High signal launch powers are often necessary in fiber optic networks to meet signal-to-noise ratio and receiver sensitivity requirements.^{1} One of the drawbacks of high launch powers is the nonlinear phenomenon of stimulated Brillouin scattering (SBS), which can be suppressed through use of advanced transmission fiber,^{1, 2} through proper choice of parameters in single-frequency amplifier design,^{3, 4, 5} or through signal broadening at the transmitter.^{6, 7, 8, 9, 10} Two standard methods have been proposed to suppress SBS through transmitter design: 1. dithering of the optical frequency of the laser transmitter, and 2. imparting a sinusoidal phase modulation to the optical carrier.^{6, 7, 8, 9, 10} Essentially, both these methods involve broadening the optical spectrum, thereby increasing the SBS threshold power.^{9} Whereas these works have generally focused on phase modulation using a single sinusoidal drive (hereafter referred to as single-tone phase modulation) and have largely ignored the phase relationships between dithering and phase modulation tones, we focus our attention on multitone phase modulation and show that the phase relationship among the various tones can impact the resulting SBS threshold by several decibels. Incorporation of a phase-locking mechanism to control the phase relationship among the various phase modulation tones may provide a significant advantage in high-power optical networks.

We begin by considering a general optical carrier with *N* phase modulation tones,

## 1

[TeX:] \documentclass[12pt]{minimal}\begin{document} \begin{equation} E\left(t \right) = E_0 \left(t \right)\prod\limits_{n = 1}^N {\exp [iA_n \cos ({2\pi f_n t + \phi _n})]}, \end{equation}\end{document} $$E\left(t\right)={E}_{0}\left(t\right)\prod _{n=1}^{N}exp[i{A}_{n}cos\left(2\pi {f}_{n}t+{\phi}_{n}\right)],$$*E*

_{0}(

*t*) is the slowly varying envelope, and

*A*

_{n},

*f*

_{n}, and

*ϕ*

_{n}are the amplitude, frequency, and phase of the

*n*th tone, respectively. Since the SBS threshold depends on the distribution of power in the optical spectrum, we must first calculate the optical power spectral density (PSD) and validate against experiment. We calculate the PSD by assuming a cw input and expanding Eq. 1 in a Fourier Bessel series. The result is a series of Dirac delta functions with weighting coefficients that depend on

*A*

_{n}and

*ϕ*

_{n}. Finally, we apply a Gaussian line shape to each peak, where the line width is obtained based on experimental measurement. In Fig. 1, we show a comparison between the optical spectrum from the experiment and our modeling for the case of a single phase modulation tone at

*f*

_{1}= 3.0 GHz and

*A*

_{1}= 5.71 rad. The input signal

*E*

_{0}(

*t*) has a measured full width at half maximum (FWHM) line width of 1.25 GHz. In Fig. 1, we show a comparison between the experimental and theoretical optical spectra for the case of three-tone phase modulation with

*f*

_{1}= 3.0 GHz,

*A*

_{1}= 5.71 rad,

*f*

_{2}= 2.75 GHz,

*A*

_{2}= 4.92 rad,

*f*

_{3}= 2.5 GHz, and

*A*

_{3}= 4.72 rad. The relative phases of the three tones are

*ϕ*

_{1}= 3.05 rad,

*ϕ*

_{2}= 3.67 rad, and

*ϕ*

_{3}= 0. We achieve excellent agreement between theory and experiment for both the single- and multitone cases.

Given an optical field with multitone phase modulation, as in Eq. 1, we calculate the SBS threshold power by integrating the total optical power spectral density over the SBS gain bandwidth *Δf*_{SBS} to obtain the power spectrum as a function of frequency,

## 2

[TeX:] \documentclass[12pt]{minimal}\begin{document} \begin{equation} \tilde P\left(f \right) = \int\nolimits_f^{f + \Delta f_{\rm SBS}} {| {\tilde E({f'})} |^2\, \textit{df}\,'}. \end{equation}\end{document} $$\tilde{P}\left(f\right)={\int}_{f}^{f+\Delta {f}_{\mathrm{SBS}}}|\tilde{E}\left({f}^{\text{'}}\right){|}^{2}\phantom{\rule{0.166667em}{0ex}}\mathit{df}{\phantom{\rule{0.166667em}{0ex}}}^{\text{'}}.$$^{11}Since we calculate the increase in SBS threshold with respect to the case without phase modulation, we must analogously calculate the power spectrum of the signal without any type of phase modulation,

## 3

[TeX:] \documentclass[12pt]{minimal}\begin{document} \begin{equation} \tilde P_0 \left(f \right) = \int\nolimits_f^{f + \Delta f_{\rm SBS}} {| {\tilde E_0 ({f'})} |^2\, \textit{df}\,'}. \end{equation}\end{document} $${\tilde{P}}_{0}\left(f\right)={\int}_{f}^{f+\Delta {f}_{\mathrm{SBS}}}|{\tilde{E}}_{0}\left({f}^{\text{'}}\right){|}^{2}\phantom{\rule{0.166667em}{0ex}}\mathit{df}{\phantom{\rule{0.166667em}{0ex}}}^{\text{'}}.$$## 4

