A theoretical study of transient stimulated Brillouin scattering in optical fibers seeded with phase-modulated light

Clint Zeringue; Iyad Dajani; Shadi Naderi; Gerald T. Moore; Craig Robin

doi:10.1364/OE.20.021196

1. Introduction

Stimulated Brillouin scattering (SBS) is the lowest threshold nonlinear effect in single-frequency continuous-wave (CW) fiber amplifiers [1]. Single-frequency, in this context, refers to a wave whose spectral linewidth is less than the intrinsic gain bandwidth associated with the spontaneous Brillouin scattering. When unseeded, the SBS process is initiated from spontaneous scattering of a thermally excited acoustic noise within the fiber medium [2]. The spontaneous sound wave interacts with the optical laser field to initiate a backward-propagating Stokes wave. When the optical laser power is sufficiently high, the process quickly becomes stimulated since the Stokes and optical laser fields beat together to create a resonant acoustic wave through the process of electrostriction. Various definitions exist for the SBS threshold and in this paper we consider the SBS threshold to occur when the time-averaged reflectivity over several transit times in the fiber is ~1%. Over the years a variety of techniques have been implemented in order to suppress SBS [3–5]. One such technique is to modulate the phase of the laser field [6]. While the effective linewidth of the laser would no longer be single-frequency, this technique has attracted attention due to recent demonstrations of beam combining of high-power fiber amplifiers modulated at the GHz level [7, 8].

In this paper, phase modulation is examined in detail through consideration of the temporal dynamics of SBS. The time-dependent system of equations describing the parametric interaction of two optical waves and a material excitation (e.g. SBS, SRS) has been examined for specific cases using both quantum mechanical and classical treatments [2, 9]. For SBS, analytical solutions for the un-modulated case can be obtained in passive media in the undepleted pump limit [2]. The system reduces to a 2 × 2 system of first-order partial differential equations. This system can then be decoupled into a second-order hyperbolic differential equation for either the acoustic wave or the Stokes wave which can be solved using Riemann’s method [10]. Numerical solutions of the triply-coupled system of equations were also presented by Boyd et al [2]. However, modulation of the signal was not considered in that work. To be certain, the SBS process has also been studied with phase modulation in the Fourier domain. While the physics in certain cases can be described in frequency space and solved as a large coupled system of nonlinear ordinary differential equations, the inclusion of products of non-phase matched terms makes this approach intractable. Typically, these products are neglected in very long fibers and/or when the frequency separation inside the laser spectral envelope is much larger than the spontaneous Brillouin bandwidth. For example, Lichtman et al. [11] examined the steady-state solution for a passive fiber pumped with laser light comprised of discrete equidistant frequencies. After neglecting terms which were not phase matched for the fiber configuration under consideration and also neglecting pump depletion through the SBS process, the solution of the discretized system was extended to include a continuous laser spectrum.

In section 2, we present the theoretical framework for phase modulation in the time domain with no assumptions on pump depletion or fiber length. The SBS process is initiated using a Langevin noise source. The numerical approach and validation is presented in section 3. In section 4, we consider sinusoidal modulations for a broad range of amplitudes and frequencies. In section 5, the effect of a white noise source on the suppression of SBS is examined for both Lorentzian and ${sinc}^{2}$ formats as a function of fiber length and modulation frequency and comparisons are made with analytical solutions. In section 6, the SBS process in a fiber of length 9 m (which is typical for a fiber amplifier) is considered for light modulated using a pseudo-random bit sequence (PRBS), and various patterns of the form $2^{n} - 1$ are analyzed and compared.

2. Theoretical framework

The SBS process is examined using a triply coupled set of partial differential equations which describes a three-wave interaction of two optical fields and an acoustic field. These equations are derived from Maxwell’s equations and the Navier-Stokes equations. We investigate here the case of a polarization-maintaining fiber seeded with light polarized along one of its axis; thus the Stokes light is polarized along the same direction. The total optical electric field is given by: $\tilde{E} = {\tilde{E}}_{L} + {\tilde{E}}_{S}$ , where ${\tilde{E}}_{L}$ and ${\tilde{E}}_{S}$ are the laser and Stokes fields, respectively. It is described by the nonlinear wave equation [2]:

\nabla^{2} \tilde{E} - \frac{n^{2}}{c^{2}} \frac{\partial^{2} \tilde{E}}{\partial t^{2}} = \frac{1}{ε_{o} c^{2}} \frac{\partial^{2} {\tilde{P}}^{(n l)}}{\partial t^{2}}

where the nonlinear polarization,

{\tilde{P}}^{(n l)}

, is due to the modification of the index of refraction as a result of electrostriction and is given by

{\tilde{P}}^{(n l)} = ε_{0} γ_{e} \tilde{ρ} \tilde{E} / ρ_{o}

. _$\tilde{ρ}$ and

ρ_{0}

are the density and background density of the fiber medium, respectively. The electrostrictive constant,

γ_{e}

, describes the change in the dielectric constant of the medium,

ε

, with respect to the change in density and is given by

γ_{e} = ρ (\partial ε / \partial ρ)

. It should be noted for the term on the right side (RHS) of Eq. (1) that only the nonlinear components oscillating at the laser and Stokes frequencies are pertinent to the SBS process.

