Single Mode Fiber VP i

Photonic Modules > Fibers > FiberNLS

FiberNLS

Nonlinear Dispersive Fiber (NLS)

Purpose

When used with sampled-mode signals, this module solves the nonlinear Schroedinger (NLS)

equation describing the propagation of linearly-polarized optical waves in fibers using the split-step

Fourier method. Depending on the signal representation, different effects are represented: if the

signals are in a Single Frequency Band (SFB), or JoinSampledBands = ON, the model takes into

account stimulated Raman scattering (SRS), four-wave mixing (FWM), self-phase modulation

(SPM), cross-phase modulation (XPM), first order group-velocity dispersion (GVD), second order

GVD and attenuation of the fiber. If the signals are in a Multiple Frequency Band (MFB), the above effects are calculated within each band. Use the module UniversalFiber (or UniversalFiberFwd) to

include interactions between MFBs and also between MFBs and Parameterized Signals. For

Parameterized Signals (CW representation) an ordinary differential equation system including Stimulated Raman Scattering (SRS) and frequency dependent attenuation is applied. For this to

operate ConvertToParameterized must be set to ON.

Keywords

Fiber, Nonlinear, Split Step, Raman, XPM, FWM, SPM, GVD

Inputs

Outputs

Parameters

In this section:

PhysicalNumerical\Split-Step ParametersNumerical\Raman Parameters

Enhanced

Port Purpose Signal/Data Type

input Input optical signal Optical Blocks

Port Purpose Signal/Data Type

output Output optical signal Optical Blocks

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PhysicalPhysicalPhysicalPhysical

Numerical\SplitNumerical\SplitNumerical\SplitNumerical\Split----Step ParametersStep ParametersStep ParametersStep Parameters

Name and Description Unit Type Volatile Value RangeDefault Value

ReferenceFrequency

Reference frequency for the specifiedparameters.

Hz float yes]0;∞[

a 193.1e12

Length

Fiber length.

m float yes[0.0;∞[

a 1.0e3

GroupRefractiveIndex

Group refractive index of thefundamental fiber mode at the reference frequency.

- float yes[0.0;∞[

a 1.47

Attenuation

Fiber attenuation per meter ifAttFileName is not specified.

dB/m float yes[0.0;10e-2]

w 0.2e-3

AttFileName

File for attenuation vs. frequency.

- inputfile yes - -

Dispersion

Dispersion coefficient in terms ofwavelength.

s/m^2 float yes [-500e-6;500e-

6]w

16e-6

DispersionSlope

Slope of the dispersion coefficient with

wavelength.

s/m^3 float yes [-50.0e3;50.0e3]

w

0.08e3

NonLinearIndex

Nonlinear refractive index measuredwith constant linear polarization. For fibers with randomly varying

birefringence this value should be reduced by a factor of 8/9.

m^2/W float yes [-50e-20;50e-

20]a

2.6e-20

CoreArea

Effective core area of the fiber for

nonlinear calculations.

m^2 float yes]0.0;∞[

a 80.0e-12

Tau1

Inverse of the negative frequencyoffset of the Raman gain peak from the Raman pump frequency.

s float yes [1e-15;50e-15]a

12.2e-15

Tau2

1/[2*pi*(width of the Raman gaincurve)].

s float yes [1e-15;50e-15]a

32.0e-15

RamanCoefficient

Fractional contribution of the delayedRaman response.

- float yes[0.0;1.0]

a 0.0

Name and Description Unit Type Volatile Value Range Default Value

StepSelectionMethod

Defines the method for selecting the size of the step in the split-step fourier method. WhenLocalErrorMethod is

selected, the step size is

- enum yes LocalErrorMethod,

NonlinearPhaseChange,ConstantStep

NonlinearPhaseChange



governed by

TargetLocalError parameter. If NonlinearPhaseChange is selected, the step size is specified by

MaxPhaseChange parameter. For ConstantStep setting,the fixed step size equal to MaxStepWidth in used in simulation.

