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1188 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 43, NO. 12, DECEMBER 2007

Output Power PDF of a Saturated Semiconductor

Optical Amplifier: Second-Order Noise

Contributions by Path Integral Method

Filip Öhman, Jesper Mørk, and Bjarne Tromborg

Abstract—Wehavedevelopedasecond-ordersmall-signalmodel

for describing the nonlinear redistribution of noise in a saturated

semiconductor optical amplifier. In this paper, the details of the

model are presented. A numerical example is used to compare the

model to statistical simulations. We show that the proper inclusion

ofsecond-ordernoisetermsisrequiredfordescribingthechangein

the skewness (third-order moment) of the noise distributions. The

calculated probability density functions are described far out in

the tails and can hence describe signals with very low bit error rate

(BER). The work is relevant for describing the noise distribution

and BER in, for example, optical regeneration.

Index Terms—Noise, optical communication, optical signal pro-

cessing, semiconductor optical amplifiers (SOAs).

I. INTRODUCTION

S

nication systems. Some examples are all-optical wavelength

conversion [1], [2], regeneration [3], [4], limiting amplification

[5], and noise suppression in spectrum-sliced wavelength-di-

vision multiplexed systems [6]. In these applications, the

performance, as measured by the bit error rate (BER), depends

on the noise distribution of the signal after the SOAs. It is

therefore important to describe the noise in an SOA in detail.

However, the saturation and nonlinear properties of the SOA

make this description complicated. Shtaif and coworkers used

a first-order perturbation analysis for examining the noise

spectra after a saturated SOA [7] and experimentally mea-

sured the noise distribution [8]. Bilenca and colleagues have

made a detailed study of noise distributions in SOAs using

multicanonical Monte Carlo simulation [9] and Fokker–Planck

equations [10]. We have previously measured and calculated

the probability density functions (PDFs) after amplification in

a saturated SOA [11]. The calculations of the PDFs were based

on statistical simulations, standard assumptions like Gaussian

or noncentral

-distributions, or used models [12] that are

not able to describe the nonlinear noise redistribution shown

in experiments [11], [13]. Large-signal simulations can, in

principle, include the nonlinear redistribution, but it is difficult

EMICONDUCTOR optical amplifiers (SOAs) have a

number of promising applications within optical commu-

Manuscript received January 18, 2007; revised June 27, 2007. This work was

supported in part by the IST Project BIGBAND, the European Network of Ex-

cellence ePIXnet, and the Danish Technical Research Council.

The authors are with COM?DTU Institute for Communication, Optics

and Materials, NanoDTU, Technical University of Denmark, DK-2800

Kgs. Lyngby, Denmark (e-mail: fo@com.dtu.dk; jm@com.dtu.dk; btrom-

borg@mail.dk).

Digital Object Identifier 10.1109/JQE.2007.906226

to reach far out in the tails of the distributions, i.e., to simulate

the rare occurrences of errors corresponding to low BERs.

One way of expanding the range of BERs by simulations is

through importance sampling [9], but efficient application of

that method is still somewhat of an art [14], [15]. In this study,

wehavedevelopeda detailedanalytical modelthat,inprinciple,

allows calculation of the PDF of a noisy signal at the output of

an SOA including the tails of very low probability densities.

The model is an extension of the standard noncentral

bution, which takes into account additional second-order noise

contributions in the sense discussed in the following.

We examine the noise properties of a generic type of SOA.

The optical amplification in the SOA is assumed to take place

in a waveguide of length

, and the electrical field in the wave-

guide is described by its complex envelope

such that

is the optical power. By solving the equations for

the propagation of the electrical field through the SOA, one can

determine the output field

field

and the spontaneous emission noise. The sponta-

neous emission is added and amplified during transmission, and

it interacts in a nonlinear manner with the signal. In this paper,

we shall only consider the case where

which we have added Gaussian noise terms. The field

can then be expanded as

-distri-

normalized

in terms of the input signal

is a CW signal to

(1)

where

(2)

is the steady-state field in the absence of noise,

ration power to be introduced later, and

the noise contributions to the phase and normalized amplitude

of th order for

.

