Characterization of Nonlinear Distortions in Fiber Optical ... · again generated, requiring a...

Characterization of Nonlinear Distortions in Fiber

Optical Transmission Systems with Mode-Division

Multiplexing

vorgelegt von

Herrn Dipl.-Ing. Georg Friedrich Rademacher

geb. in Minden / Westfalen

Von der Fakultät IV - Elektrotechnik und Informatik - der

Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Ingenieurwissenschaften

- Dr.-Ing.-

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender des Prüfungsausschuss: Prof. Dr.-Ing. Rolf Schuhmann

Gutachter: Prof. Dr.-Ing. Klaus Petermann

Gutachter: Prof. Dr.-Ing. Peter Krummrich

Gutachter: Prof. Dr.-Ing. Christian Alexander Bunge

Tag der wissenschaftlichen Aussprache: 18.06.2015

Berlin 2015

I would like to thank Prof. Klaus Petermann for his ongoing encouragement that led

to the present work. His knowledgeable and cheerful way of guiding everybody in this

institute makes this a great place to be.

Furthermore, I would like to thank Prof. Peter Krummrich from TU Dortmund. Numer-

ous discussions during project meetings where very helpful in the progress of this project. I

am grateful that Prof. Krummrich, Prof. Christian Alexander Bunge and Prof. Rolf Schuh-

mann agreed to be members of my Dissertation committee.

I would also like to thank Dr. Stefan Warm, with whom I spend a lot of time in defining

the direction of my research and discussing my results.

My sincere thanks goes to all my colleagues from the HFT institute: Dimi, Mahmoud,

Abdul, Edgar, Karsten, Lars, Benni, Isaac, Christos, Xiamong, Matthias and Henrietta.

Thanks a lot for the fun time in this institute, including lots of coffee and (non-) political

discussions.

I am also thankful to the Deutsche Forschungsgemeinschaft (DFG) for funding the re-

search project that was the basis for this thesis.

Last but not least I would like to thank my lovely wife Geraldine for being so under-

standing and supportive, especially during the last months of my thesis writing. Also she

helped a lot by proof reading several different states of this thesis.

iii

Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 History of Optical Communication . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Mode-Division Multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Objectives of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Wave Propagation in Optical Fibers 9

2.1 Linear Propagation in Optical Fibers . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Single- and Multi-Mode Fibers . . . . . . . . . . . . . . . . . . . . 9

2.1.2 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.3 Fiber Modes in Step-Index Fibers . . . . . . . . . . . . . . . . . . 11

2.1.4 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.5 Fiber Modes in Graded-Index Fibers . . . . . . . . . . . . . . . . . 17

2.1.6 Numerical Calculation of Fiber Modes . . . . . . . . . . . . . . . . 22

2.1.7 Accuracy of the Infinite Parabolic-Index Assumption . . . . . . . . 24

2.1.8 Few-Mode Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2 Nonlinear Propagation in Multi-Mode Fibers . . . . . . . . . . . . . . . . 33

2.2.1 Generalized Nonlinear Schrödinger Equation for Multi-Mode Fibers 34

2.2.2 Single-Mode NLSE . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2.3 Two-Mode NLSE . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2.4 Four-Wave Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2.5 Consequence of Intermodal FWM for the NLSE . . . . . . . . . . 43

2.2.6 Manakov Schrödinger Equation for Multi-Mode Fibers . . . . . . . 45

3 Capacity Limits of Multi-Mode Fibers 47

3.1 Nonlinear Gaussian Noise in Single-Mode Fibers . . . . . . . . . . . . . . 48

3.2 Nonlinear Gaussian Noise in Multi-Mode Fibers . . . . . . . . . . . . . . . 52

3.2.1 Verification of the Multi-Mode Nonlinear Gaussian Noise Model . . 54

v

3.3 Capacity Comparison: SMF - MMF . . . . . . . . . . . . . . . . . . . . . 61

3.4 Capacity of a Standard MMF . . . . . . . . . . . . . . . . . . . . . . . . . 64

4 Few-Mode Fiber Systems 714.1 DMD Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2 Single Wavelength MDM Transmission . . . . . . . . . . . . . . . . . . . 73

4.2.1 Influence of the DMD Management on the Nonlinear Impairments . 75

4.3 WDM-MDM Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3.1 Impact of the WDM Channel-Granularity on the Intermodal Non-

linear Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5 Summary and Outlook 835.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