The evolution of the density, $\tilde{ρ}$ , of the acoustic field is described by the wave equation [2]:

\frac{\partial^{2} \tilde{ρ}}{\partial t^{2}} - \frac{Γ_{B}}{q^{2}} \nabla^{2} \frac{\partial \tilde{ρ}}{\partial t} - v_{S}^{2} \nabla^{2} \tilde{ρ} = - \frac{1}{2} ε_{o} γ_{e} \nabla^{2} 〈 {\tilde{E}}^{2} 〉 + \tilde{f}

where

Γ_{B}

is the phonon decay rate,

v_{S}

is the speed of sound, and

q

is the acoustic wave number. Conservation of energy and momentum dictate that the acoustic and optical wave numbers and angular frequencies are related through:

Ω = ω_{L} - ω_{S}

and

q = k_{L} + k_{S}

, where

Ω

_,

ω_{L}, ω_{S},

are the acoustic, laser, and Stokes angular frequencies, respectively.

k_{L}

and

k_{S}

are the laser and Stokes wave numbers. For the first term on the RHS of Eq. (2), only the product of optical fields oscillating at

Ω

is pertinent. The second term accounts for the initiation of the SBS process from a Langevin noise source. It can be expressed in the following form:

\tilde{f} = - 2 i Ω f (z, t) e^{i (q z - Ω t)} + c . c .

where

f

is a Gaussian random variable with zero mean and is

δ

correlated in space and time. Therefore, it obeys the following relation:

〈 f (z, t) f^{*} (z, t) 〉 = Q δ (z - z') δ (t - t')

where

Q

is a parameter describing the strength of fluctuations. This strength parameter is determined using thermodynamics arguments to be

2 Γ_{B} k T ρ_{o} / v_{S}^{2} A

, where

A

is the effective interaction area [2].

Substituting the total optical field into Eq. (1), and using the slowly-varying envelope approximation, one can obtain the equations which describe the evolution of the amplitudes of the laser and Stokes waves, $A_{L}$ and $A_{S}$ , in the fiber:

\frac{c}{n} \frac{\partial A_{L}}{\partial z} + \frac{\partial A_{L}}{\partial t} = \frac{i ω γ_{e}}{2 n^{2} ρ_{o}} ρ A_{s}

- \frac{c}{n} \frac{\partial A_{S}}{\partial z} + \frac{\partial A_{S}}{\partial t} = \frac{i ω γ_{e}}{2 n^{2} ρ_{o}} ρ^{*} A_{L}

where

ω \approx ω_{L} \approx ω_{S}

, and

ρ

is the amplitude of the acoustic wave. Phase modulation effects are included through the initial conditions and the boundary condition for the electric field at the input end of the fiber:

A_{L} (0, t) = A_{L}^{0} e^{i φ (t)}

,

A_{L} (z > 0, 0) = 0

, where

φ (t)

is the phase modulation function and

A_{L}^{0}

describes the input electric field amplitude of the laser. The Stokes field is counter-propagating and is zero everywhere at

t = 0

. It is subject to the following conditions:

A_{S} (z, 0) = 0

,

A_{S} (L, 0) = 0,

where

L

is the length of the fiber.

It should be noted that implicit in the derivation of Eqs. (5-6) is the assumption that the higher order terms arising from the second-order derivative in time of the nonlinear polarization in the case of phase modulation are small. Specifically, this is justified as long as $d φ (t) / d t$ is much smaller than the optical frequency, i.e. $d φ (t) / d t < < ω$ .

Since the phonons are highly damped, they propagate over very short distances. Consequently, in the phonon equation, the spatial variation of the amplitude is neglected leading to:

\frac{\partial^{2} ρ}{\partial t^{2}} + (Γ_{B} - 2 i Ω) \frac{\partial ρ}{\partial t} + (Ω_{B}^{2} - Ω^{2} - i Ω Γ_{B}) ρ = ε_{o} γ_{e} q^{2} A_{L} A_{S}^{*} - 2 i Ω f

where we have introduced the resonant acoustic frequency of the medium,

Ω_{B} = 2 n v_{S} ω_{L} / c

.

It should be obvious that in the absence of external effects such as temperature variation, stress, fiber impurities, and other sources of variation for the acoustic velocity that only acoustic frequencies at or near the resonant frequency of the medium contribute significantly to the growth of the acoustic wave. Keeping this in mind, we solve Eq. (7) at resonance, $Ω = Ω_{B}$ , leading to:

\frac{\partial^{2} ρ}{\partial t^{2}} + (Γ_{B} - 2 i Ω_{B}) \frac{\partial ρ}{\partial t} - i Ω_{B} Γ_{B} ρ = ε_{o} γ_{e} q^{2} A_{L} A_{S}^{*} - 2 i Ω_{B} f

In comparing Eq. (8) to the corresponding equation in Ref [2], the second-order derivative in time is retained in our case. This term accounts for fast phase modulation on the order of Brillouin frequency. This term was not included in Ref [2] as that study did not consider phase modulation. As an analogy, Damzen et al. discuss the need for retaining the second-order derivative in the description of SBS driven with short pulses [12]. Note also the inclusion of the term

Γ_{B} \partial ρ / \partial t

in Eq. (8). This is because the exclusion of this term while including the second-order derivative term can lead to solutions that grow in time. These solutions are obviously not physically viable. Equations (5), (6), and (8), along with the boundary and initial conditions completely describe the three-wave interaction.

