depicted in Fig. 2(a). Using $E_{\mathrm{in}}=E_0{e}^{i{\omega }_{0}t}$, E_out and I_out of each case are decomposed into discrete frequency components as follows:

where J_n(x) is the first kind Bessel function of order n. It is noticeable that ω_m-component in I_out of the MZI is simply proportional to J ₁(b)sin a.

 

Fig. 2. (a) Structure of a sinusoidally intensity-modulated waveform. (b) Refractive index (n), gain of electric field, and the induced refractive index change (Δn) of an optical medium under consideration as a function of optical frequency. ω_m is the modulation frequency.

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We assume that the modulated waves are propagated through an optical fiber of length L, where a narrowband gain is applied for the slow light propagation. The ordinary refractive index (n), the applied gain (of the electric-field), and the induced refractive index (Δn) are depicted in Fig. 2(b) with respect to optical frequency (ω), where the propagation loss is ignored. For simplicity, the gain is supposed to be symmetric across ω₀, which is accompanied with an anti-symmetric Δn by Kramers-Kronig relations. With g_n and Δφ_n representing the gain and the relative phase shift (compared to that at ω=ω ₀) of n^th order harmonic component of ω_m, respectively, the optical waves output from the medium are described as follows:

where φ ₀ (≡n₀ ω₀ L/c) is the optical phase shift at ω=ω₀.

It is notable that the higher-order harmonic terms (2ω_m, 3ω_m ‖) in Eq. (4) and (8) become negligible in the case of a small modulation amplitude (b<π/4). Comparing I_out’s before and after the amplification, one can find out a few interesting points as follow:

1) The gain of the DC part has little dependence on ω_m in the case of small b, and is smaller than that of CW waves (g ² ₀) in both cases.

2) The gain of the dominant AC part (ω_m-component) is g0g1 in both cases which depends on ω_m by g₁.

3) The gain of the AC part gradually decreases to g0 as ω_m goes to infinity (i.e. g1 goes to 1), which is half the gain of the CW wave in an exponential scale.

The phase shift of the ω_m-component in Eq. (6) and (8) is the same as the relative phase shift Δφ₁, which is given by

where τ, Δn _{1, c} are the time delay of the dominant AC part (ω_m-component), the phase index change at ω=ω₀+ω_m, and the speed of light in vacuum, respectively. Since Δn₁ is the only term related to the gain in Eq. (9), the pure time delay Δτ by the slow light effect in the medium becomes

Therefore, the time delay of the intensity-modulated waveform in the slow light medium is proportional to the phase index change. Only when the relation between Δn and ω is linear, one has Δn₁/ω_m=dn/dω and Eq. (10) leads to a familiar result of Δτ=Δn_gL/c with the group index change Δn_g. Eq. (10) can be applied for the direct measurement of the phase index variation near the sharp resonance of the slow light scheme such as the SBS in optical fibers.

3. Experiments and results

We constructed an experimental setup as shown in Fig. 3. A 1550 nm laser diode (LD) was used as a light source, and the fiber under test (FUT) was a 2.1 km single-mode fiber with the Brillouin frequency ν_B ~10.844 GHz at the operation wavelength. The output from the LD was divided by a 50/50 coupler. In one arm, an electro-optic modulator (EOM1) and a microwave generator were used to generate first-order sidebands detuned from the carrier wave by the Brillouin frequency (ν_B) of the FUT. The output from the EOM1 was launched into another electro-optic modulator (EOM2) based on the MZI to apply an intensity modulation using an RF generator and a DC bias control. In the other arm, a Brillouin pump wave was prepared using an Er-doped fiber amplifier (EDFA), and launched into the FUT through a circulator in the opposite direction to the probe wave. In the FUT, the powers of the pump and the probe waves were 13 dBm and -14 dBm, respectively. The Brillouin gain for a CW (un-modulated) probe wave in the FUT was about 20.5 dB. After the Brillouin interaction, an optical filter was used to remove the carrier and the anti-Stokes sideband, and the time waveform of the probe (Stokes) wave was recorded using a photodiode (PD) and an oscilloscope with the signal power kept constant by a variable optical attenuator (VOA).