A Phase Matching 85A.1 Intramodal Phase-Matching . . . . . . . . . . . . . . . . . . . . . . . . . . 85

A.2 Intermodal Phase-Matching: Two-Mode Process . . . . . . . . . . . . . . . 85

A.3 Intermodal Phase-Matching: Generalized . . . . . . . . . . . . . . . . . . 86

B Derivation of the NL Gaussian Noise Model 87B.1 Single-Mode NL Gaussian Noise Model . . . . . . . . . . . . . . . . . . . 87

B.2 Multi-Mode NL Gaussian Noise Model . . . . . . . . . . . . . . . . . . . 89

B.3 Two-Mode NL Gaussian Noise Model . . . . . . . . . . . . . . . . . . . . 90

B.4 Simplified MM NL Gaussian Noise Model . . . . . . . . . . . . . . . . . . 93

C Split-Step Fourier Method 95

D Parameter Lists 97

E Calculation of the ROSNR from the NL GN Model 103

vi

Chapter 1

Introduction

1.1 Motivation

Optical communication is the key technology behind the endless growth of the internet

that connects the modern globalized world. Today, it is taken for granted that audio and

video communication between parties that are scattered all around the world are possible at

any given time and at almost no cost. Video platforms like Netflix [1] and Youtube [2] are

just two examples for services that allow end-users to consume an enormous diversity of

video content at any given time. Companies like Google [3] offer more and more services

that are available exclusively online with the very abstract definition of the cloud [4] as

computational resource and data storage behind it. Figure 1.1 shows the development of

the average monthly data traffic in northern America from 2003 until 2011 and the future

predictions until 2018 [5, 6, 7, 8]. The past development as well as the predictions show an

exponential growth that needs to be satisfied by each component of the telecommunication

infrastructure, including optical backbone transport networks.

1.2 History of Optical Communication

Two innovations, each leading to a Nobel Prize, enabled the success of fiber optic commu-

nication technologies. The invention of the double hetero structure laser [11] in the 1960s as

a coherent light source and the development of fibers with low attenuation [12] made it pos-

sible to optically transmit large amounts of data over long distances. The high frequency of

light as a signal carrier further offers the potential for a very large transmission bandwidth.

Figure 1.2 shows the capacity evolution of fiber optic communication systems, while each

mark symbolizes a new record that was reported in the corresponding year [10]. Until

the early 1990s, single-channel, direct detection systems were the state of the art, where

1

2 CHAPTER 1. INTRODUCTION

Figure 1.1: Traffic development in the United States of America. Exponential growth has been observed in the

past and is expected for the future [5, 6, 7, 8, 9, 10]

digital information (0s and 1s) were transmitted by modulating the amplitude of a laser

source. Capacity increases were mostly facilitated by improving the fiber quality and the

electronics for transmitters and receivers. For long-haul transmission links, a complete sig-

nal regeneration was necessary after a certain link length. Thus, the signal was received and

again generated, requiring a complete optical-electro-optical (OEO) conversion, consum-

ing large amounts of power. The invention of the Erbium Doped Fiber Amplifier (EDFA)

in the late 1980s [13] made this OEO conversion obsolete by performing an optical am-

plification to compensate for the attenuation that a signal experiences during transmission.

Furthermore, EDFAs make it feasible to use a larger part of the available frequency band for

long-haul networks, as they can simultaneously amplify different data streams at different

carrier wavelengths (Wavelength Division Multiplexing, WDM). This multiplexing tech-

nique enables the transmission of larger amounts of data while at the same time keeping the

speed requirements on the electronics of each transmitter and receiver constant. Coherent

transmission techniques in combination with high-speed digital signal processing matured

in the early 2000s [14], leading to an increase of the spectral efficiency by not only mod-

ulating the amplitude of a laser source but also the phase, enabling so-called higher order

modulation formats [15]. Together with WDM and the exploitation of the two polarizations

of a light source [16], the continuous improvement of coherent technologies allows data

rates of more than 100 Tb/s that are available in current transmission systems [17].

1.3. MODE-DIVISION MULTIPLEXING 3

1

10

100

1000

10000

100000

1970 1980 1990 2000 2010 2020

Capa

city

(Gb/

s)

Year

NO WDM WDM

Coherent Modulation /Polarization

EDFA/WDM

Direct-detection

Figure 1.2: Capacity development of optical long-haul transmission systems. The overall exponential growth

has been feasible by several breakthrough inventions, including Erbium Doped Amplifiers (EDFA), Wavelength

Division Multiplexing (WDM), Polarization Division Multiplexing (PDM) and coherent modulation formats.