3. Numerical approach and validation

In order to study the effects of a phase-modulated laser at high reflectivity, we perform numerical integration of the coupled set of equations using the method of characteristics [13]. The optical fields are counter propagating and can be solved along the characteristics _{$d z / d t = c / n$} and _{$d z / d t = - c / n$} for the laser field and Stokes, respectively, leading to:

\frac{d A_{L}}{d t} = i σ ρ A_{s}

\frac{d A_{S}}{d t} = i σ ρ^{*} A_{L}

where

σ = ω γ_{e} / 2 n^{2} ρ_{0}

. As discussed above, the phonon equation is devoid of the spatial variation. It can be expressed as:

\frac{α}{Γ_{B}} \frac{d^{2} ρ}{d t^{2}} + (α - i) \frac{d ρ}{d t} - i \frac{Γ_{B}}{2} ρ = χ A_{L} A_{S}^{*} - i f

where

χ \equiv ε_{o} γ_{e} q^{2} / 2 Ω_{B}

,

α \equiv Γ_{B} / Ω_{B}

.

Equations (9)-(11) may be solved using standard ODE numerical routines. In the discrete case the amplitude of the noise term used in Eq. (11) is given by:

f_{j, k} = \sqrt{\frac{n Q}{{(Δ t)}^{2} c}} S_{j, k}

where

j, k

describe the grid points of intersection along the characteristics in space and time respectively,

S_{j, k}

is a complex Gaussian random distribution function with zero mean and unit variance, and

Δ t

is the time-step. The appropriate boundary and initial conditions for the laser field are:

A_{L} (j = 0, k) = A_{L}^{0} e^{i φ (k)}

and

A_{L} (j > 0, k = 0) = 0

, where

j = 0

corresponds to

z = 0

, and

k = 0

to

t = 0

. For the Stokes field, they are:

A_{S} (N_{j}, k) = 0

and

A_{S} (j, 0) = 0

, where

N_{j}

corresponds to

z = L

_. Finally, the initial and boundary conditions for the phonon field are

ρ (j = 0, k) = {ρ^{'}}_{0, k}

and

ρ (j, k = 0) = {ρ^{'}}_{j, 0}

, where

ρ^{'}

is the solution to Eq. (11) without the source term and second-order order derivative which is insignificant without the driving force. It is given by:

{ρ^{'}}_{j, k} = \sqrt{\frac{n Q}{c Γ_{B}}} S_{j, k}

In order to characterize the numerical stability, we solve these equations using both Euler and the modified Euler method for an un-modulated laser field. The two methods are compared by investigating the conservation equation for the SBS process in a passive fiber along the fiber length using 100 spatial points. The time-step and the spatial step are related through $Δ z = c Δ t / n$ · Since, for all practical purposes, the Stokes and laser frequencies are equal, the following conservation equation is obeyed [14]:

< {| E_{L} (z, t) |}^{2} > - < {| E_{S} (z, t) |}^{2} > = C

where

C

is constant. Here the brackets indicate the time-average in the long time limit where

Γ_{B} t > > 1

. It should be noted that this conservation equation is simply a statement of the conservation of the number of photons for counter-propagating waves in the absence of loss.

The parameters used in our simulations are shown in Table 1 . Most of these parameters are typical of silica fibers. The phonon lifetime, $τ_{p}$ , is related to $Γ_{B}$ by $Γ_{B} = 1 / τ_{p}$ . The fiber length and diameter are taken to be 5 m and 10 µm, respectively. Figures 1(a) and 1(b) compare the agreement of the numerical integration with the conservation relation as expressed by Eq. (14) for the Euler and modified Euler methods, respectively, over a broad range of reflectivity. We note that the vertical axis is normalized to the largest computed value at the respective reflectivity. It is clear that in the low reflectivity limit and all the way up to the SBS threshold, defined nominally at a reflectivity of ~1%, both numerical methods agree extremely well with the conservation equation. For the case corresponding to the highest reflectivity of 21%, the modified Euler technique has a maximum variation less than 1% compared to a 6% variation using just the one-step Euler method. We see that both methods indicate variation near the input end of the fiber where the reflectivity is highest. All results presented hereafter are done with the modified Euler technique.

Table 1. Fiber simulation parameters. The density, index of refraction, Brillouin shift, and phonon lifetime parameters are based on approximate values for optical fiber at room temperature and a laser wavelength of 1064nm [14].

View Table

Fig. 1 Normalized conservation equation along fiber length for different reflectivities using a) one-step Euler Method (top) and b) modified Euler method (bottom). $< R >$ defines the time-averaged reflectivity over several transit times in the long time-limit $Γ_{B} t > > 1$ .

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The accuracy of the numerical solver can also be tested by comparing the SBS reflectivity to the analytical approximation in the low-SBS gain limit for an un-modulated laser. The analytical solution is similar to that obtained for the three-wave interaction describing Raman gain [9] and is given by:

A_{S} (z, τ) = \frac{i γ_{e} ω}{2 n ρ_{0} c} A_{L} \int_{0}^{τ} d τ' \int_{0}^{L} d z' e^{- Γ_{B} (τ - τ') / 2} f^{*} (z', τ') I_{0} ({[G Γ_{B} (z - z') (τ - τ') / L]}^{1 / 2})

where

I_{0}

is the modified Bessel function of order zero,

τ = t - z n / c

is the local time variable, and

L

is the length.

G = g_{\max} I L

, where

I

is the intensity of the laser and

g_{\max}

is the peak Brillouin gain factor, corresponds to the logarithm of single-pass intensity amplification. In the long time limit, Eq. (15) reduces to:

R = {(\frac{γ_{e} ω}{2 ρ_{o} n c})}^{2} \frac{Q L}{Γ_{B}} e^{G / 2} [I_{o} (G / 2) - I_{1} (G / 2)]

where

I_{1}

is the modified Bessel function of order 1 and

R

is the reflectivity (i.e.