The optical output power

is the satu-

andare

is equal to, i.e.,

(3)

where

of noise. In the calculations of the PDF for

imation

tion of

, while the approximation

leads to a noncentral

cludes the second-order term

cludethe

-term,i.e.,itdoesnotcomprisethefullsecond-order

is the steady-state power in the absence

, the approx-

gives a Gaussian distribu-

-distribution for . The latter in-

, but it does not in-

0018-9197/$25.00 © 2007 IEEE

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ÖHMAN et al.: OUTPUT POWER PDF OF A SATURATED SOA1189

noise contribution. The

the gain saturation caused by the copropagating amplified spon-

taneous emission (ASE). The main results of this paper are the

derivation of an analytical expression for the PDF of

second order and the presentation of approximate methods for

calculating thePDF.Wealsopresentexamplesthatcompare the

distribution with the results obtained from a direct large-signal

simulation of the time-domain equations for the field and the

carrier density in the SOA. The distributions agree over the

range in which we can obtain the simulated PDF within a rea-

sonable computation time. Furthermore, we demonstrate that

the full second-order calculation may give a BER, which at the

level of 10

deviates by an order of magnitude from a non-

central

calculation.

Our present analysis assumes a CW incoming signal, but, in

order to deal with most SOA applications (i.e., amplification

of digital signals, wavelength conversion, and regeneration), it

mustbeextendedtoincludemodulatedsignals.Thiscanbedone

following our approach, but the calculations become substan-

tially more numerically demanding. Our analysis also assumes

copropagation of signal and ASE. In order to take into account

counterpropagating ASE or reflections from theSOA facets, the

model must include an extra field equation for the backward

propagating field as in laser modeling. It is beyond the scope

of this paper to extend the formalism to cope with modulated

signals, counterpropagating ASE, or cases of low signal power,

where the noise cannot be considered as a perturbation. How-

ever,weshowthatsecond-ordereffectsbecomeimportantwhen

calculating low BERs, and we expect this conclusion to persist

in analyses that go beyond our simplifying assumptions.

The paper is structured as follows. In Section II, the basic

model is presented, while the details of the PDF calculations

are presented in Section III and the numerical examples are an-

alyzed and discussed in Section IV. The final conclusions are

drawn in Section V. Some of the more detailed derivations are

collected in the three Appendices.

-term contains, among other effects,

to

II. NOISE MODEL FOR THE SOA

The analysis of noise in the SOA is performed in two dif-

ferent ways: a perturbation analysis to second order in the noise

contributionsandalarge-signalsimulation.Bothapproachesare

based on a model for the SOA that is described in this section.

Themodelisastandardrateequationforthecarrierdensityin

anSOAandapropagationequationfortheelectricfield

as described in [16]. The noise is incorporated in the equations

by Langevin forces, in accordance with [7] and [17]. The re-

sulting equations for carrier density and electric field are

,

(4)

(5)

where

is the active volume,

the effective cross-section area of the active region,

photon energy,

is the linewidth enhancement factor, and

is the waveguide loss. The time variable is a shifted time coor-

dinate,

, where

and

is the group velocity. The propagation is unidirectional

is the injected current,is the elementary charge,

is the spontaneous carrier lifetime,is

is the

is the real time coordinate

and perfect anti-reflection coatings are assumed, i.e., the reflec-

tivities of the facets are zero. The gain

a linear function of the carrier density, and it is assumed that the

carrier frequency

is chosen at the gain peak. The gain is then

is approximated as

(6)

where

sity at transparency.

The functions

describes the spontaneous emission noise and

the carrier density noise imposed by carrier injection and re-

combination noise. The work in [7] presents a detailed analysis

of the influence of carrier density noise on the relative intensity

noise (RIN) spectrum of the output signal of a saturated SOA.

The analysis shows that, for input powers of about 10 % of the

saturation power, the carrier noise only gives a small contribu-

tion to the RIN at low frequencies compared with the carrier

bandwidth. At higher input powers, the carrier noise becomes

increasingly important and may even lead to quantum optical

squeezing effects for input powers above the saturation power

[18]. In this paper, we only consider cases of moderate satura-

tion, and we shall therefore assume that

The Langevin function

for spontaneous emission is con-

sidered as a Gaussian noise source with correlation relations

is the differential modal gain andis the carrier den-

andare Langevin noise terms, where

describes

.

(7)

The Fourier transform of

emission spectrum

is the local spontaneous

(8)

where

dealing with a narrow frequency range around the carrier

frequency

, we can often assume that the spectrum is

white, i.e., constant in frequency, and use the approximation

. The corresponding

is the population-inversion factor. When we are

is then

(9)

However, a constant, unlimited noise spectrum means that the

noise power is infinite, which leads to divergent terms in the

second-order noise contributions. We shall therefore use the

form

(10)

where

the steady-state gain

, we use the approximation

simulations, we use the time step

of the signal. By assuming that the signal is a sample-and-hold

signal, i.e., the signal is constant between sampling points, the

correlation in time becomes

is finite and where the gain factor in

. For the population-inversion factor

is

. In the

for the discrete sampling

.