SymmetricSplitStep

Type of the split-step Fourier method.(context sensitive)

- enum yes NO, YES No

MaxStepWidth

Maximum step width allowed in the split-step Fourier method.

m float yes]0.0;∞[

a 1.0e3

MinSplitStepWidth

The minimum allowed step width in the split-step Fourier method.(context sensitive)

m float yes[0.0;∞[

a 1.0

InitialSplitStepWidth

Initial step width for local error method in the split-step Fourier algorithm.

(context sensitive)

m float yes]0.0;∞[

a 1.0e3

TargetLocalError

Maximum tolerable local error in the split-step Fourier method. The

step width is chosen in such a way that the local error on a single simulation step is within [TargetLocalError/2;

2*TargetLocalError]. Smaller values of TargetLocalError result in smaller steps, which provide better accuracy, but require longer

computation time. The upper and lower limits of the step width are set by the parametersMaxStepWidth and

MinSplitStepWidth, respectively.(context sensitive)

- float yes]0;∞[

a 1.0e-3

MaxPhaseChange

Maximum tolerable

phase change acrossone step of the split-step Fourier method. The step width is chosen in such a way that the

phase of the optical field in the nonlinear step does not change by more than MaxPhaseChange over

deg float yes]0;∞[

a 0.05



Numerical\Raman ParametersNumerical\Raman ParametersNumerical\Raman ParametersNumerical\Raman Parameters

EnhancedEnhancedEnhancedEnhanced

the step. Smaller values

of MaxPhaseChange result in smaller steps, which provides better accuracy, but requires longer computation

time. The upper and lower limits of the step width is set by the parameters MaxStepWidth and MinSplitStepWidth,

respectively.(context sensitive)

Name and Description Unit Type VolatileValue

Range

Default

Value

FiberParts

For Raman interaction between Parameterized signals. The number of sub-integrations used along the fiber by the ODE solver.

- int yes[1;1e9]

a 10

MaxStepsNumber

For Raman interaction between Parameterized signals. The maximum number of ODE integration steps allowed for one fiber part.

- int yes[1;1e9]

a 100000

RelativeAccuracy

For Raman interaction between Parameterized signals. The required accuracy that the ODE solvershould reach before stopping, unless MaxStepsNumber is exceeded.

- float yes [1e-

20;1e-1]a

1e-6

InitialStepWidth

For Raman interaction between Parameterized signals. The duration of the first timestep used by the ODE solver.

s float yes [1e-

15;1e-9]a

10e-12

MinimumStepWidth

For Raman interaction between Parameterized signals. The minimum duration of the timestep used by the ODE solver.

s float yes [0;1e-9]

a

0.0

Name and Description Unit Type VolatileValue Range

Default Value

JoinSignalBands

Join all sampled signal bands to one signal band.

- enum yes NO, YES Yes

AddNoiseBinsToSignal

Incorporate Noise Bins into signal bands if applicable.

- enum yes NO, YES Yes

ConvertToParametrized

Defines if input signals are converted to

Parameterized signals. Must be set to YES if Raman interaction is to be calculated, even if input signals are already Parameterized.

- enum yes NO, YES No

Active

Defines if the module is active or not.

- enum yes On, Off On



Description

The sampled bands must contain a single signal polarization. However, unpolarized noise can be

propagated in Noise Bins. This is useful for saturating optical amplifiers with both polarizations of

noise. However, this unpolarized noise should not be added to the sampled band, as only a linear-

polarization sampled signal can be handled. This requires the Global Parameter InbandNoiseBins =

ON, and the fiber parameter AddNoiseBinsToSignal = OFF.