 

Fig. 3. Experimental setup for the measurement of the slow light effect on intensity-modulated waveforms: LD, laser diode; EOM, electro-optic modulator; EDFA, Er-doped fiber amplifier; VOA, variable optical attenuator; PD, photodiode; FUT, fiber under test.

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At first, the Brillouin gain spectrum (BGS) of the FUT was measured, sweeping the microwave frequency in the EOM1 without any modulation in the EOM2. The result matches well with a Lorentzian fit as shown in Fig. 4(a), where the Brillouin gain bandwidth (full width at half maximum in dB-scale) is about 40 MHz.

 

Fig. 4. (a) Brillouin gain spectrum of the fiber under test. (b) Measured waveforms of the intensity modulated probe waves at f_m=20 MHz with the parameters of a=2.03 and b=π/8. The red and the black curves correspond to the case with and without the pump wave, respectively.

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With the microwave frequency set to the ν_B of the FUT, an intensity modulation was applied to the EOM2 with the modulation parameters of a=2.03 and b=π/8 (refer to Fig. 1(b) and Eq. (4)). The modulation frequency f_m (≡ω_m/2π) was swept from 2 to 80 MHz by 2 MHz step, and the time waveforms were recorded for each case. The measured probe waveforms with (red) or without (black) the pump wave are depicted in Fig. 4(b) at f_m=20 MHz, where a time delay of about 10 ns and the change of the AC/DC ratio are observed when the pump is on.

Figure 5(a) shows the gain of the DC and the AC parts of the probe wave with respect to f_m. It is remarkable that the gain of the DC part is maintained with little deviation to a specific value slightly smaller than the CW gain (g ² ₀~20.5 dB), while that of the AC part gradually decreases to half the original gain (in dB-scale) as explained by Eq. (8). The blue curve is the simulation result of the AC gain based on Eq. (8) and the Lorentzian fit in Fig. 4(a), and the red curve represents the best fitting result with the gain bandwidth set to 34 MHz.

The measured time delay and the corresponding phase index change based on Eq. (10) are plotted as a function of f_m in Fig. 5(b), where another notable feature of the modulated waveform is seen that the time delay gradually decreases to zero as f_m becomes larger (a similar shape to the BGS), which is different from the group index variation that becomes negative as the frequency detuning is increased [18]. The blue curve is a simulation result of Δn based on Eq. (10) and the Lorentzian fit in Fig. 4(a), where the peak value and the overall shape match decently with the experimental result except some deviation in the higher frequency region. Similar to the case of Fig. 5(a), the best fitting result (red curves) was achieved with the gain bandwidth set to 34 MHz. We believe that the discrepancy came from the effect of pump depletion resulting in the broadening of the Brillouin gain bandwidth.

Although the drop of the gain in the AC part and the decrease of the time delay play a role of degrading factor in the signal processing based on the Brillouin slow light, they can be easily mitigated by introducing the arbitrary control of the gain bandwidth [5].

 

Fig. 5. (a) Gain of the DC and the AC parts with respect to the modulation frequency (b) Time delay of the modulated waveform and the corresponding phase index change with respect to the modulation frequency. Note that the blue curves are the simulation results using the Lorentzian fit in Fig. 4(a), and the red curves are the best fitting results with the gain bandwidth set to 34 MHz in both figures.

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4. Conclusion

We have demonstrated a theoretical analysis and the experimental confirmation of the effects of Brillouin slow light on sinusoidally intensity-modulated waveforms. It has been shown that the DC and the AC parts of the waveform experience different Brillouin gains as a function of the modulation frequency and that the optical time delay is proportional to the phase-index change generated by the slow light configuration. The group index measurement by the delay of the modulated waveform is valid only when the modulation frequency is very small enough to ignore the effect of dispersion. Additionally, it is notable that the approach of this work is applicable to phase-modulated signals which are also composed of sinusoidally modulated waves with different frequencies.