Correlations between intensity fluctuations within stimulated Brillouin waveforms generated by...

Optics Communications 242 (2004) 267–278

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Correlations between intensity fluctuations within stimulatedBrillouin waveforms generated by scattering of

Q-switched pulses in optical fiber

Juan Correa, Enrique Manzano, Ryan Tracy, John R. Thompson *

Department of Physics, DePaul University, 2219 North Kenmore Avenue, Chicago, IL 60614, USA

Received 2 June 2004; received in revised form 9 July 2004; accepted 4 August 2004

Abstract

We present a detailed experimental study of the statistical properties of pulsed stimulated Brillouin scattering in opti-

cal fiber using single-mode pump pulses with durations comparable to the phonon lifetime but much shorter than the

fiber transit time. Under these conditions the temporal dynamics of the scattered light is stochastic and results in the

generation of scattered intensity waveforms with complex structures. We provide experimental evidence and simple

theoretical arguments to interpret the energy fluctuations when varying fractions of the scattered waveforms are inte-

grated and statistically analyzed. Two-point intensity correlation functions are used to determine to what extent the

complex intensity waveforms represent sequences of statistically independent scattering events.

� 2004 Elsevier B.V. All rights reserved.

Keywords: Brillouin scattering; Nonlinear optics; Statistical optics

1. Introduction and background

The Central Limit Theorem implies that a ran-

domly fluctuating physical variable that represents

the sum of many statistically independent contri-

butions will have a probability distribution that

0030-4018/$ - see front matter � 2004 Elsevier B.V. All rights reserv

doi:10.1016/j.optcom.2004.08.008

* Corresponding author. Tel.: +1 773 3251375; fax: +1 773

3257334.

E-mail address: [email protected] (J.R. Thompson).

is approximately Gaussian in shape. The widthof this probability distribution will become pro-

gressively narrower in relation to its mean as the

number of independent contributions increase [1].

In this article, we discuss experiments that investi-

gate the pulse energy fluctuations associated with

spontaneously initiated stimulated Brillouin scat-

tering (SBS) in optical fiber when varying fractions

of the complex SBS waveforms are integrated andstatistically analyzed. We use measured intensity

correlations and a simple statistical model to probe

ed.

mailto:[email protected]

268 J. Correa et al. / Optics Communications 242 (2004) 267–278

the independence of intensity fluctuations in differ-

ent sections of the SBS waveforms.

Brillouin scattering of optical pulses was first re-

ported for bulk crystals of quartz and sapphire just

a few years after the birth of the laser and, almosta decade later, in low loss glass optical fiber [2,3].

Since that time SBS has enjoyed a long history

of research by many different groups for both ap-

plied and fundamental reasons. SBS has been used

or proposed for pulse compression and beam

clean-up of high power pulsed lasers [4–8], to pro-

duce narrow-band optical filters for communica-

tion systems [9], for signal conditioning ofmicrowave frequency communication signals

[10,11], and for fiber sensors [12]. SBS has also

been extensively studied because of its complex

stochastic and deterministic dynamics [13–20].

This article focuses on fundamental aspects of

the statistical properties of spontaneously initiated

SBS in low loss optical fiber.

Spontaneous Brillouin scattering occurs whenan intense pump wave scatters off of high fre-

quency acoustic waves that are excited by thermal

or quantum fluctuations of the medium. The scat-

tered light travels in the backward direction with

respect to the pump due to phase matching con-

straints, and the scattered light is down-shifted in

frequency from the pump by an amount equal to

the acoustic frequency in order to conserve energy.For glass optical fibers the acoustic wave fre-

quency is typically in the 10–20 GHz range for

near infrared pump light so that at room tempera-

ture the average thermal energy available for a

vibrational mode is much greater than the phonon

energy. Therefore, the random acoustic distur-

bances that initiate Brillouin scattering in glass fi-

ber at room temperature are thermally excited[13,14,21]. The thermally generated acoustic waves

can be represented as a zero-mean Gaussian ran-

dom process with fluctuations that are uncorre-

lated in space and time. For this reason the

waveforms produced by spontaneously initiated

SBS can have exceedingly complex temporal struc-

ture and display large scale intensity fluctuations.