(11)

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1190IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 43, NO. 12, DECEMBER 2007

This correlation corresponds to a

also limits the amount of noise power

-shaped spectrum, which

(12)

Note that the unnormalized

sampling interval is sufficiently short to make the noise spec-

trum much wider than any filtering in the system, this assump-

tion gives first-order results very similar to the white noise as-

sumption.

The injected field is chosen to be of the form

function is used. As long as the

(13)

where

Gaussian stochastic variables with zero mean and variance

They are sampled with time interval

tion relations

is constant and the real functionsand are

.

and satisfy the correla-

(14)

with

ratio by noting that the input power has a noncentral

bution with mean and variance given by

givenby(11).is relatedtothe input signal-to-noise

-distri-

(15)

(16)

where

we must choose

. By comparing (13) with (1), we see that

(17)

(18)

to ensure that (1) is satisfied to second order.

A. Large-Signal Simulations

In order to have a comparison for our second-order model,

we have implemented a brute-force large-signal model, which

uses statistical methods for simulating the PDF of the signal.

The model has been presented in detail in [13] and is based on

the work in [17]. The rate equations (4) and (5) are integrated

numerically by discretizing the signal in time and the SOA in

the -direction.Noisetermswithstatisticsaccordingto(14)and

(7) are added to the signal field at the input and for each SOA

section, respectively.At theoutput of thedevice, thesignal field

is filtered using an optical filter and then detected assuming an

idealnoiselessdetectorincludingalow-passelectricalfilterwith

filter function

. The statistics of the detected signal are then

extracted, and the PDF is estimated by making a histogram of

the signal.

B. Perturbation Expansion

The perturbation analysis of (4) and (5) is based on the as-

sumption that the Langevin noise term

with the signal field and can be considered as a perturbation.

The first step in the analysis is to derive the steady-state solu-

tion for

. As mentioned above, we will throughout the

paper assume that

.

Equations (6) and (4) lead to the following equation for the

gain:

is small compared

(19)

where

is the saturation power

(20)

and

is the unsaturated gain, i.e., the steady-state gain when

as

(21)

The steady-state solution to (19) becomes

(22)

where

duced in (2). For

is the steady-state normalized field amplitude intro-

, integration of (5) gives

(23)

Equations (22) and (23) can be solved numerically for given

input field

.

In (1), the envelope field was factorized as

the factor

describes the perturbations due to noise as

, where

(24)

By inserting

equation

into (5), it follows that satisfies the

(25)

Equations (19) and (25) can be solved by inserting the expan-

sions (24) for

and

(26)

forthegainandequatingtermsofthesameorderin

first-order equations become

.The

(27)

(28)

(29)

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ÖHMAN et al.: OUTPUT POWER PDF OF A SATURATED SOA1191

where

tions for the second-order terms become

and . The equa-

(30)

(31)

(32)

The frequency-domain solution of these equations are given in

Appendix A.

Equations (27)–(32) also apply to the case where the in-

coming field is modulated, provided that

and are the time-dependent field and gain solutions to (5)

and (19) in the absence of noise

.

III. PDF OF A FILTERED OUTPUT SIGNAL

The output field from the SOA is assumed to be filtered by

an optical filter with a bandwidth that is much smaller than the

bandwidth of the spontaneous emission spectrum. The filter is

described by a time response function

to be real to ensure that the filtering is symmetric around the

carrier frequency

. The filtered output field is then

, wheremeans convolution in the time domain.

For the filtered output power, we obtain an expression similar to

(3) as

, which we assume

(33)

We notice that the filtering induces a second-order phase-to-in-

tensity conversion. The expression reduces to (3) when

.

TheaimofthissectionistocalculatethePDFof

thepowerisassumedtobestationary,itissufficienttodetermine

the PDF for

. The procedure is first to derive an expression

for the moment generating function (MGF)

.Since

(34)

The PDF of

form

is then obtained from the inverse Laplace trans-

(35)

where

for

noise functions

nienttointroduceavectorspacespannedbythevectors

and, where

The vectors are assumed to satisfy the orthogonality relations

is a real number for which

.Inordertodescribethesetoffrequency-domain

has no singularities

, it is conve-

,is real, andis the label or .