This is one of the simplest fiber models in the range. It is the basis of many fiber simulations. However, more complex fiber models are provided to offer more flexibility in terms of input data

format, interactions between signal representations, bidirectional Raman amplification, dispersion

decreasing fiber (UniversalFiber or UniversalFiberFwd), polarization mode dispersion

(FiberNLS_PMD), fast simulation of jitter in RZ systems (JitterLongHaul, JitterShortHaul), and

simulation with aperiodic waveforms (TimeDomainFiber).

Note: For simulations where the nonlinear effects are unimportant, simply set NonLinearIndex

to zero and ignore the parameters CoreArea, Tau1, Tau2, RamanCoefficient, and all settings in

the Numerical Parameters category.

The simulation speed of this module can be improved using parallel computations on a multi-core central processing unit (CPU) or a supported graphical processing unit (GPU), see Chapter 3, “User

Interface Reference” in the VPItransmissionMaker™/ VPIcomponentMaker™ User Interface Reference for details on supported hardware and GUI controls for parallel simulations.

The FiberNLS module can use parallel computations for the split-step Fourier algorithm. Both FFT

calculation and multiplication of the sample arrays will be done in parallel.

When both multithreading within modules and GPU-assisted computation are switched on and

supported by the hardware, the module will use the GPU, as it usually reduces the simulation time.

Note: Only calculations with sampled bands will be parallelized, both on the CPU and GPU.

Split-Step Fourier Implementation

In this section:

Dispersion OperatorNonlinear Operator (with no Raman effect)Split-Step Fourier Method

Nonlinear Operator (with Raman effect)

The generalized nonlinear Schrödinger equation is used to describe the inband effects:

where denotes the slowly-varying complex-envelope of the electric field of the light wave,

characterizes its power, is the nonlinearity operator, and is the dispersion

operator.

Dispersion Operator

(1)



Note: For linear fibers, this is the only part of the theory that is required, as a single-section model will be applied if the NonLinearIndex is zero.

By removing changing the time reference to remove the intrinsic fiber delay, the dispersion

operator can be written as:

where:

The dispersion parameter, Dλ, has to be entered in SI units [10

-6 s/m

2= 1 ps/(km · nm)]

The dispersion slope parameter, Sλ, also has to be entered in SI units [10

3 s/m

3= 1 ps/

(km · nm2)]

Note: Setting the DispersionSlope to zero does not give a constant pulse spreading on each

WDM channel. This is because the dispersion is specified in terms of wavelength, but

implemented in terms of optical frequency. This is apparent when Dispersion and

DispersionSlope are set to zero, but there remains some pulse spreading.

The parameter AttFileName can be used to specify a file that contains the Attenuation

parameter [dB/m] vs. the optical frequency [Hz]. The file format is:193e12 0.2e3

193.1e12 0.2e3

193.2e12 0.2e3

Nonlinear Operator (with no Raman effect)

If stimulated Raman is to be excluded from the simulation (parameter RamanCoefficient = 0) the

nonlinear operator is simply given by:

(2)

β2 [s

2/m] describes the first order group-velocity dispersion (GVD), and is related to the

Dispersion parameter at the reference wavelength (= c/ReferenceFrequency) by

(3)

β3 [s

3/m] the second order GVD slope, and is related to the DispersionSlope S

λ= dD

λ/dλ by:

(4)

α [1/m] the attenuation constant. The Attenuation parameter a [dB/m] is related to α by

. (5)

(6)



and depends on the nonlinear index n2 (parameter NonlinearIndex, see Polarization

Considerations), the effective CoreArea Aeff

, as well as on the reference frequency fref

and the

velocity of light in vacuum c.

Split-Step Fourier Method

The split-step Fourier method divides the fiber into alternate sections of two types. The first type

represents the dispersion in the frequency domain, the second represents the nonlinearity in the

time domain. The choice of step-size is crucial in achieving a balance between computation time

and accuracy.