For more details see [9, 10]

Several studies have been proposed in order to estimate the maximum capacity of single-

mode transmission systems [18, 19, 10], concluding that the physical capacity limits of the

current single-mode fiber infrastructure will be reached soon if the exponential growth of

the world wide data traffic continues.

1.3 Mode-Division Multiplexing

Figure 1.3 summarizes the five modulation dimensions for an electromagnetic wave that

propagates in an optical fiber, as proposed in [20]. The dimension time that is connected

to the maximum symbol rate is limited by the speed of the optp-electronics, used in the

transmitter and receiver of the transmission link. Quadrature is the dimension that uses

amplitude and phase (I and Q component) of the electromagnetic wave. The limits of this

dimension are defined by the signal-to noise ratio (SNR) which is available at the receiver

and is therefore defined by the length of the transmission link and the quality of the used

components. The frequency dimension is defined by the usable bandwidth in a fiber and

the amplifiers. This dimension is not yet fully exploited as most transmission system only

use the C-Band [21], while more frequency bands become available when applying different

amplifier types [22], allowing some potential capacity increase. Polarization is already fully

used, as an electromagnetic wave has only two orthogonal polarizations that can possibly


Modulation Dimensions

Time Frequency

Polarization

Space

Quadrature

Figure 1.3: Modulation dimensions of an electromagnetic wave that travels in an optical fiber [20]

be modulated.

Space is the only dimension out of these five which has not been exploited until now

[20]. Space-division multiplexing is a very abstract term that can refer to different appli-

cations. For instance, if one shared laser is used to generate the carrier signals for several

transmitters, each transmitting a signal into a different Single-Mode Fiber (SMF), this can

already be considered as space-division multiplexing. Similar combinations of components

could be amplifiers that share a pump laser or receivers sharing a local oscillator.

A consequent use of space-division multiplexing (SDM) further needs a shared transmis-

sion medium. Figure 1.4 shows three examples of such media. 1.4 (a) depicts a single-mode

fiber bundle that can be considered as one transmission medium for SDM. When consider-

ing a fiber bundle for SDM transmission, shared components inside transmitters, amplifiers

or receivers can be introduced and the spatial separation of the channels prevents coupling.

However, a complete integration of transmitters, amplifiers or receivers is not feasible, and

the per-fiber capacity is not increased compared to single-mode fibers. Therefore, the costs

and also the energy consumption per bit of SDM transmission systems with fiber bundles

might not be significantly lower than those of parallel single-mode systems.

Figure 1.4 (b) shows a multi-core fiber (MCF) with seven fiber cores, each supporting

one fiber mode. This type allows a capacity increase proportional to the number of cores.

MCFs have been demonstrated with up to 19 cores [23], while it might be difficult to pro-

duce reliable fibers with a much higher core number due to physical constraints on the

maximum fiber diameter. MCFs can be designed with strongly and weakly coupled cores.

Weak coupling has the advantage that each fiber core can be considered as an independent

1.3. MODE-DIVISION MULTIPLEXING 5

(a) (b) (c)

Figure 1.4: Suitable fibers for space-division multiplexing: (a) single-mode fiber bundle (b) multi-core fiber (c)

multi-mode fiber

transmission channel. Strong coupling further requires mechanisms to equalize for the cou-

pling. However, it has been shown that strong coupling between the cores of an MCF can

lead to an enhanced performance due to lower nonlinear signal degradation [24].

Figure 1.4 (c) shows a multi-mode fiber (MMF) as the third fiber type for SDM. Ideal

MMFs allow multiple independent transmission paths. These paths are commonly called

eigenmodes or in short modes. These eigenmodes are the solutions of a wave-equation, the

number of solutions is therefore discrete and countable. Common multi-mode fibers guide

around 50 spatial modes per polarization, but it is possible to design fibers that support sev-

eral hundred or only as little as three spatial modes. Fibers that support only a few modes

are called few-mode fibers. They gained a lot of interest in the last years as they can be

used to proof certain concepts in SDM and are relatively easy to handle. Since a practical

multi-mode fiber is not ideal, production tolerances, macro- or micro bends and thermal ef-

fects can lead to linear coupling effects between the eigenmodes of a fiber. In general, each

of the modes of a MMF has a different propagation constant, leading to different group

velocities of the signals that travel in each of the fibers’ modes. In contrast to fiber bundles

or Multi-core fibers, the electromagnetic field distributions of the MMF’s eigenmodes spa-

tially overlap each other. This leads to a potential nonlinear interaction between different

fiber modes, so-called intermodal nonlinear interaction. These nonlinear interactions are

the main focus of this work. Since the investigations in this thesis are limited to multi-

mode fibers, the term mode-division multiplexing (MDM) is used in order to emphasize the

transmission of data over the eigenmodes of a multi-mode fiber.