Theoretical and experimental work on SBS usingcontinuous-wave (cw) laser light in the absence

of feedback, has shown that these large scale inten-

sity fluctuations can persist even under conditions

of significant pump depletion [13–15,18]. When

feedback comes into play cw SBS shows temporal

patterns that can be explained in terms of the lon-

gitudinal modes of the fiber resonator, but there

can also be more complex dynamical behavior thatappears to be deterministic chaos. In fact there has

been substantial discussion in the research litera-

ture about the conditions under which the tempo-

ral dynamics are stochastic or deterministic and

about the conditions required for deterministic

chaos to be observed. The very careful experimen-

tal and theoretical work of Dammig et al. seems to

have largely settled these questions. Their workshows that SBS is very sensitive to feedback so

that relatively small amounts of it are sufficient

to change the dynamics from purely stochastic to

largely deterministic, and behavior that appears

to be deterministic chaos can result from thermal

drift of the fiber resonator [13–20]. Feedback does

not have to be ‘‘designed’’ into the system to be

important; it can come from Fresnel reflectionsoff the fiber end faces or, as a couple of recent

studies have shown, by repeated Rayleigh scatter-

ing in high loss fibers used as cw laser resonators

[22,23].

There have been a number of studies on dynam-

ical aspects of pulsed SBS in bulk media and in

waveguides, with limited attention given to charac-

terizing the statistical properties of the scatteredlight [24–35]. Some of the early studies of pulsed

SBS in optical fiber focused on understanding

the temporal structure of noise initiated scattering.

Much of this work explored the relationship be-

tween the time scale of the intensity modulations

of the scattered light and the phonon lifetime for

various scattering strengths [24–29]. Recent theo-

retical and experimental work by Ogusu and Lihave looked at the temporal dynamics of pulsed

SBS and the interplay of SBS with Kerr nonlinea-

rities for propagation through glass optical fiber,

propagation through fiber Bragg gratings, and

propagation through fiber ring resonators. Much

of their work looks at the temporal dynamics un-

der circumstances where the temporal width of

the optical pulse is longer than the transit timethrough the fiber [30–32]. There is a limited

amount of early experimental and theoretical work

addressing the statistical properties of transient

Q-SwitchedPump Pulses

Pump PulseDiagnostics

ND Filter

1/2 Wave Plate

Linear Polarizer

20X Objective

40X Objective

71 m SMPMOptical Fiber

Etalon

Lens

Output BeamDiagnostics

InGaAs PD

Isolater

Beamsplitter

Beamsplitter

Mirror

(a)

Q-SwitchedPump Pulses

InGaAs PD

ND Filter

Aperture 20X Objective

Lens

(b)

J. Correa et al. / Optics Communications 242 (2004) 267–278 269

SBS in gas and liquid filled cells. This work inves-

tigated phase fluctuations in SBS by using two

identical Brillouin cells as phase-conjugate mirrors

in an interferometer which converted phase fluctu-

ations into energy fluctuations recorded by calo-rimeters. This work also investigated localized

temporal fluctuations in the transverse beam pro-

file of pulses generated by SBS in gas or liquid

filled cells [33–35].

In this article we report an experimental study

of the energy statistics of complex intensity wave-

forms produced by transient SBS in low-loss silica

glass optical fiber when the duration of the excitinglaser pulse is much less than the single-pass transit

time of the fiber so that the effects of feedback can

be ignored. In particular we focus on how the en-

ergy statistics evolve as we integrate a larger frac-

tion of the complex SBS waveforms. We will use

the measured ‘‘slice’’ energy statistics, simple theo-

retical arguments, and measured intensity correla-

tion functions to show that the observed statisticalbehavior provides a good example from nonlinear

optics of the Central Limit Theorem. To the best

of our knowledge, such a study has not been car-

ried out by other groups working on pulsed SBS

in optical fiber.

40X Objective

DispersingPrisms

71 m SMPMOptical Fiber

Fig. 1. Experimental setup: ND, neutral density; PD, photo-

diode; SMPM, single mode polarization maintaining. (a) Setup

for integrating and collecting SBS intensity waveforms. (b)

Setup for collecting transmitted pump waveforms.

2. Experiment apparatus and procedures

The experimental setup for this work is shown

in Fig. 1. For clarity the setup is represented by

two schematics: part (a) for the portion of theapparatus used to detect the backward propagat-

ing SBS waveforms; and part (b) for the portion

of the apparatus used to study the depletion of

the transmitted pump pulses by SBS. There were

three types of data collected: large sets of SBS slice

energies for statistical analysis, SBS waveform

ensembles to estimate intensity correlation func-

tions, and transmitted pump pulse temporal enve-lopes to study pump depletion by SBS. The setup

and procedures for each type of data will be dis-

cussed below after detailed information about

the pump laser is given.