R = {| A_{S} (z = 0) |}^{2} / {| A_{L} (z = 0 |}^{2}

).

Figure 2 shows the results of this comparison for the fiber parameters listed in Table 1 using 100 spatial points.

Fig. 2 Comparison between the reflectivity for undepleted pump as obtained from Eq. (16) and numerical integration of the equations using a distributed fluctuating noise source. The reflectivity in the case of the numerical integration is averaged over several transit times in the long time limit.

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In the limit where single-pass SBS gain is small, we obtain excellent agreement between the analytical approximation and the numerical integration of the coupled system. For the numerical solution, we observe logarithmic growth in the reflectivity over the entire gain range and departure from the undepleted pump approximation near $G ~ 25$ which is in excellent agreement with previous results [2].

Another useful comparison to measure the accuracy of the discrete noise term and the numerical integration is to compare the full width at half maximum (FWHM) of the power spectral density (PSD) of the Stokes field at the input end, $P_{S} (z = 0, ω)$ , as obtained from the numerical simulations to that obtained using the undepleted pump limit. It is well-known that, as the gain increases, the Brillouin gain bandwidth narrows. In the undepleted pump limit, the relation is of the form:

P_{S} (z = 0, ω) \propto [\exp (\frac{G {(Γ {}_{B}/ 2)}^{2}}{ω^{2} + {(Γ_{B} / 2)}^{2}}) - 1]

Figure 3 shows the comparison with the plots normalized to the phonon decay rate

Γ_{B}

. The numerical solution exhibits narrowing of the Stokes bandwidth. We see that in the low SBS gain limit we have good agreement with Eq. (17) and a departure from the analytical approximation as the single-pass SBS gain is increased. This to be expected since Eq. (17) is only valid for an undepleted pump. We also notice that we are able to numerically resolve the spontaneous Brillouin bandwidth. These results are in agreement with Ref [2]. It is not clear to us which numerical method was used in Ref [2] to solve the triply coupled system.

Fig. 3 Comparison between analytical SBS gain bandwidth for undepleted pump using Eq. (17) and numerical integration of the equations using a distributed fluctuating noise source. There is a departure as the single-pass SBS gain is increased. We also observe an asymptotic convergence to the spontaneous Brillouin bandwidth at low gain.

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4. Sinusoidal phase modulation

In this section we explore the SBS suppression due to phase modulating a monochromatic field with a single-frequency sinusoidal modulation function with modulation amplitude $γ$ and modulation frequency $Δ ω$ . The modulation function is given by:

φ (t) = γ \sin (Δ ω t)

For this type of modulation function the resulting power spectral density of the optical field is well-known. The time dependence of the input electric field is of the form:

{\tilde{E}}_{L} (0, t) = A_{L}^{0} [\begin{matrix} J_{0} (γ) \cos (ω t) + J_{1} (γ) \cos (ω + Δ ω) t - J_{1} (γ) \cos (ω - Δ ω) t \\ + J_{2} (γ) \cos (ω + 2 Δ ω) t - J_{2} (γ) \cos (ω - 2 Δ ω) t + ..... \end{matrix}]

Therefore the spectral power exhibits a series of discrete sidebands at integer multiples of the modulation frequency. The PSD of the

n^{t h}

sideband is given as:

P_{L, n} \propto J_{n}^{2} (γ)

where

J_{n}

is the

n^{t h}

order Bessel function of the first kind.

We characterize the SBS threshold as a function of phase modulation amplitude and frequency for a specific fiber length of 5 m and a core diameter of 10 µm. In all cases, we define the SBS threshold as the input field power for which the time-averaged reflectivity over several transit times (transit time is given by $τ_{t r} = n L / c$ ) is ~1%. In order to capture the effects of phase modulation, we ensure a temporal spacing corresponding to the lesser value of 50 points per modulation periods or 50 points per phonon lifetime. In addition, the total simulation time encapsulated at least 20 transit times. The SBS threshold enhancement normalized to the un-modulated threshold is shown in Fig. 4 for various combinations of $γ$ and $Δ ω$ using the fiber parameters shown in Table 1. The x-axis in the figure is normalized to the spontaneous Brillouin bandwidth.

Fig. 4 Numerical results (solid dots) of SBS threshold enhancement factor vs. normalized modulation frequency for various modulation amplitudes. Solid curves represent best fit for numerical results. As expected, the results indicate asymptotic convergence to the theoretical approximation of Eq. (21) in the large modulation frequency limit.

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An analytical expression for the SBS threshold can be obtained using a heuristic treatment in the frequency domain. For large modulation frequencies, $Δ ω > > Γ_{B}$ , the Stokes modes resulting from the phase modulation of the optical field act independently and the SBS threshold is determined by the sideband with the highest spectral power. To that end the SBS threshold enhancement factor as compared to an un-modulated wave is given by [15]:

\frac{P_{t h}}{P_{t h}^{0}} = \frac{1}{J_{n, \max}^{2} (γ)}

where

P_{t h}

is the SBS threshold,

P_{t h}^{0}

is the SBS threshold for the case of an un-modulated field, and

J_{n, \max} (γ)

is the Bessel function of the 1st kind corresponding to the sideband with the maximum value. The SBS enhancement factor as provided by Eq. (21) is plotted in Fig. 4 for comparison to the numerical simulations.