The step size, Δz (the length of fiber represented by one nonlinearity and one dispersion operator)

is determined according to the following parameter settings:

When using the local error method, the following should be observed:

where Δφnl

(radians) is the maximum acceptable nonlinear phase shift1. The parameter

MaxPhaseChange sets the maximum acceptable phase change (but in degrees) and is normally

within the range of 0.06…0.2 degs.

Note: For both local error and nonlinear phase change methods, the step size is limited from

above and below by settings of the parameters MaxStepWidth and MinSplitStepWidth,

respectively.

If the desired step size is less than MinSplitStepWidth, the module will stop with an error

message. Normally, the minimal step size should be set as a ‘guard condition’ to ensure that the simulation will not take an excessive amount of time.

If the desired step size is larger than MaxStepWidth, the module will use the maximal allowed

step size. This is normally set to be far less than the walk-off length (the fiber distance that

produces a relative phase shift of 2π radians between the carriers) between two optical carriers due to dispersion. Obviously, the wider the WDM spectrum within a SFB, the shorter

the step length should be. This actually shows that it is better to model a few WDM channels

in an SFB, than all channels (this is called the Mean Field Approach, and is discussed in the VPItransmissionMaker™Optical Systems User’s Manual).

If StepSelectionMethod = LocalErrorMethod, the module takes the step according to the

algorithm described in [1], maintaining the relative error at each step within range [ε/2; ε],

where the desired error value ε is defined by the parameter TargetErrorValue. This mode is

rather similar to adaptive step algorithms routinely used in ordinary differential equations.

+ Local error method has greater order with respect to step size than the other optionsavailable for the fiber modules, which means that it will be more efficient (require less steps) for high-accuracy simulations. However, due to a larger number of computations per

step, low-accuracy simulations can be somewhat longer.

+ In some cases, such as propagation of several frequency components in a shifted-

dispersion fiber, local error method tends to make larger errors in low-power regions of the

spectrum (like the FWM products).

If StepSelectionMethod = NonlinearPhaseChange, the step size, Δzφ

, determined by not

allowing the nonlinear phase shift (proportional to optical power) to exceed a defined value,

MaxPhaseChange. This gives a step size

(7)



Assuming a propagation of optical signals in +z direction and a symmetrical split-step algorithm

(parameter SymmetricSplitStep is set to YES, or if the parameter StepSelectionMethod is set to

LocalErrorMethod2), the mathematical formalism of the procedure can be described as follows:

If the parameter SymmetricSplitStep is set to NO, the equation changes to:

The nonlinearity operator is applied on the field at locus z0. Therefore, the exact dependence of

operator on the time-dependent amplitude in the interval Δz is replaced by its start value:

Nonlinear Operator (with Raman effect)

If the parameter RamanCoefficient is not zero, the nonlinearity operator is extended to:

where (parameter RamanCoefficient) represents the fractional contribution of the delayed

Raman response which is approximated by:

In Equation 11, (parameter Tau1) and (parameter Tau2) are two adjustable parameters to

provide a good fit to the Raman-gain spectrum:

The commonly used values are =0.18, τ1=12.2 fs, τ

2=32 fs giving an offset of 13 THz and a

width of 10 THz. The approximated Raman gain and a realistic spectrum which has been measured in silica fibers are shown in Figure 1.

If StepSelectionMethod = ConstantStep, the constant step size equal to the value of the

MaxStepWidth parameter will be used.

(8)

(9)

(10)

(11)

, t > 0 (12)

Tau1 (s) is the inverse of the frequency offset (in radians/s) between the Raman pump and

peak Raman gain,



Figure 1 Example of the measured (thick line) and approximated (thin line, default parameter setting) Raman spectra for pure silica at 1500 nm

Parameterized Signals and Noise Bins: Raman scattering

In this section:

Polarization Considerations

For Parameterized Signals and Noise Bins, the effect of stimulated Raman scattering is modeled by

the following ordinary differential equations:

where Aiis the amplitude of the i th Parameterized Signal, N

k is the amplitude of the k th Noise Bin,

and αi, α

kare representing their attenuation coefficients, respectively. In Equation 12, f

rdescribes

(13)

the fractional contribution of the delayed Raman response to the nonlinearity coefficient γ, and

represents the imaginary part of the Fourier-transformed Raman response function

at the frequency difference between the Parameterized Signals (fi− f

m). In the lower equation,

denotes the imaginary part of the Fourier transformed Raman response function at

the frequency difference between the Noise Bins and the Parameterized Signals.