The pump laser used to drive the SBS process is

a Q-switched, diode-pumped Nd:YAG laser. The

laser operates on a single transverse and longitudi-

nal mode at a center wavelength of 1064 nm. The

pulse repetition rate is 1 kHz, and the pulses have

smooth Gaussian shaped envelopes with a full-width at half-maximum of approximately 30 ns.

The pulses are also linearly polarized since one

face of the Nd:YAG crystal is Brewster cut. The

energy fluctuations of the Q-switched pulses are

generally quite small with a standard deviation

that is less than one percent of the mean pulse en-

ergy. Single longitudinal mode operation of the la-

ser is obtained by using a combination of twomode selection techniques. First, a 2 mm uncoated


optical flat is inserted in the cavity to reduce the

number of lasing modes in each pulse to a few

adjacent modes. The second technique, known as

prelasing, involves turning down the losses on

the acousto-optic Q-switch until there is a weakcontinuous wave background between the pulses.

For a modest range of losses this continuous wave

background is single mode and acts as a seed to

lock the forming pulses onto a single mode [36].

Between 85% and 90% of the total optical energy

was contained in the pulses for all of the reported

data. This energy fraction was determined by mak-

ing average pump power measurements with theAOM operated in two different modes: a mode

where the acoustic waves are periodically shut off

to allow pulses to form, and a mode where the

acoustic waves were continuously on so that only

the weak cw background was present. For the por-

tion of the pump beam coupled into the fiber, the

average cw background power was always less

than 0.1 mW while the peak power of the pumppulses were between 6 and 20 W. When the etalon

and Q-switch losses were properly adjusted, we

could obtain stable single-mode oscillation for up

to 2 h at a time before bimodal operation and

mode hopping occurred. Single longitudinal mode

operation of the laser during data collection was

verified by continuously monitoring the output

of a pulsed laser spectrum analyzer. One couldalso detect periods of multimode operation by

monitoring the input pump pulse temporal enve-

lope, which was smooth when the laser oscillated

on a single mode but deeply modulated in multi-

mode operation, with a beat frequency of approx-

imately 1.25 GHz. Finally, it should be noted that

the efficiency of SBS is very sensitive to the mode

structure of the pump pulses. The SBS process ismost efficient when high-quality single-mode

pump pulses are used, and the presence of rela-

tively small side modes resulted in a substantial

reduction in the generated SBS signal strength.

This fact provided yet another way to monitor

the pump pulse mode structure during data

collection.

As mentioned in the Section 1, the presence offeedback can have dramatic effects on the tempo-

ral dynamics of SBS. We believe that the effects

of optical feedback are not important in our exper-

iments since the physical length of the pump pulse

in the fiber (csp/n � 6 m for our 30 ns pulses) is

roughly 10 times smaller than the fiber length

(71 m). In addition, the pump and SBS pulses

are only coupled when the relationship betweentheir propagation directions are correct. For these

reasons we did not make any special efforts to

cleave the fiber ends at a large angle to discourage

reflections that were coupled into a guided mode.

We used the diffuse Fresnel reflection of the pump

pulse off the fiber input to determine when the SBS

scattering occurred in relation to the entry of the

peak of the input pump pulse into the fiber. Thisbrings up yet another reason why we can ignore

the effects of feedback: all of the measurements

on individual SBS pulses were made during a time

interval that was less than a single round trip time

of the pump pulse within the 71 m fiber.

The SBS slice energy statistics were measured

using the apparatus shown in Fig. 1(a). The optical

isolator prevented back-scattered light from enter-ing the laser cavity. The half-wave plate and pola-

rizer were used to vary the input power to the fiber.

The polarizer transmission axis was fixed at an an-

gle that was parallel to one of the principal axes

of the elliptical core polarization maintaining fiber.

The half-wave plate could then be used to rotate

the incident pump polarization and vary the

amount of light that was transmitted through thepolarizer and subsequently coupled into the fiber.

The 71 m optical fiber supported a single trans-

verse mode at the pump and SBS wavelengths

and was polarization maintaining. The fiber loss

coefficient was between 2 and 3 dB/km at the

pump wavelength, as specified by the manufac-

turer. The effective mode area was approximately

7.6 lm2, which is slightly larger than the area ofthe elliptical core. This estimate of the effective

area is based on earlier stimulated Raman scatter-

ing and parametric four-wave mixing experiments

and simulations. The SBS gain coefficient for our

fiber should be close to the published value for sil-

ica glass which is 5 · 10�11 m/W [21]. The back-

scattered SBS waveform exited the fiber input

and a portion of it was directed to the detectionsystem by the beamsplitter between the input cou-

pler and the linear polarizer. A neutral density fil-

ter prevented saturation or damage of the InGaAs

-50 0 50 100 150 200 250 3000.0

0.2

0.4

0.6

0.8

1.0(a)