When considering modulation frequencies within the Brillouin bandwidth, $Δ ω / Γ_{B} < 1$ , little enhancement is expected regardless of modulation amplitude due to a large degree of overlap between the SBS spectra in the sidebands. We see from Fig. 4 that the expected enhancement in this regime is minimal for all modulation amplitudes. For the range of modulation frequencies $0.5 Γ_{B} \leq Δ ω \leq 5 Γ_{B}$ , the SBS threshold increases with modulation frequency. This regime also describes the transition from strong to weak overlap among the SBS in the optical sidebands. When considering even larger modulation frequencies, $ω_{F M} > 5 Γ_{B}$ , additional enhancement in SBS suppression is minimal since in this regime the Brillouin gain overlap among the sidebands is very small. As expected, in the large modulation frequency limit, the threshold enhancement approaches the approximation of Eq. (21). Inspection of Eq. (21) indicates that the general trend is an increase in the enhancement factor is expected with an increase in $γ$ . In practice, there is an upper limit to the voltage that can be applied to the electro-optic modulator to achieve the desired $γ$ .

Qualitatively, the effect of the overlap can be seen by comparing the SBS spectra near SBS threshold at $γ = 1.435$ for $Δ ω = 2 Γ_{B}$ and $Δ ω = 20 Γ_{B}$ . Using Eq. (19), one may verify that this specific modulation depth results in three equal intensity lobes near the center of the spectrum for the laser field. The PSD of the laser and Stokes fields for these two cases are shown in Fig. 5 . For simplicity, the spectra are normalized about their respective carrier frequencies. It is important to note that the large amount of power near the carrier Stokes frequency in Fig. 5(a) (i.e. $Δ ω = 2 Γ_{B}$ ) is a result of appreciable overlap in the Brillouin gain bandwidth of the sidebands and non-phase matched cross interactions among the pump and Stokes fields as they propagate inside a relatively short fiber. As a result, the central Stokes encounters greater SBS and is higher than the two inner bands by 12 dB. On the contrary, in the case where the pump bands are separated by $Δ ω = 20 Γ_{B}$ , the overlap is minimal and the Stokes spectrum closely resembles that of the laser with three equal intensity sidebands (i.e. the carrier frequency and the two inner sidebands). In this case the difference is less than 1.5 dB as shown in Fig. 5(b). These features cannot be explored using Eq. (21) since cross interactions were ignored in the derivation of that equation.

Fig. 5 Results of the PSD for the laser and Stokes fields near SBS threshold for the case $γ = 1.435$ with a) $Δ ω = 2 Γ_{B}$ and b) $Δ ω = 20 Γ_{B}$ . Both curves are normalized about their respective optical carrier frequencies. The results show that in the small modulation regime interactions among the various sidebands tend to feed the carrier frequency leading to a higher central Stokes mode. The PSD was generated using the total simulation time.

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For large modulation frequencies our results are in agreement with previously published experimental results [15]. We note that the departure from the simple analytical solution for small modulation frequencies can also be conceivably captured by solving a nonlinear system of equations in the frequency domain. However, in this case, one has to track the cross interactions among the laser and Stokes sidebands. Furthermore, the stochastic nature of the SBS process is not accounted for, which can be important when considering random phase fluctuations in the input laser.

5. White noise modulation

We now consider the effect of white noise phase modulation on the SBS process [7,8]. A description of this technique is provided in Ref [16]. A white-noise source (WNS) is sent through a filter with a pass band control mechanism, and then through an RF amplifier. The amplified RF signal drives an electro-optic modulator which increases the linewidth of the seed laser; producing a “broadband” output. The initial WNS is delta correlated in the sense that:

〈 φ (t) φ (t^{'}) 〉 = C δ (t - t^{'})

〈 φ (t) 〉 = 0

where

C

is a constant. The combination of band pass filter and RF amplification provides the desired lineshape and linewidth. We will consider two different lineshapes. In one case, the lineshape mimics a

{sinc}^{2}

function. In the other case, it is a Lorentzian which (along with a Gaussian lineshape) is a more traditional lineshape for a WNS phase modulation scheme. As will be discussed below, the latter is related to the often quoted formula for SBS suppression due to a phase modulated (or broadband) laser [14]:

\frac{P_{t h}}{P_{t h}^{0}} = 1 + \frac{Δ ω}{Γ_{B}}

where

Δ ω

is the effective linewidth,

P_{t h}

is the SBS threshold of the broadband laser, and

P_{t h}^{0}

is the SBS threshold for a single-frequency laser (i.e. laser with a linewidth much smaller than the Brillouin gain bandwidth).

Figure 6 shows an example of the PSD for a modulation depth of $π$ and a FWHM of 2 GHz. The intensity is plotted on a logarithmic scale. The ${sinc}^{2}$ behavior of the envelope is a result of a rectangular band pass filter response. The envelope shows the expected distribution on a typical optical spectrum analyzer (OSA) with GHz resolution. We show up to 5 nulls of the ${sinc}^{2}$ envelope on each side of the carrier frequency in Fig. 6(a). A “zoomed-in” region of the primary envelope is shown in Fig. 6(b).

Fig. 6 PSD of optical field driven with a white-noise source (WNS) passed through a 2 GHz band-pass filter and possessing a ${sinc}^{2}$ envelope. Figure 6(a) shows up to 5 nulls on each side of the carrier frequency. Figure 6(b) is the “zoomed-in” region of the primary envelope.