This system of ordinary differential equations is solved using a 4th-order Runge-Kutta equation

solver. The parameters MaxStepsNumber, MaxPhaseChange, RelativeAccuracy, InitialStepWidth,

MinimumStepwidth are used to tune the equation solver. The parameter FiberParts can be used to

subdivide the fiber length into several sections. For each of these parts the integration algorithm is

restarted.

The demonstration Optical Systems Demos > Characterization > Raman Scattering > Raman Power Transfer illustrates this mode of operation.

Polarization Considerations



SPM is different for linear, circular and random polarizations. In a real fiber, Polarization-Mode

Dispersion (PMD) leads to a random evolution of polarization (between linear and circular) along the

fiber, which means that the strength of the SPM varies along the fiber (linear polarization produces

a stronger peak field than circular polarization). Averaging over all polarization states (assuming

complete polarization mixing, i.e., when the fiber is much longer than the beat length and

correlation length) gives a well-known factor of 8/9. This factor is often already included in the

measured value of the nonlinear refractive index, n2. In this case, the n

2 value can be directly used

for the parameter NonlinearIndex. However, if you use the n2 value that has been measured

without this correction, i.e. for constant linear polarization, the measured n2

should be multiplied by

a factor 8/9 before being input as a parameter.

To model the average effect of XPM between channels (using a SFB), where the polarizations of the

channels rotate with respect to one another over the length of the fiber, the nonlinear factor should

be reduced by a factor 2/3 (or 3/4 if the measured n2 value already includes the 8/9 factor). For

worst-case calculations (where the polarization remains linear along the fiber), the nonlinear

refractive index should not be reduced. Note that it is not possible to apply valid correction factors

simultaneously to SPM and XPM using this model. For accurate simultaneous simulation of the

mentioned effects the more complex fiber models FiberNLS_PMD, UniversalFiber or

UniversalFiberFwd should be used.

Signal Representation

In this section:

Block

Block

Single Frequency Bands (SFBs) are simulated using the split-step Fourier method. All signals are assumed to be polarization aligned within the band.

Multiple Frequency Bands (MFBs) are individually simulated using the split-step Fourier method. No

interaction between the bands is calculated. That is, Raman power transfer, XPM and FWM between

the bands are not calculated. MFBs may be converted to an SFB by setting JoinSignalBands to ON.

Parameterized Signals are computed solving the above system of ordinary differential equations

using a 4th-order Runge-Kutta equation solver. ConvertToParameterized should be set to ON. Any

sampled signals will be converted to Parameterized Signals if ConvertToParameterized is set to ON.

For parameterized signals, the values of a number of physical parameters are saved to the tracking

data (see Chapter 5 of the VPItransmissionMaker™Optical Systems User’s Manual). They can be

visualized using the LinkAnalyzer module. The fiber parameter GroupRefractiveIndex is used to

calculate the fiber delay (TransitTime in LinkAnalyzer).

The accumulated self-phase modulation in the signal tracking data is characterized by the phase

shift and is calculated according to [2]

where is the input signal power and is the effective fiber length.

Noise Bins

The system of ordinary differential equations includes the interaction between Noise Bins and

Parameterized Signals when ConvertToParameterized is set to ON.

Boundary Conditions

Periodic. For Aperiodic Boundary Conditions see TimeDomainFiber.

(14)



Reinitialization Behavior

Multiple Runs

Reinitialization is performed after each run.