Figure 1.5 shows a generalized block diagram of an SDM transmission link. A transmit-


Trans-mitter

Transmission Medium

Receiver/Digital Signal

Processing

Introduction of Spatial Diversity

SpatialFiltering

Amplifi-cation

Figure 1.5: Block Diagram of an optical transmission system with space-division multiplexing

ter generates one or more independent optical signals per spatial path, potentially including

multiple wavelength channels in a WDM scheme. These signals are then multiplexed onto

different wavelengths and different spatial paths of either the fibers of a bundle, the cores of

a MCF or the modes of a MMF.

Several different methods have been proposed as spatial multiplexers. Multiplexing into

a single-mode fiber bundle is straight forward by standard splicing procedures. In con-

trast, intense research has been done in order to optimize multiplexing into MCF and MMF.

For MCF, tapered cladding structures have been proposed [25], where for an n-core MCF,

n single-mode fibers are adiabatically tapered together until the core distances between the

single-mode fiber cores fit the core arrangement in the MCF. Lens-based as well as 3D wave

guides that are inscripted into special polymer structures have been proposed [25]. The first

approaches to excite individual modes of a MMF were realized by using phase-plate-based

devices to convert the beams of a single-mode fiber into field distributions that are similar

to the eigenmodes of the multi-mode fiber [26]. These couplers are rather lossy, insertion

losses in the range of 10 dB have been reported [27]. Spot-based mode multiplexers [28],

including photonic lanterns [29], have been presented almost being loss-less, while excit-

ing linear combinations of the fiber modes. Integrated mode-multiplexers on silicon basis

have been successfully demonstrated with relatively large loss [30]. A low loss multi-mode

grating solution, based on silicon optical techniques has further been designed [31]. Several

amplification strategies have been demonstrated for multi-core [32] and multi-mode [33, 34]

transmission systems. Demultiplexing can be performed with the same components that are

used for multiplexing.

When using a transmission medium with strong modal coupling, signal processing, usu-

ally carried out as Digital Signal Processing (DSP), is required to equalize this coupling

by applying Multiple-Input-Multiple-Output (MIMO) algorithms to the received signals

[35, 27]. These algorithms are well known from wireless communication [36]. Their com-

plexity is highly dependent on the difference of the group velocity between different spatial

paths, leading to constraints in the maximum transmission distance [37].

1.4. OBJECTIVES OF THIS WORK 7

Input Power (a.u.)

Spe

ctra

l Effi

cien

cy (a

.u.) Linear Channel

Nonlinear Channel

Figure 1.6: Schematic representation of the system performance, expressed as the spectral efficiency, at a

given input power for a linear and a nonlinear channel: while the system performance in a linear channel can be

arbitrarily increased with an increased input power, a nonlinear channel will have a maximum spectral efficiency

at a certain input power for a given system design [18, 19].

1.4 Objectives of this Work

Propagation effects of signals in fibers can generally be separated into linear and nonlinear

effects. Linear transmission effects in multi-mode fibers have been intensively studied since

more than 40 years [38, 39, 40]. However, the application area of multi-mode fibers is

traditionally the short-haul regime [21, pp. 214]. Addressing each fiber mode with different

signals in long-haul transmission systems is a new domain and might require the adaption

of already existing theories for the linear signal propagation.

Kerr-effect-based nonlinear signal distortions have been studied intensively in single-

mode fiber based transmission systems and have been identified as the major limitation for

the system performance that can be expressed e.g. as the spectral efficiency [18, 19, 21].

Figure 1.6 shows a schematic representation of the spectral efficiency at a given input power

for a linear and a nonlinear channel [19]. In linear transmission systems, the major limi-

tation is caused by additive noise [18, 41]. Consequently, the spectral efficiency can be

raised by increasing the power of the transmitted signal. Optical fibers are in general a non-

linear transmission channel [18, 19]. In such channels, the spectral efficiency can only be

raised with increasing the input power until a certain threshold is reached, while a further

power-increase lowers the system performance as nonlinear signal distortions become dom-

inant. Thus, it is necessary to determine the optimum input power for a maximum system

performance.