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One can also shape the spectrum into a Lorentzian lineshape by modifying the filter response and modulation depth. To shape the spectrum into a Lorentzian with the desired FWHM, the RF signal is convoluted with a $sinc$ function and the appropriate RF gain needed to approximate a Lorentzian lineshape is applied. To this end, a simple root-solving technique with the merit function being the FWHM of the resultant Lorentzian lineshape was employed. Figure 7 shows the resultant lineshape for, a desired FWHM of 2 GHz using the algorithm just described. Due to the randomly generated noise, there is no defined spacing between any of the spectral modes and the spectrum is nearly continuous (in this case with a Lorentzian envelope). Simulations were performed on a passive fiber using the same parameters listed in Table 1.

Fig. 7 Resultant power spectral density (PSD) of optical field driven with a white-noise source (WNS) tailored to produce a Lorentzian lineshape with FWHM of 2 GHz.

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The SBS process under phase modulated conditions was previously studied in the frequency domain by Lichtman et al. [11] in the un-depleted pump limit. Furthermore, there were two main assumptions made in deriving the analytic expressions for the Brillouin gain: (1) The spacing between optical modes as a result of the phase modulation was large and/or (2) the fiber was long. With these assumptions the phase-mismatched terms arising from the modulated laser and Stokes fields become insignificant and the effective Brillouin gain, $g_{e f f}$ , was derived to be:

g_{e f f} (ω_{S}) = g_{0} (Γ_{B} / 2) L \int_{- \infty}^{\infty} \frac{I (ω) d ω}{{(Γ_{B} / 2)}^{2} + {(ω - ω_{S} - Ω_{B})}^{2}}

where

I (ω)

describes the intensity distribution of optical modes in the frequency domain as a result of the phase modulation scheme and

g_{0} = γ_{e}^{2} ω^{2} / ρ_{0} n c^{3} v_{S} Γ_{B}

. Therefore,

g_{e f f}

is a convolution of the laser spectrum and the spontaneous Brillouin linewidth. The SBS threshold is determined by the maximum value of

g_{e f f}

which occurs near the resonance condition. We carried out the integration for the single-frequency, i.e.

I (ω) = I^{(0)} δ (ω - ω_{L})

, as well as the two WNS formats described above. For the Lorentzian lineshape, Eq. (24) is recovered. We will derive here, the enhancement factor for the

{sinc}^{2}

format of the WNS. The maximum effective Brillouin gain,

g_{e f f, \max}

, is given by:

g_{e f f, \max} = g_{0} (Γ_{B} / 2) L \int_{- \infty}^{\infty} \frac{A \sin^{2} [(ω - ω_{L}) π / Δ ω] d ω}{[{(Γ_{B} / 2)}^{2} + {(ω - ω_{L})}^{2}] {[(ω - ω_{L}) π / Δ ω]}^{2}}

The normalization constant,

A

, can be determined by integrating the intensity distribution over the frequency spectrum and setting it equal to

I^{(0)}

, i.e. by setting the intensity in the

{sinc}^{2}

case equal to the intensity in the single frequency case. This leads to

A = I^{(0)} / Δ ω

. By making the substitution

w = (ω - ω_{L}) π / Δ ω

and performing some manipulations, one can express Eq. (26) as:

g_{e f f, \max} = \frac{g_{0} Γ_{B} π I^{(0)} L}{2 {(Δ ω)}^{2} a^{2}} \int_{- \infty}^{\infty} (\sin c^{2} w - \frac{\sin^{2} w}{a^{2} + w^{2}}) d w

where

a^{2} = {(Γ_{B} π / 2 Δ ω)}^{2}

. The integral of the

{sinc}^{2}

function from

- \infty

to

\infty

is

π

. The second integral can be broken into two parts. One part can be integrated through a trigonometric substitution while the second can be integrated through analytic continuation to the complex plane and the application of the Residue Theorem. The following equation is then obtained:

g_{e f f, \max} = \frac{g_{0} Γ_{B} π^{2} I^{(0)} L}{2 {(Δ ω)}^{2} a^{2}} (1 - \frac{1}{2 a} + \frac{e^{- 2 a}}{2 a})

The SBS threshold enhancement factor for the

{sinc}^{2}

format of the WNS is the ratio of its

g_{e f f, \max}

to that of the single-frequency case. It can be readily shown, that the latter is

2 g_{0} L I^{(0)} / Γ_{B}

. This leads to the following expression:

\frac{P_{t h}}{P_{t h}^{0}} = \frac{π}{(e^{- π Γ_{B} / Δ ω} - 1) \frac{Δ ω}{Γ_{B}} + π}

In addition, it is straightforward to show that for large

Δ ω / Γ_{B}

the enhancement factor approaches

0.64 Δ ω / Γ_{B}

. In other words, this treatment indicates that for large

Δ ω / Γ_{B}

, the

{sinc}^{2}

format provides 0.64 the SBS suppression as the Lorentzian format.