Pulse Time (ns)

Pulse Time (ns)

-50 0 50 100 150 200 250 3000.0

0.2

0.4

0.6

0.8

1.0(b)

PulseTime (ns)-50 0 50 100 150 200 250 300

Inte

nsity

(A

rb.U

nits

)In

tens

ity (

Arb

.Uni

ts)

Inte

nsity

(A

rb.U

nits

)

0.0

0.2

0.4

0.6

0.8

1.0 (c)

Fig. 2. Sample SBS waveforms for three values of input pump

peak power: (a) 6.26 W, (b) 12.5 W, and (c) 18.8 W. Each

waveform is the scattering that resulted from a single pump

pulse. The zero on the time scale marks the time when the peak

of the input pump pulse entered the fiber.


photodiode used to detect the SBS waveform. An

etalon (4 mm thick, 90% reflectance both sides)

was used to pass the SBS pulse and block any

pump light that resulted from Fresnel reflections

off the fiber ends. A 100 mm focal length lenswas used to image the scattered light onto the

100 lm diameter active area of a high speed InG-

aAs photodiode (400 ps rise time). The output of

this photodiode was connected to either a high

speed digital oscilloscope (up to 10 GSa/s and 3

GHz analog bandwidth) or a fast gated integrator

with computer interface.

The SBS pulses were initiated by spontaneousscattering of the pump pulses off of thermally gen-

erated acoustic waves in the optical fiber. The

newly generated Brillouin pulse counter-propa-

gated through the pump pulse and was amplified

by stimulated scattering while the pump and Bril-

louin pulses overlapped in the fiber. The resulting

SBS intensity waveform that exits the fiber input

has a complex shape even though the pump pulseis quite smooth and regular. Sample SBS wave-

forms generated by scattering from individual

pump pulses are shown in Fig. 2 for three different

input pump powers. The detailed structure of these

waveforms changed randomly from shot to shot,

and there was substantial shot to shot timing jitter

in the leading edge of the SBS waveform. At the

lowest input pump power the shot-to-shot timingjitter in the leading edge of the SBS waveform

was roughly 40 ns for a waveform with a total

duration of hundreds of nanoseconds. At the high-

est input pump power the shot-to-shot timing jitter

in the leading edge of the SBS waveform was

approximately 4–5 ns for a waveform with a total

duration of tens of nanoseconds. It is clear that the

SBS waveforms do not reproduce the input pumppulse shape and are not simply a compressed ver-

sion of the input pump pulse. They appear to be

a series of intensity spikes that represent different

scattering episodes from the same input pump

pulse as it propagates through the fiber. Sections

of the first two sample SBS waveforms appear to

consist of trains of intensity spikes at roughly reg-

ular spacing. The average temporal spacing be-tween intensity spikes is most likely determined

by a combination of the phonon damping time

and the pump intensity profile [24–26,28]. As the

pump sweeps through a section of fiber where

there is a thermally excited acoustic wave of the

right frequency to initiate Brillouin scattering,

scattering occurs until the damping has erased


the organized motion of the acoustic wave or until

the pump intensity drops below the threshold to

produce detectable scattering.

As a first attempt at understanding whether or

not the sequence of intensity spikes in an SBSwaveform were statistically independent, we stud-

ied the pulse energy statistics as different fractions

of the SBS waveforms were integrated. The inte-

gration period ranged from 5 ns, the width of the

narrowest observed intensity spikes, to a duration

that was comparable to the total duration of the

SBS waveform. An electronic gate pulse generated

by the gated integrator defined the integrationinterval, and its leading edge was kept fixed at a

position that excluded the shot to shot timing jitter

of the leading edge of the SBS waveform. The

trailing edge of the gate pulse was progressively ex-

tended to later times until most of the SBS wave-

form was integrated or, for the weakest

scattering data, the slice energy noise showed little

change as the gate width was increased.We also collected ensembles of 1000 SBS wave-

forms at different input pump powers to use for

estimating the intensity correlation function. In

this work, the intensity correlation function is de-

fined as the normalized correlation coefficient be-

tween the intensity fluctuations at two different

times in the SBS waveform as a function of the

time delay between the two points:

CIðt0; sÞ ¼Iðt0Þ � Iðt0Þh ið Þ Iðt0 þ sÞ � Iðt0 þ sÞh ið Þh i

r0rs;

where I(t0) is the intensity at the ‘‘initial’’ time t0, sis the delay from the initial time, r0 is the standarddeviation of the SBS intensity at the initial time, rsis the standard deviation at the delayed time, and

the angle brackets denote averaging over theensemble of collected waveforms [37]. The position

of t0 was fixed at a time near the leading edge of

the SBS waveforms, consistent with the placement

of the leading edge of the gate pulse in the slice en-

ergy noise measurements. CI(t0,s) was calculated

from an ensemble of waveforms for various

choices of the delay s. The waveform ensembles

were collected using the apparatus shown in Fig.1(a). The output of the InGaAs photodiode was

connected to a fast digital oscilloscope, and digi-

tized SBS waveforms were transferred to the com-

puter via a GPIB interface. A set of 1000 SBS

waveforms, each one the result of scattering from

a single pump pulse, was collected for several input

pump powers. The digital oscilloscope samplingrates for these experiments was either 2.5 or 5

GSa/s so that the temporal resolution was more

than adequate to capture the fine structure of the

SBS waveform. The pump pulse was continuously

monitored during waveform collection to ensure

that there were no lapses into multimode

operation.

Finally, we collected average transmitted pumpwaveforms for different input pump power levels

to determine the amount and nature of pump

depletion due to SBS. This data was collected

using the apparatus shown in Fig. 1(b). The dis-

persing prisms allow us to separate and detect

any scattered light due to stimulated Raman scat-

tering in the forward direction. The aperture is

positioned to select the transmitted pump pulseand block the Stokes pulse generated by Raman

scattering. The neutral density filter ensures that

the InGaAs photodiode is not saturated, and a

250 mm focal length lens is used to image the

transmitted pump beam onto the active area of

the photodiode. The photodiode has a rise time

of 200 ps and is connected to the same digital oscil-

loscope used to monitor the input pump pulse andSBS waveforms. Each transmitted pump wave-

form was averaged over 500 shots of the laser.

As in the earlier measurements, the pump pulse

was continuously monitored to ensure stable sin-

gle-mode operation during periods of data

collection.

3. Experiment results and interpretation

The relative noise of the energy contained in a

temporal slice of the SBS waveform as a functionof the duration of the slice is shown in Fig. 3 for

three different input pump powers. Measured rela-

tive noises and relative noises obtained from a sim-

ple statistical model (described below) are shown

on each graph. The relative noise for each data

point is the ratio of the standard deviation of the

slice energy to the mean slice energy, expressed

GateWidth (ns)

GateWidth (ns)

GateWidth (ns)

0 30 60 90 120 150 180

Rel

ativ

e N

oise

(%

)R

elat

ive

Noi

se (

%)

Rel

ativ

e N

oise

(%

)

0

20

40

60

80

100(a)

0 20 40 60 80 1000

10

20

30

40

50

60(b)

0 20 40 60 80 1005

10

15

20

25

30

35(c)

Fig. 3. Slice energy noise versus slice duration for three values

of input pump peak power: (a) 6.14 W, (b) 12.5 W, and (c) 18.4

W. Triangles mark the measured data points, and crosses mark

the slice noises calculated using the statistical model described

in the text.


as a percentage in the figure, for an ensemble of

7200 slice energies collected at a fixed input pump

power and gate width. The lowest input pump

power corresponds to very weak Brillouin scatter-

ing for which there is negligible pump depletion.

The highest input pump power corresponds to

strong Brillouin scattering for which there is signif-

icant pump depletion and the onset of stimulated

Raman scattering in the forward direction. The

basic trend displayed in each of the graphs inFig. 3 is the same: the energy fluctuations are larg-

est for the narrowest slice and steadily decrease as

more of the SBS waveform is integrated. Looking

at the structure of the sample SBS waveforms in

Fig. 2, one can see that increasing the slice dura-

tion corresponds to adding up the energy in more

of the intensity spikes in the total SBS waveform.

We can represent the total SBS waveform as acomposite of energy units that correspond to indi-

vidual intensity spikes. If the fluctuations of differ-

ent energy units are statistically independent then,

by virtue of the Central Limit Theorem, we expect

that their sum will become progressively less noisy

as more units are included in the sum.