Restart of Simulation or Reset during Simulation

Reinitialization is performed.

Reset during Simulation

Reinitialization is performed.

Module Deactivation

If parameter Active is OFF, the optical input signals will be passed through the module without any

changes.

Validation

In this section:

(1) Spectral broadening of a Gaussian pulse due to Self-Phase Modulation (SPM)(2) Temporal evolution of solitons

i) Fundamental solitonii) Second order Soliton

This section is intended to prove the agreement of known theoretical predictions for optical fibers.

Since there are numerous theoretical aspects to compare with the behavior of computational

applications, a choice of two test runs has been made focusing on both the nonlinear and dispersive

properties implemented in the module.

The two test runs explained below concentrate on the temporal and/or spectral evolution of single

pulses in single channels.

(1) Spectral broadening of a Gaussian pulse due to Self-Phase Modulation (SPM)

a) Theory

It is known [2], that in the limit of (no GVD) the nonlinear Schrödinger equation

becomes

where the normalized amplitude U (z,τ) is related to the field amplitude A(z,τ) by

The nonlinear length LNL

is introduced by

where γ is defined as described above and P0 is the peak power of the Gaussian pulse. The solution

(15)

(16)

(17)



to (15) is given by

with a phase argument

The effective fiber length zeff

becomes zeff

= z if α = 0 (no fiber loss) is assumed. The phase shift is

time-dependent and gives rise to a frequency chirp δ ω(t). The latter is given for a Gaussian pulse

by

where is the spectral width of the initial pulse, and Φmax

is the maximum phase shift

which is reached at the pulse center [2]. The spectral broadening is thus a function of the pulse

power and the propagation length z.

b) Simulation

TThe most important parameters are: PeakPower, NonlinearIndex, Length.

After every fiber length , the maximum phase shift is increased by one radian which, in

turn, gives rise to an interference pattern stemming from the superposition of the chirped pulse’s

frequency components. The number of peaks in the spectrum of the pulse M is approximately given by the relation

c) Results

The simulation has been performed with fiber lengths giving 0 π, 0.5 π, …, 3.5 π maximum phase

shift. The spectral plots resulting from this simulation are shown in Figure 2 and are in good

agreement with [2].

(18)

(19)

(20)



Figure 2 Spectrum of a Gaussian pulse as it propagates along the Fiber (Simulation Results). Parameters used: SampleRate 1024e9, BitRate 1e9, Bits 1, FWHM 1/(16*BitRate), PeakPower 0.1,

PRBSType ONE.

(2) Temporal evolution of solitons

Solitons are of major interest in optical fiber communications. The term “soliton” refers to special

kinds of waves that can propagate undistorted over long distances and remain unaffected after

collision with each other. Ideal Solitons exist in fibers with anomalous group-velocity dispersion (D

> 0, β2< 0, cf. (4)) and vanishing loss. Solitons are well suited for functional tests of this module,

since both nonlinear and dispersive properties enter into the numerical simulation.

i) Fundamental soliton

a) Theory

The analytic solution representing solitons is obtained, e.g., applying the inverse scattering method [2] [p. 142] to the nonlinear Schrödinger equation. This method leads to a general solution for

solitons of different orders. The fundamental soliton (first order) will propagate undistorted without

change in shape for arbitrarily long distances. A higher-order soliton exhibits an oscillating pattern of its temporal and spectral evolution.These spatial variations are periodic with length of z

0 (soliton

period) for solitons with an integer order.