Kerr-nonlinearities were given comparatively little attention in multi-mode fibers, as


they practically pose only a performance limitation in long-haul transmission systems that

were so far designed with single-mode fibers [21, pp. 913]. Four-wave mixing between

frequency components that travel in different modes of a multi-mode fiber has been shown

for very short fiber length [42]. Recently, the observation of four-wave mixing and cross-

phase modulation were also reported in few-mode fibers of several km length [43, 44].

As few-mode fiber-based transmission systems become more and more attractive for long-

haul transmission, numerical models for the signal-propagation in such system have been

proposed [24, 45, 46, 47, 48, 49].

The core focus of this thesis is to offer a thorough analysis of the nonlinear signal prop-

agation in few- and multi-mode fibers and its impact on the performance of mode-division

multiplexed transmission systems. The goal is thereby divided into two parts:

• Identification of Kerr-effect based capacity limitations in multi-mode fiber basedMDM systems

• Analysis of nonlinear propagation effects that arise from the implementation of mode-division multiplexing compared to single-mode fiber systems, especially the interplay

between linear and nonlinear propagation effects.

This thesis is divided into five chapters. This introduction is followed by a chapter which

provides the theoretical background to the theories that are used in the further work. The

third chapter is focused on an analytical approach to estimate the transmission capacity of

idealized mode-division multiplexed transmission links. The fourth chapter concentrates on

the numerical simulation of complete optical MDM systems with impairments that might

occur during deployment in realistic scenarios. The final chapter summarizes the main

findings of this thesis and offers an outlook on future work.

Chapter 2

Wave Propagation in Optical Fibers

2.1 Linear Propagation in Optical Fibers

2.1.1 Single- and Multi-Mode Fibers

Standard single-mode and step-index multi-mode fibers are designed as a radial symmetrical

cylinder containing two concentrically arranged regions with different indexes of refraction,

called core and cladding, as shown in figure 2.1 (a) and (b). The core’s index of refraction

n1 is slightly larger than the cladding’s index of refraction n2. An electromagnetic wave

can be guided by this structure due to the total internal reflection inside the core, which

is a result of the different indexes of refraction of the core and the cladding. The coating

acts as a protection against scratches and water ingression and it insures additional physical

stability. The third fiber type, shown in figure 2.1 (c) shows a graded-index multi-mode

fiber. The refractive-index profile is different from the former two fiber types in view of the

fact that it does not only have two distinct regions with constant indexes of refraction, but

the index-profile inside the fiber core follows a certain function, while it is constant inside

the cladding. This has certain advantages as it will be shown in the following sections.

2.1.2 Material Properties

Optical fibers for long-haul applications are made out of glass, chemically known as silicon

dioxide (SiO2). Silicon dioxide is, as most solids, a dispersive material, meaning that dif-

ferent spectral components, which propagate inside the material, travel at different speeds.

Starting point for the following analysis is an x-polarized plane wave with definition of the

electric field as:

Ex(t) = Re(E0ej(ωt−βz)) (2.1)

9

10 CHAPTER 2. WAVE PROPAGATION IN OPTICAL FIBERS

Radius

n1

n2

Radius

n1

n2

Radius

n (r)

n2

(a) (b) (c)Core Cladding Coating

Figure 2.1: Cross section and refractive index profile of a (a) single-mode fiber, (b) step-index multi-mode fiber

and (c) graded-index multi-mode fiber.

with E0 describing the fields amplitude and initial phase, ω the angular frequency of the

wave and β the propagation constant. Planes of constant phase of such a monochromatic

wave propagate at the speed of the phase velocity:

vph(ω) =k0c

β=

c

n(ω)(2.2)

with n being the refractive index of the material at the angular frequency ω and c is the

speed of light in vacuum [50]. Pulses are composed of several different spectral compo-

nents. The information that is carried by a pulse travels with the speed of the group-velocity

[51, p. 38]:

vgr(ω) =

(dβ

dω

)−1=

cd(ωn(ω))

dω

=c

N(2.3)

with N being the group index [51, pp. 38]:

N =d(ωn(ω))

dω= n(ω) + ω

dn(ω)

dω= n(λ)− λdn(λ)

dλ(2.4)