We conducted a study using the time-dependent SBS model to examine the validity of Eq. (24) and Eq. (29). The advantage to modeling the system of equations in the time-domain is that the cross-interactions and phase-mismatched terms are handled naturally as opposed to modeling the system in the Fourier domain where keeping track of these interactions can become difficult. Figure 8 shows the enhancement in the SBS threshold as a function of the normalized FWHM of the Lorentzian spectral lineshape for different fiber lengths varying from 2 m to 85 m. The normalization of the latter is with respect to the spontaneous Brillouin bandwidth. For each fiber length, the enhancement factor represents the SBS threshold at a particular FWHM divided by the SBS threshold for the no modulation case (i.e. single-frequency case) at that length. The solid line represents the enhancement factor according to Eq. (24). The ten least steep lines correspond to fiber lengths in the range 2 m to 11 m in increments of 1 m. As shown, there is a strong dependence on the fiber length for relatively short lengths. At a fiber length of 85 m, the enhancement factor is in accordance with Eq. (24). No further enhancement is obtained by increasing the fiber length further; thus indicating Eq. (24) represents the theoretical upper limit. The discrepancy between the analytical solution of Eq. (24) and the simulated results for short fiber lengths is an artifact of the approximations made in deriving Eq. (24) which do not account for cross-interactions between phase mismatched terms as the fields propagate down the fiber. For sufficiently long fiber, the overall contribution of the phase mismatched terms will become very small compared to the phase matched terms; thus the SBS threshold enhancement factor will approach that provided by Eq. (24). Since fiber amplifiers are typically 5-10 m long, there should be a significant departure from the “expected” SBS suppression in amplifiers. This departure is in general agreement with recent experiment results [17].

Fig. 8 Enhancement factor vs. normalized FWHM for the Lorentzian white noise at different lengths of fiber. The ten least steep lines correspond to fiber lengths in the range 2 m to 11 m in increments of 1 m. For long fiber, the enhancement factor approaches the theoretical limit provided by Eq. (24). For the parameters given in Table 1, a normalized linewidth of 25 corresponds to 1.4 GHz.

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A similar set of simulations were conducted for the non-optimized case ( ${sinc}^{2}$ format of the WNS). The results are shown in Fig. 9 . The solid line represents the theoretical upper limit as determined by Eq. (29). To be certain, this equation does not strictly represent a straight line but for the range of values plotted, it closely follows a linear dependence. As in the case of the Lorentzian lineshape, for sufficiently long fiber, the enhancement factor closely resembles the derived analytical formula. In this case, this analytical formula is provided by Eq. (29). It is interesting to note that even at a fiber length of 2 meters, the enhancement factor for the ${sinc}^{2}$ format is still ~0.64 that of the Lorentzian; albeit both enhancement factors are lower than their respective formulas as provided by Eq. (24) and Eq. (29).

Fig. 9 Enhancement factor vs. normalized linewidth for the ${sinc}^{2}$ white noise at different fiber lengths. For long fiber, the enhancement factor approaches the theoretical limit provided by Eq. (29).

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6. Pseudo-random bit sequence (PRBS)

Another effective scheme for SBS suppression is to apply a PRBS phase modulation format. The modulation formats of a WNS and PRBS are considerably different. When the phase modulator is driven by a PRBS, the resultant optical power spectral density contains discrete periodic features that are a function of the modulation frequency and the PRBS pattern length. A PRBS pattern is typically denoted as $2^{n} - 1$ . The power $(n)$ indicates the shift register length used to create the pattern. The $2^{n} - 1$ patterns contain every possible combination of $n$ number of bits 0 and 1 except the null pattern. For example, a $2^{3} - 1$ pattern contains a periodic structure of 7 bits. In this study, we take the 0 and 1 to represent modulation depths of 0 and $π$ , respectively. The temporal spacing between each bit is determined by the modulation frequency of the PRBS generator. Figure 10 shows the normalized power spectral density of an optical field phase modulated with a PRBS $2^{3} - 1$ at 2 GHz. The spectrum contains a series of modes with a spacing of $Δ v = v_{p m} / (2^{n} - 1)$ , where $v_{p m}$ is the PRBS modulation frequency. In this case, the modes are separated by $Δ v = 2 / (2^{3} - 1) \approx 0.29$ GHz. In addition, the envelope carries a ${sinc}^{2}$ dependence with nulls at integer multiples of the modulation frequency $v_{p m}$ .

Fig. 10 a) Normalized power spectral density and b) “zoomed-in” region of optical field phase-modulated with PRBS pattern $2^{3} - 1$ with a modulation frequency of 2 GHz.

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We used the $2^{3} - 1$ pattern in the time-dependent model to study the SBS as a function of modulation frequency using the fiber parameters provided in Table 1 for a fiber of length 9 m and a core diameter of 10 µm. The fiber length is typical of that used in high power fiber amplifiers. The results are shown in Fig. 11 . A maximum enhancement factor of approximately 6 is obtained. There is a rollover in the enhancement at ~1.5 GHz. At this modulation frequency, the separation among the adjacent sidebands lying within the envelope is appreciably more than twice the spontaneous Brillouin bandwidth. Consequently, the sidebands generated by the PRBS act independently and thus the SBS threshold is dictated mainly by the modes carrying the highest amount of optical power. Any further increase in the modulation frequency beyond this point is minimal.

Fig. 11 Enhancement factor vs. modulation frequency for PRBS pattern $2^{3} - 1$ for a 9 m fiber.

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We also conducted a study using the PRBS patterns $n = 5, 7, 9, 17, 31$ for the same length of fiber. Previous experimental results showed that the pattern $n = 7$ was much superior to $n = 17, 31$ in high power amplifiers at modulation frequencies of the order of GHz [18]. The modulation frequency was varied all the way up to 5 GHz (i.e. approximately 90 times the spontaneous Brillouin bandwidth). The results of this study are shown in Fig. 12 . It is clear from this figure that the $n = 5, 7, 9$ PRBS patterns are superior to $n = 17, 31$ for all modulation frequencies considered in these simulations. Qualitatively, one can explain these results by looking at the instantaneous phase of the optical field over several phonon lifetimes.