One way of determining whether or not the en-

ergy fluctuations of different sections of the SBSwaveform are statistically independent is to com-

pare the measured relative noise data to a simple

statistical model. In this model we imagine break-

ing up each SBS waveform into a series of 10 ns

slices, a width comparable to the typical phonon

decay time. The slice energies are assumed to be

independently fluctuating random variables with

identical means and standard deviations. The ratioof the standard deviation to the mean energy for

an individual slice is assigned a value that matches

the measured value for a 10 ns slice. The ratio for

the sum of N slices is then calculated using the fol-

lowing relationship:

rN

lN¼ 1ffiffiffiffi

Np r0

l0

� �;

where r0/l0 is the measured ratio of the standard

deviation to the mean for a 10 ns slice. This simple

relationship follows directly from the assumption

of statistical independence [38]. The relative noises

calculated using this model are shown in Fig. 3

along with the measured data. Although there

are some differences between the model and meas-ured values, the level of agreement is quite good gi-

ven the simplicity of the model and supports the

idea that the energy fluctuations in different sec-

tions of the SBS waveform are statistically


independent or perhaps weakly correlated. Later

in this section we will present measured intensity

correlation functions that provide possible reasons

for the differences between the statistical model

and measurements seen in Fig. 3.We turn now to the slice energy histograms

shown in Fig. 4, which are derived from the same

data sets used to calculate the relative noises

shown in Fig. 3. Each histogram is generated from

a set of 7200 raw slice areas that were divided by

the mean slice area so that the mean of the scaled

data was one. This is the reason the horizontal axis

is labeled ‘‘normalized slice area.’’ Fig. 4(a) showsthe histogram for the lowest input pump power

and a 5 ns slice width. This histogram is highly

skewed, with a long tail extending to slice energies

that are many times the mean energy. (The nega-

tive values on the graph are due to the effects of

0 2 4 6 8

Cou

nts

0

200

400

600

800

1000

1200(a) (

0.0 0.5 1.0 1.5 2.0

Cou

nts

0

200

400

600

800(c)

Normalized Slice Area


(

Fig. 4. Sample slice energy histograms for narrow and wide slices at t

6.14 W, 180 ns slice; (c) 18.4 W, 5 ns slice; (d) 18.4 W, 90 ns slice. Not

the same input pump power.

the integrated electronic noise, which fluctuated

about an average value of zero, pushing a few of

the very weak SBS slice areas below zero. Less

than 1% of the data points were negative for this

histogram, and these points were much smaller inmagnitude than the mean so that the effect on

the relative noise was negligible.) By way of con-

trast, Fig. 4(b) shows the histogram for the same

input pump power but a slice width of 180 ns. This

histogram is much closer to being symmetric, like a

Gaussian distribution, with energies that are much

more tightly confined around the mean. Frames (c)

and (d) of Fig. 4 show histograms for the highestinput pump power for slice widths of 5 and 90

ns, respectively. Both histograms for the highest

pump power are symmetric, but the increase in

the sharpness of the distribution is clearly evident

for the longer duration slice. These changes in

0 2 4 6 8

Cou

nts

0

200

400

600

800

1000

1200b)



0.0 0.5 1.0 1.5 2.0

Cou

nts

0

200

400

600

800d)

wo different input pump peak powers: (a) 6.14 W, 5 ns slice; (b)

e the contrast between the narrow and wide slice histograms for

0 10 20 30 40 50 60

Cor

rela

tion

Coe

ffic

ient

Cor

rela

tion

Coe

ffic

ient

Cor

rela

tion

Coe

ffic

ient

0.0

0.2

0.4

0.6

0.8

1.0(a)

0 10 20 30 40 50-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0(b)

Delay (ns)

Delay (ns)

Delay (ns)

0 5 10 15 20 25 30-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0(c)

Fig. 5. Intensity correlation functions for three values of input

pump peak power: (a) 5.99 W, (b) 12.0 W, and (c) 18.0 W.


shape and/or sharpness of the histograms as slice

width is increased are consistent with the idea that

the intensity fluctuations in different sections of the

SBS waveform are independent or only weakly

correlated.To provide experimental evidence that the

trends seen in the slice energy statistics can be ex-

plained in terms of the Central Limit Theorem, we

turn to measurements of the intensity correlation

function for the SBS waveforms. The results of

these measurements for three different input pump

powers are shown in Fig. 5. As mentioned earlier,

in calculating the intensity correlation function theinitial point had a fixed position near the leading

edge of the SBS waveform while the delayed point

was systematically swept to later times to calculate

the correlation coefficient for varying delays be-

tween the two points. The structure of the intensity

correlation function does not seem to be sensitive

to the particular choice of the initial point�s loca-tion in the leading section of the SBS waveform.We chose an initial point that excludes most of