The group-velocity generally changes with the frequency since the refractive index of

the material is a function of the frequency. To describe this effect, it is customary to define

the material dispersion coefficient DM [50]:

DM = −λ

c

d2n(λ)

dλ2(2.5)

The previous derivation (2.2) - (2.5) requires a precise description of the relation between

the index of refraction of the material with either the angular frequency ω or the wavelength

2.1. LINEAR PROPAGATION IN OPTICAL FIBERS 11

λ. The Sellmeier equation is a very accurate approximation of the index of refraction [51,

pp. 39]:

n2(ω)− 1 =∑i

Biω2i − ω2

(2.6)

Each Bi and its corresponding angular frequency ωi originates from an absorption reso-

nance of different strength and origin.

As waves, propagating in media, generally experience attenuation, it is customary to

define an imaginary part of the index of refraction [51, pp. 55]:

n = n′ − jn′′ (2.7)

with the field attenuation parameter

α = k0n′′ (2.8)

it is possible to describe the magnitude of a wave after transmitting through a fiber of

length L as [51, pp. 55]:

E(L) = E0e−αL (2.9)

where E0 is the initial amplitude at the beginning of the fiber. Attenuation mainly orig-

inates from material absorption and Rayleigh scattering in the wavelength region that is

considered in this work [51, pp. 55]. The attenuation is usually specified in the logarithmic

unit dB/km, with the connection between the linear and the logarithmic unit as:

αdB = 8.686α (2.10)

αdB is minimal in the order of about 0.2dB/km at a wavelength of 1.55µm. Thus,

1.55µm is the preferred wavelength for long-haul transmission systems.

2.1.3 Fiber Modes in Step-Index Fibers

This paragraph is intended to provide a deeper understanding of the term fiber mode by

deriving field distributions and propagation constants for single- and multi-mode step-index

fibers. A more detailed derivation can be found in many books, e.g. [21, pp. 81]. The start-

ing point is a harmonic wave that propagates along the z-axis of an optical fiber. Neglecting

the time-dependance ejωt, the electric field of such a wave can be described as:

E⃗(x, y, z) = E⃗(x, y)exp(−jβz) (2.11)


with the propagation constant β in the range:

k0n2 < β < k0n1 (2.12)

The electrical field E⃗(x, y, z) fulfills the following wave equation in the core and the

cladding:

∆Ex,y + k20n

2iEx,y = 0 (2.13)

Combining eq. (2.11) and eq. (2.13) leads to the following equation:

∆tEx,y + (k20n

2i − β2)Ex,y = 0 (2.14)

with the transverse Laplace operator ∆t = ∂2

∂x2+ ∂

2

∂y2

Optical fibers for communications are usually weakly guiding fibers, meaning that the

core and cladding indexes of refraction are very similar:

n1 − n2n1

≪ 1 (2.15)

Neither the wave equation (2.14) nor the boundary conditions at the core-cladding in-

terface couple the x- and y components of the electric field [21, pp. 81]. This justifies

the approach of linearly polarized waves for the electric field. The x-polarized field can

exemplary be written in cylindrical coordinates as:

Ex = ψ(r, ϕ)exp(−jβz) (2.16)

Considering a step-index fiber with the core-cladding boundary at r = a, eq. (2.17)

leads to:

ψ(r, ϕ) =

⎧⎨⎩ψ1(r, ϕ) if r ≤ aψ2(r, ϕ) if r > a (2.17)Equation (2.14) can then be written as:

∆tψi + (k20n

2i − β2)ψi = 0 (2.18)

for i = 1, 2 and the boundary conditions at the core-cladding interface [21, pp. 85]:

ψ1|r=a = ψ2|r=a (2.19)

∂ψ1∂r

|r=a =∂ψ2∂r

|r=a (2.20)


After the following normalizations:

Numerical Aperture: NA =√(n21 − n22) (2.21)

Normalized Frequency: V = k0a√(n21 − n22) = k0aNA (2.22)

Normalized propagation constant: B =β2

k20− n22

n21 − n22≈

βk0

− n2n1 − n2

(2.23)

Core parameter: u = V√1−B = a

√k2on

21 − β2 (2.24)

Cladding parameter: v = V√B = a

√β2 − k2on22 (2.25)

eq. (2.18) can be written as:

a2∆tψ1 + u2ψ1 = 0 for r ≤ a (2.26)

a2∆tψ2 − v2ψ2 = 0 for r > a (2.27)

(2.28)

with the solutions:

ψ1(r, ϕ) = A1Jl(ru

a)

{cos(lϕ)

sin(lϕ)

}(2.29)

ψ2(r, ϕ) = A2Kl(rv

a)

{cos(lϕ)

sin(lϕ)

}(2.30)

For each circumferential order l > 0, eq. (2.29) and (2.30) have two orthogonal solutions

due to the cos and sin terms.