A fundamental soliton is given by the analytic expression

with the normalized field amplitude U (ξ, τ), cf. (16), and

(22)



Here, N denotes the soliton order with

and LD is the dispersive length

P0 refers to the peak power, T

0 is given by T

FWHM, and the approximate relation for the pulse

width (sech pulse)

Therefore, for given values of γ and β2 and the pulse width T

FWHM, the peak power of a soliton of N

th order is uniquely defined.

b) Simulation

Propagation of first and second order solitons has been simulated using the setup which uses a

Pulse Sechant Transmitter, a Nonlinear Dispersive Fiber (NLS) module, and visualizers for the time and frequency domain. The first-order pulse should maintain its shape while propagating along the

fiber. This has been tested for up to 100 times the soliton period (approx. 10,000 km fiber length).

The results have been evaluated by comparing the data obtained with the analytical

expression (22).

c) Results

The simulation data obtained for 0…5000 km fiber length has been found to agree quite well

with (22). The pulse shapes for the numerical and theoretical results are shown in Figure 3. No

difference can be seen.

Figure 3 Pulse shape of a fundamental soliton after z=0 km and z=5000 km

The following parameter settings have been used for the simulation:

(23)

(24)

(25)

(26)



The parameter Length was chosen according to multiples of the soliton period z0. MaxStepWidth was

set to the fiber length. To obtain a soliton of order N, the parameter PeakPower has to be set

according to equation (24), depending on the settings of the other parameters:

The parameter DispersionSlope has been adjusted so that the second order GVD β3 vanishes:

ii) Second order Soliton

In contrast to the fundamental soliton, there is a periodical change in temporal shape and spectrum

of higher-order solitons due to propagation along the fiber. As an example, a second order soliton is simulated.

a) Theory

The periodic evolution of a higher-order soliton is ruled by the group velocity dispersion β2 of the

fiber and the pulse width T0. The analytic solution found for the 2nd order soliton is given by

shows a periodic behavior with period ξ0

= π / 2. The correct implementation of

dispersive (GVD) and nonlinear effects (SPM) in the fiber module may be observed by launching a

2nd order soliton into the fiber and by comparing the results with the theoretical predictions.

b) Simulation

In the diagrams shown in Figure 4, the soliton pulse evolves from its initial state to a split state in

the time domain and back again to the initial shape after one soliton period .

c) Results

The period length was found to coincide with the predicted theoretical value.

Example

Optical Systems Demos > Characterization > Dispersion & Kerr > Four Wave Mixing

illustrates the split-step Fourier method and the oscillatory growth of Four-Wave Mixing products along a fiber.

SampleRate 1024e9 NonlinearIndex 2.6e20

BitRate 1e9 CoreArea 80.0e12

Bits 1 MaxStepWidth set to fiber length

ReferenceFrequency 193.1e12 MaxPhaseChange 0.05

Length varying, see text FWHM 1/(16*BitRate)

Attenuation 0 PeakPower (see text)

Dispersion 16.0e6 DispersionSlope (see text)

. (27)

. (28)



Figure 4 Time evolution of a 2nd order soliton (N = 2). The fiber length is expressed in terms of

the periodic length . Parameter Settings: SampleRate 1024e9, NonlinearIndex 2.6e

20, BitRate 1e9, CoreArea 80.0e12, Bits 1, MaxStepWidth 1.0e3, RefFrequency 193.1e12,

MaxPhaseChange 0.05, Length (see text), FWHM 1/(16*BitRate), Attenuation 0, PeakPower (see

previous text), Dispersion 16.0e6, DispersionSlope (see previous text).

References

Setting to Δφnl

to 2π radians gives the Nonlinear Length, which is a useful metric.

The local error method implicitly uses symmetrical split-step algorithm, see details in [1].

Photonic Modules > Fibers > FiberNLS

[1] O. V. Sinkin, R. Holzlönner, J. Zweck, C.R. Menyuk, “Optimization of the split-step Fourier method in modeling optical-fiber communications systems,” J. Lightwave Technol., vol. 21, no. 1, pp. 61–68,

January 2003.

[2] G.P. Agrawal, Fiber-Optic Communication Systems, 2nd edition. Wiley Interscience, 1997, pp. 62-63.

[3] G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, 1995.

1

2

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