Fig. 12 a) SBS threshold enhancement factor vs. modulation frequency for different PRBS patterns b) Sample phase as a function of time over several phonon lifetimes for $n = 7, 17, 31$ at 2 GHz modulation frequency. The time is normalized to $2 π τ_{p}$ .

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In Fig. 12(b), we plot this phase over ten lifetimes for $n = 7, 17, 31$ at a modulation frequency of 2 GHz. Clearly, the PRBS patterns for $n = 17, 31$ result in long dwell times where the field is un-modulated over the phonon lifetime. During these un-modulated windows, a significant amount of SBS is generated. Alternatively, it can be argued that for $n = 17, 31$ the period of these sequences at GHz level modulation is much longer than both the phonon lifetime and the roundtrip time of the light in a fiber of length 9 m. In contrast, the PRBS for $n = 5, 7, 9$ induces several $π$ phase shifts within a phonon lifetime. This significantly disrupts the transient buildup of the SBS process and results in a relatively large enhancement in the SBS threshold. Consequently, as shown in Fig. 12(a), an SBS threshold enhancement factor of >45 can be obtained in a fiber of length 9 m at a modulation frequency of 5 GHz. This length of fiber is typical of that used in high power fiber amplifiers. Note that the $n = 9$ pattern overtakes $n = 5$ at ~2.2 GHz, and $n = 7$ at ~4 GHz. As is in the case of $n = 3$ , a rollover in the enhancement factor is expected for these patterns as the modulation frequency is increased further.

7. Conclusion

In summary we have presented simulations from a time-dependent model to study the effects of phase modulation on SBS in fibers where noise is initiated from a Langevin background noise and the effect of modulation is considered by imposing a time-dependent boundary condition at the input end of the fiber. The triply coupled nonlinear system was solved using the method of characteristics. The numerical solutions were validated by comparing to known solutions for the un-modulated laser case and to the case of a sinusoidal modulation. For the latter, our model captures the cross interactions among the laser and the Stokes sidebands. The SBS process was also considered for phase modulation through a WNS and comparisons were made with formulas derived by analytical means. We showed that significant departures occur from these formulas for fiber lengths <10 m which are typical of lengths used in fiber amplifiers. This can have a significant impact on the beam combining of kilowatt class fiber lasers as broader linewidths than anticipated are required to mitigate SBS; thus imposing stricter requirements on optical path matching. Finally, we showed for a fiber length typical of fiber amplifiers, the PRBS patterns at or near $n = 7$ provide the best mitigation of SBS. We will present experimental verification of the main conclusions of this paper in future publications.

Acknowledgment

We would like to thank Dr. Arje Nachman of the Air Force Office of Scientific Research (AFOSR) for funding this effort and Dr. Joshua Rothenberg from Northrop Grumman for helpful discussions.

References and links

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4. C. Robin and I. Dajani, “Acoustically segmented photonic crystal fiber for single-frequency high-power laser applications,” Opt. Lett. 36(14), 2641–2643 (2011). [CrossRef] [PubMed]

5. C. Zeringue, C. Vergien, and I. Dajani, “Pump-limited, 203 W single-frequency monolithic fiber amplifier based on laser gain competition,” Opt. Lett. 36(5), 618–620 (2011). [CrossRef] [PubMed]

6. F. W. Willems, W. Muys, and J. S. Leong, “Simultaneous Suppression of Stimulated Brillouin Scattering and interferometric noise in externally modulated lightwave AM-SCM systems,” IEEE Photon. Technol. Lett. 6(12), 1476–1478 (1994). [CrossRef]

7. G. D. Goodno, S. J. McNaught, J. E. Rothenberg, T. S. McComb, P. A. Thielen, M. G. Wickham, and M. E. Weber, “Active phase and polarization locking of a 1.4 kW fiber amplifier,” Opt. Lett. 35(10), 1542–1544 (2010). [CrossRef] [PubMed]

8. C. X. Yu, S. J. Augst, S. M. Redmond, K. C. Goldizen, D. V. Murphy, A. Sanchez, and T. Y. Fan, “Coherent combining of a 4 kW, eight-element fiber amplifier array,” Opt. Lett. 36(14), 2686–2688 (2011). [CrossRef] [PubMed]

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$ρ_{ο}$	$2201 k g / m^{3}$	$v_{S}$	$5.9 x 10^{3} m / s$
_$ω$	$1.77 \times 10^{15} r a d / s$	$A$	$7.85 x 10^{- 11} m^{2}$
$γ_{e}$	$1.95$	$Ω_{B}$	$10.1 \times 10^{10} r a d / s$
$n$	$1.45$	$L$	$5 m$
$T$	$300 K$	$2 π τ_{p}$	$17.5 n s$

A theoretical study of transient stimulated Brillouin scattering in optical fibers seeded with phase-modulated light

Abstract

1. Introduction

2. Theoretical framework

3. Numerical approach and validation

4. Sinusoidal phase modulation

5. White noise modulation

6. Pseudo-random bit sequence (PRBS)

7. Conclusion

Acknowledgment

References and links

Cited By

Figures (12)

Tables (1)

Equations (29)

Optics Express