Equation (2.29) and (2.30) combined with the boundary conditions (2.19) and (2.20)

lead to the following characteristic equation for the deduction of the propagation constant

β and its normalized counterpart B, since u and v only contain β as an unknown variable

[21, pp. 85]:

−uJl+1(u)Jl(u)

+vKl+1(v)

Kl(v)= 0 (2.31)

with

V 2 = u2 + v2 (2.32)


0 2 4 6 8-30

-20

-10

0

10

20

30

u

Cha

ract

eris

tic F

unct

ion

s(u)

l = 0l = 1l = 2

Figure 2.2: Characteristic function (2.31) as a function of the core parameter u for a fiber with V = 8.2.

Different plots represent different circumferential orders l. Each zero corresponds to one guided wave. The

radial order p of the guided wave increases with every zero point.

Figure 2.2 shows a plot of the characteristic function:

s(u) =−uJl+1(u)Jl(u)

+vKl+1(v)

Kl(v)(2.33)

for an exemplary step-index fiber with core-radius of a = 10µm and numerical aperture

NA = 0.2 at wavelength λ = 1.55µm (V = 8.2) as a function of the core-parameter u.

Each graph shows a different circumferential order l. Each zero of eq. (2.31) leads to up

to four guided waves, as it will be discussed in the next section. For each circumferential

order l, a discrete number of zeros can be counted, leading to a radial order p, while counting

starts at the lowest zero.

LP-Mode Notation

Each fiber mode can be uniquely distinguished with the LP notation, where LP stands for

linearly polarized. The first index represents the circumferential order and the second index

the radial order. a and b stand for the orientation of the mode, that is defined by either

the cos or the sin dependance of the field that exist for l > 0 in eq. (2.29) and (2.30).

For l = 0, only one orientation exists. A spatial mode is uniquely identified with this

LP-denomination, e.g. LP01 or LP11a or LP11b. Furthermore, each spatial mode exists

in two orthogonal polarizations, since eq. (2.14) has two independent solutions for both

Ex and Ey. Since the LP-denomination is not consistently defined in the literature, table


LP Spatial and Polarization Modes LP Spatial Mode LP mode

LP01xLP01

LP01LP01yLP11ax

LP11aLP11

LP11ayLP11bx

LP11bLP11by

Table 2.1: Concept of the LP denomination as it is used in this work for six exemplary LP spatial and polariza-

tion modes.

(a) (b) (c)

Figure 2.3: Schematic representation of the power distributions of three exemplary linear polarized fiber modes,

according to eq. (2.17): (a) the fundamental LP01 mode, (b) a LP11 mode and a LP24 mode. The color

represents intensity, while red indicates the highest intensity. Each graph is scaled to its maximum intensity.

2.1 summarizes the denominations, used in this work, exemplary for six different (spatial /

polarization) modes, similar to [27].

Fibers where only one zero can be found in the characteristic equation (2.33) only sup-

port the LP01 mode and are called single-mode fibers, while this mode still exists in the two

polarizations x and y.

The total number of guided modes for a step-index fiber can be approximated with [38,

pp. 303]:

M =V 2

2(2.34)

Where M includes all spatial and polarization modes. Figure 2.3 (a) - (c) show the

intensity distributions for three exemplary fiber modes that can be calculated with eq. (2.29)

and (2.30).


2.1.4 Dispersion

The group-delay is the time that a signal takes to propagate through a fiber section of unit

distance. It can be calculated as the inverse of the group-velocity, defined in eq. (2.3), for

each mode with the index lp:

τlp =dβlpdω

(2.35)

Combined with eq. (2.23), this leads to:

τlp =d(k0(Blp(n1 − n2) + n2))

dω(2.36)

To further simplify eq. (2.36), it is assumed that n1 and n2 have the same dependance

on ω (no profile dispersion), leading to:

τlp = (n1 − n2)d(k0Blp)

dω+d(k0n2)

dω=n1 − n2

c

d(V Blp)

dV+

1

cN2 (2.37)

Equation (2.37) contains the derivative of the normalized propagation constant Blp with

respect to the normalized frequency V . Figure 2.4 shows the normalized propagation con-

stant Blp as a function of the normalized frequency V for several fiber modes. It can be

seen that each mode’s slope is different at a given V , leading to a different derivative in

eq. (2.37) and therefore in general to a different group-velocity. This phenomenon is called

modal dispersion and causes an arrival time difference of pulses that travel in different fiber

modes and thereby limiting the maximal pulse-repetition rate. Consequently, single-mode

fibers were so far the first choice for long-haul transmission, as they only support one fiber

mode and thus do not exhibit modal dispersion.