the shot-to-shot jitter in the leading edge of the

SBS waveform just as we did for the leading edge

of the gate pulse in the slice energy noise measure-

ments described earlier. In all of the graphs, near

neighbors have intensity fluctuations that are

highly correlated. The strong positive correlation

between near neighbors tends to fade after the de-lay reaches the 5–10 ns range. For the lowest input

pump power, the correlation function has an ex-

tended shoulder beyond the initial period of strong

positive correlation where the correlation coeffi-

cient is small and positive. The extended duration

of positive correlations between intensity fluctua-

tions in different parts of the SBS waveform may

explain why the measured slice energy noises aresystematically larger than the noises calculated

using the simple statistical model since positive

correlations between different slices will enhance

the fluctuations over the case when the slices are

uncorrelated. For the intermediate input pump

power, there is a long interval of negative intensity

correlations beyond the initial strong positive cor-

relations. These negative correlations may explainwhy the measured slice energy noises are systemat-

ically lower than the calculated values since nega-

tive correlations between different slices will

suppress the fluctuations below the value for

uncorrelated slices. For the highest input pump

power the strong positive correlations between

nearest neighbors is followed by relatively strong

negative correlations for a narrow range of delays

followed by very weak correlations for larger de-

lays that fluctuate between positive and negative

Time (ns)

0 20 40 60 80 100

Nor

mal

ized

Int

ensi

ty

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 6. Transmitted pump temporal profile for three values of

input pump peak power: solid curve, 6.10 W; dashed curve, 12.2

W; dash-dot curve, 18.3 W.


values. The comparison between the measured and

calculated slice energy noises is arguably the best

for this power. The measured value dips below

the calculated value for intermediate slice dura-

tions, which could be the consequence of the shortinterval of negative correlations discussed above

[37,38]. We can summarize these results by noting

that for all three input pump powers the regime of

strong correlation, either positive or negative, is

confined to points that are within roughly 10–20

ns of one another, which is comparable to the pho-

non decay time typical for silica fibers. So for long

integration intervals we are indeed adding up sec-tions of the waveform that are independently fluc-

tuating or only weakly correlated.

There is one final feature of the intensity corre-

lation functions that merit comment: the weak

oscillations seen for time delays falling after the

initial region of strong correlations. These oscilla-

tions are most prominent in Fig. 5(a), which is the

intensity correlation function in the absence ofpump depletion. This feature could be the result

of the roughly regular spacing of intensity peaks

in the SBS waveforms [26]. The correlation func-

tion represents the ensemble average of the overlap

of an SBS waveform with a delayed version of it-

self. When the delay corresponds to an integer

multiple of the average spacing between adjacent

intensity spikes, one expects to observe peaks inthe correlation function. However, the amplitude

of the correlation peaks will be diminished by shot

to shot jitter in the SBS waveforms.

We also made measurements to determine the

degree of depletion of the transmitted pump by

SBS. The normalized average temporal profile of

the transmitted pump is shown in Fig. 6 for several

different input pump powers that match the pumppowers for the SBS data presented in this article.

Each of the transmitted pump waveforms was

averaged over 500 shots of the laser. Looking at

the figure, we note that the trailing edge of the

pump pulse is eaten away by the SBS process as

the input pump power is increased. This is consist-

ent with the idea that Brillouin scattering tends to

be initiated near the peak of the pump pulse, sinceSBS is an intensity dependent process, and then

counter-propagates through the trailing edge of

the pump pulse growing by stimulated scattering

as it sweeps through the pump. For the highest

pump power, the peak of the transmitted pump

pulse is just beginning to flatten due to the onset

of significant stimulated Raman scattering in the

forward direction. Any further increase in pump

power would result in the preferential depletion

of the peak of the pump pulse by stimulated Ra-man scattering. It should be noted that all of our

SBS data was taken for input pump powers below

the threshold for significant Raman induced pump

depletion. Finally, it should be pointed out that

there is negligible pump depletion for the lowest

input pump power. We verified this by overlaying

the normalized input and transmitted pump

pulses. The shapes of the two pulses were essen-tially identical for the lowest pump power.