(36)

(37)

(38)

A set of noise functions is then described by a vector

components

with

(39)

(40)

i.e., the vector

the noise added along the length of the amplifier (39) and the

noise of the input signal (40), which is independent of . Using

this notation, it is shown in Appendix A that the filtered output

power can be written as

describes the real and imaginary parts of both

(41)

where

erator

is the vector with components (70) and is the op-

(42)

is the operator with matrix elements given by (71)–(73).

The probability of observing a particular noise vector

assumed to be given by the Gaussian functional (see, e.g., [19,

eq. (12.41)])

is

(43)

where

.

elements given in Appendix B. For

the Gaussian noise distribution (43), the MGF (34) is the path

integral

is a normalization constant that ensures that

is the reciprocal of the diffusion operator with matrix

given by (41) and for

(44)

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1192 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 43, NO. 12, DECEMBER 2007

The integral can be shown to be given by the explicit form (see,

e.g., [20, eq. (1.12)])

(45)

The MGF can be calculated directly by discretizing the space

and frequency intervals.

From the MGF, it is straightforward to derive the mean value

and the central moments (see, for example, [21]) by using the

cumulant-generating function (CGF)

(46)

From the CGF, the cumulants of order

can be calculated by

(47)

The first-order cumulant is identical to the mean value, and or-

ders two and three are identical to the respective central mo-

ments defined by

(48)

The distribution thus has the mean, variance, and third-order

moments

(49)

(50)

(51)

The first term in (50) corresponds to the spontaneous–spon-

taneous beat noise while the second term represents the

signal–spontaneous beat noise.

It is shown in Appendix C that (45) reduces to the familiar

MGF for a noncentral

-distribution when the operator

(41) is zero as

in

(52)

The distribution has the mean value and variance

(53)

(54)

By (70) and (77)–(79), the parameter

can be written as

(55)

The expression (55) (multiplied by four) can be shown to be the

frequency integral over the RIN spectrum.

Thus far, we have treated the optical field and power, but, in

order to compare with measurements, we need to introduce the

conversion to the electrical domain by a detector. The detector

model used here is an ideal noiseless square-law detector with

unit responsivity and a limited frequency response. The effect

on the PDF can thus be calculated by introducing a second filter

acting on the optical power, which is the same as filtering the

electrical current from the detector. The time-dependent current

is then

(56)

where

the MGF expressed by (45) agrees with (56) if a new oper-

ator

, with matrix elements given in Appendix A (74)–(76),

and a new vector

, with components as shown for

but multiplied by

, are substituted for

tively, in (45). The operator

is very similar to

trical filtering means that the matrix elements are multiplied by

.

In Section IV, we present a numerical analysis of the effect

on the PDF of

and the electrical filter.

is the filter function of the detector. The result for

in (70)

, respec-and

, but the elec-

IV. NUMERICAL EXAMPLES AND DISCUSSION

Here, the model is used to analyze a few specific examples.

The examples are chosen to be similar to the ones in [11] in

ordertobe able tocompare withthemeasurementsinthatwork.

The PDFs from the second-order model are calculated by dis-

cretizing space and frequency and numerically solving (45) and

(35). The matrix calculations, and especially calculating the de-

terminant, requires a large computer memory when a fine dis-

cretizationisused.Inordertoreducetherequirementsandallow

better numerical resolution, we instead calculate an appropriate

number of terms in the sums of the expansions in (94) and (95)

[with

replaced byin (94)], as described in Appendix C.

For each higher order moment to be included in the investiga-

tion, another term has to be included in the sums when calcu-

lating the MGF. In this investigation, we have limited ourself to

thethird-order momentand, hence, three termsof (94) and (95).

The parameters used in the calculations are a mixture of

known physical parameters for the measurements of [11] (i.e.,

length of SOA, bias current, input powers and detection band-

width), reasonable guesses (e.g., coupling losses, waveguide

losses, linewidth enhancement factor, and carrier lifetime) and

fitted parameters (i.e., small-signal gain, saturation power, and

input signal-to-noise ratio). The fitted parameter values are

chosen to give a qualitatively reasonable fit to the experimental

results in [11] for both the standard deviation and the skewness,

which is defined as the normalized third-order central moment

of the distributions. No quantitative fitting procedure has been

carried out. The chosen parameter values are shown in Table I.

The filters used in the calculations are eighth-order Butterworth

filters.

First, we will compare the PDFs. Fig. 1 shows the PDFs cal-

culated with

and without

of the second-order terms, as well as the simulations. The use-

fulness of analytical expressions for the PDFs is clearly seen

when comparing the tails of the measured and simulated PDFs.

the proper inclusion

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