[TeX:] \documentclass[12pt]{minimal}\begin{document} \begin{equation} \Delta P_{\rm SBS}^{th} = - 10\log _{10} \left({\frac{{\tilde P^{\max}}}{{\tilde P_0^{\max}}}} \right). \end{equation}\end{document} $$\Delta {P}_{\mathrm{SBS}}^{th}=-10{log}_{10}\left(\frac{{\tilde{P}}^{max}}{{\tilde{P}}_{0}^{max}}\right).$$Using two tones, *f*_{1} and *f*_{2}, we show the impact that the phase differences between the tones can have on the SBS threshold. We denote the transmitted optical field by

## 5

[TeX:] \documentclass[12pt]{minimal}\begin{document} \begin{eqnarray} E(t) &=& E_0 \exp [iA_1 \cos (2\pi f_1 t + \phi _1) ]\nonumber\\ &&\times\,\exp [ iA_2 \cos (2\pi f_2 t + \phi _2)] \nonumber\\ &=& E_0 \exp [ iA_1 \cos (\phi _1)\cos (2\pi f_1 t)\nonumber\\ && -\, iA_1 \sin (\phi _1)\sin (2\pi f_1 t) ]\nonumber\\ &&\times \exp [ iA_2 \cos ({\phi _2})\cos ({2\pi f_2 t})\nonumber\\ && -\, iA_2 \sin ({\phi _2})\sin ({2\pi f_2 t}) ]. \end{eqnarray}\end{document} $$\begin{array}{ccc}\hfill E\left(t\right)& =& {E}_{0}exp[i{A}_{1}cos(2\pi {f}_{1}t+{\phi}_{1})]\hfill \\ & & \times \phantom{\rule{0.166667em}{0ex}}exp[i{A}_{2}cos(2\pi {f}_{2}t+{\phi}_{2})]\hfill \\ & =& {E}_{0}exp[i{A}_{1}cos\left({\phi}_{1}\right)cos\left(2\pi {f}_{1}t\right)\hfill \\ & & -\phantom{\rule{0.166667em}{0ex}}i{A}_{1}sin\left({\phi}_{1}\right)sin\left(2\pi {f}_{1}t\right)]\hfill \\ & & \times exp[i{A}_{2}cos\left({\phi}_{2}\right)cos\left(2\pi {f}_{2}t\right)\hfill \\ & & -\phantom{\rule{0.166667em}{0ex}}i{A}_{2}sin\left({\phi}_{2}\right)sin\left(2\pi {f}_{2}t\right)].\hfill \end{array}$$*A*

_{1,2}but also on the phase difference between the tones

*ϕ*

_{2}−

*ϕ*

_{1}, it is clear that this relative phase can have an important effect on the SBS threshold.

In Fig. 2 we show a contour plot of the enhancement in SBS threshold as a function of the amplitude *A*_{2} and phase *ϕ*_{2} of the second tone, and setting the amplitude *A*_{1} = 1 rad and phase *ϕ*_{1} = 0 rad of the first tone. We assume *f*_{1} = 2.5 GHz and *f*_{2} = 3.0 GHz. This contour plot clearly shows that by proper choice of phase difference between the two tones, one can significantly increase the SBS threshold compared to a randomly selected phase difference. Please note that it is often standard practice to set the strength of the second tone to be around 2.4 rad (corresponding to the root of the zero-order *J*_{0} Bessel function), because around this value the increase in SBS threshold is relatively insensitive to phase. However, we see that by locking the phases properly, even modest increases in the value of *A*_{2} can lead to substantial increases in the SBS threshold.

Figure 2 shows a similar contour plot for *A*_{1} = *A*_{2} on the *y* axis and the phase difference *ϕ*_{2}−*ϕ*_{1} on the *x* axis. Again, we have *f*_{1} = 2.5 GHz and *f*_{2} = 3.0 GHz. This contour plot also shows the significance of the relative phase between the two tones. Here, we see that if we combine phase locking with equal drive amplitudes, the SBS threshold can be increased by 17 dB using *A*_{1} = *A*_{2} less than 5 rad.

Figure 3 shows validation of SBS threshold calculation as a function of phase difference for the case of two-frequency phase modulation, assuming *f*_{1} = 2.5 GHz, *f*_{2} = 3.0 GHz, and *A*_{1} = *A*_{2} = 1.2 rad. Agreement between modeling and experiment is very good in both cases. Here the experimental measurements were made using the experimental scheme depicted in Fig. 4. The output of a narrow line-width laser was coupled into a 10-GHz electro-optic phase modulator that was electrically driven with two independent rf sources, paired via a 10-GHz rf amplifier. While the rf signals were driven at the same amplitude *A*_{1} = *A*_{2} = *A*, the frequencies *f*_{1} = 2.5 GHz and *f*_{2} = 3.0 were chosen in accordance with Eq. 5, but optimized based on Fig. 2 to yield an additional 17 dB of SBS suppression. The dual-tone, phase modulated optical signal was then launched into subsequent stages to measure the optical spectra and relative SBS threshold with respect to an unmodulated optical signal. Figure 3 shows validation for the same setup, but as a function of amplitude, assuming a fixed phase difference of 120 deg.

In conclusion, we show that phase locking between the multiple tones of a multitone phase modulator can substantially increase the SBS threshold. This result is of particular value in optical networks requiring high launch powers.