The information about pump depletion can be

used to explain features of the SBS intensity statis-

tics that are quantified in Figs. 3 and 5. Starting

with Fig. 3, note that there is a systematic shift

to lower relative noises as the input pump power

is increased, which can be seen by comparing data

points for the same slice widths from frames (a)–(c). In the absence of pump depletion in the weak

scattering limit, spontaneous scattering dominates

the statistics of the resulting intensity spikes. The

fluctuating amplitude of the thermally generated

acoustic waves that drive the spontaneous scatter-

ing can be modeled by a zero-mean Gaussian ran-

dom process; consequently, for sufficiently weak

scattering the distribution of energies contained


in individual intensity spikes should be negative

exponential. The result is nearly full-scale energy

fluctuations when the slice width is comparable

to the width of an individual intensity spike

[13,14]. For strong scattering the intensity of indi-vidual spikes begins to saturate as the pump is de-

pleted, and stimulated scattering is now the

dominant contributor to the growth of intensity

spikes. The resulting intensity spikes will have sub-

stantially larger energies with less spread in energy

about the mean, giving rise to a symmetric proba-

bility distribution with a narrow width. For longer

duration slices the relative energy noise drops asthe pump power increases because we are summing

energies of individual slices that are more stable.

Finally, let us consider the effects of pump

depletion on the intensity correlation functions

shown in Fig. 5. At high input pump powers,

strong scattering in the early sections of the fiber

tends to deplete the pump so that only weak

intensity spikes are generated later in the fiber.The region of strong negative correlation seen in

the intensity correlation function for the higher

input pump powers is a consequence of the effects

of pump depletion. Intensity spikes in the early

part of the SBS waveform are effectively compet-

ing for the energy supplied by the pump, and this

competition demands that they be negatively cor-

related. For weak SBS the pump remains strongthroughout the fiber so that different intensity

spikes are not competing for the energy supplied

by the pump, and one would expect the correla-

tion function to drop to zero, or perhaps display

weak oscillations about zero, after the narrow re-

gion of strong positive correlation between near-

est neighbors. However, the data in Fig. 5(a)

shows an extended region of weak positive corre-lations rather than an abrupt drop to zero, the

prominent ‘‘shoulder’’ beyond the main peak of

the correlation function. We believe that this pos-

itive shoulder may reflect the impact of small

shot-to-shot fluctuations in the pump pulse peak

amplitude on the SBS waveform fluctuations.

Pump pulses with larger peak amplitudes will

lead, on average, to trains of larger intensityspikes within an SBS waveform. The reverse will

be true for pump pulses with smaller peak ampli-

tudes. Thus, the successive spikes in an SBS wave-

form will tend to rise and fall in concert with the

rise and fall of the pump pulse peak amplitude.

Once pump depletion comes into play, this fea-

ture disappears because of the strong competition

between individual spikes within an SBSwaveform.

4. Conclusions

In this article we have presented a detailed

experimental study of the statistical properties of

pulsed SBS in optical fiber for pulse widths com-parable to the phonon lifetime but much shorter

than the fiber transit time. Under these conditions

the temporal dynamics of the scattered signal

should be purely stochastic, with fluctuations

determined by the thermally generated sound

waves that initiate the scattering and by any resid-

ual amplitude fluctuations of the single mode

pump laser. We have provided experimental evi-dence and simple theoretical arguments to assess

whether or not different sections of a SBS wave-

form represent statistically independent scattering

episodes from the same input pump pulse. The

temporal structure of the SBS waveforms suggest

this interpretation, especially for weak scattering,

since the scattered light from a smooth single-

peaked pump pulse consists of a long series ofirregular intensity spikes with durations less than

the pump pulse width. Our analysis indicates that

this interpretation is a reasonable approximation

that can explain the general trends of the statistics

of the integrated SBS waveform energy, but that

there are intensity correlations that cause modest

deviations of the statistics from the idealistic case

of completely independent fluctuations. We be-lieve these correlations between the intensity fluc-

tuations within an SBS waveform are the

consequence of the finite damping time for the

thermally excited acoustic waves, the influence of

residual pump pulse amplitude fluctuations, and

the competition that is induced by pump

depletion.

In the future we plan to extend this work in sev-eral ways. First, we are working on a theoretical

model for pulsed SBS that will incorporate ther-

mal and pump fluctuations. We hope to perform


simulations and experiments that thoroughly

investigate the interplay of these two sources of

noise in pulsed SBS. It might also be interesting

to investigate the interplay of Brillouin and Ra-

man scattering to determine how the statisticalproperties of the scattered light, in the forward

and backward directions, are impacted by compet-

ing nonlinear processes. We have recently com-

pleted an experimental and theoretical study of

noise shaping by stimulated Raman scattering in

optical fiber that should help with this investiga-

tion [39,40].

Acknowledgements

This work is supported by the National Science

Foundation through grant PHY-0140305. J.R.

Thompson was also supported by a faculty re-

search leave granted by the University Research

Council at DePaul University.

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