Besides modal dispersion, each spectral component inside a fiber mode travels at a dif-

ferent phase velocity. According to eq. (2.3), the overall chromatic dispersion can be cal-

culated as the derivative of the group-delay with respect to the wavelength λ:

dτlpdλ

=(n1 − n2)

cλ

d2(V Blp)

dV 2 Waveguide Dispersion

− λc

d2n2dλ2

Material Dispersion

(2.38)

Equation (2.37) shows that the total chromatic dispersion for each fiber mode is com-

posed of the material dispersion that was discussed in the previous section and the so-called

waveguide dispersion [51, pp. 39]. While the material dispersion is a given material prop-

erty, it is possible to alter the waveguide dispersion by changing the shape of the fiber’s

refractive-index profile. For single-mode fibers, it is possible to design fibers with negative

chromatic dispersion in order to compensate the overall dispersion of a transmission link

[21, pp. 98].


1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

V

B

01

02

11

51

21

Figure 2.4: Normalized propagation constant Blp vs. normalized frequency V for a weakly guiding step-index

fiber. The numerical aperture NA = 0.1 and the wavelength λ = 1.55µm are kept constant while the core

radius a is varied from 4 to 20 µm in order to vary the normalized frequency V . The numbers specify several

exemplary LP modes.

2.1.5 Fiber Modes in Graded-Index Fibers

In order to minimize the group-delay differences between signals that travel in different

fiber modes, it is beneficial to use a non step-index refractive-index profile inside the core

[21, pp. 85]. In general, such an index profile can then be described as:

n2(r) =

⎧⎨⎩n21(1− 2∆f(r)) for r ≤ an22 for r > a (2.39)with the profile-function f(r) and ∆ = n

21−n222n21

≈ n1−n2n1 . A very prominent cate-gory of graded-index fibers are power-law fibers where the profile function is defined as

f(r) =(ra

)g. The calculation of the propagation constants and eigenmodes for graded-index fibers is considerably more complicated than it is for step-index fibers. Numerical

solving strategies are generally needed to determine propagation constants and field distri-

butions of graded-index fibers.

However, [38, pp. 471 ff.] gives an approximate solution for both, the field distributions

and the propagation constants for the special case of the infinite parabolic-index fiber that

is valid when neglecting profile dispersion (d∆dλ ≈ 0) [38, pp. 471 ff.]. This fiber is moreof a theoretical concept, where the radius of the fiber core extents to infinity with a profile-

function1 f(r) =(ra

)2. The propagation constant for each mode can then be given as [38,1see also figure 2.9 (a)


10 20 30 405.87

5.88

5.89

5.9

5.91x 106

Mode Number

Pro

paga

tion

Con

stan

t ( β

0) (1

/m)

Figure 2.5: Propagation constants of 42 fiber modes (including orientation and polarization) of an exemplary

infinite parabolic-index fiber, calculated with eq. (2.40). The modes are sorted by their propagation constants

and form groups of modes with equal propagation constants. The fiber parameters can be found in table: 2.4,

while the labeling is explained in table 2.2.

pp. 471 ff.]:

βlp = n1k0

(1−

√2∆

n1k0a(4p+ 2l − 2)

)1/2(2.40)

where ∆, n1 and a according to eq. (2.39). l and p are the circumferential and radial

orders of the corresponding fiber mode. Equation (2.40) suggests that it is possible to find

degenerate modes with equal propagation constants:

4p+ 2l − 2 = m,with m = 2, 4, 6, ... (2.41)

where m is the principal mode number. Modes with the same principal mode number

m, and therefore equal propagation constants, form mode-groups. Figure 2.5 shows the

propagation constant of 42 spatial and polarization modes of an exemplary multi-mode

fiber with infinite parabolic-index profile, calculated with eq. (2.40). The fiber parameters

can be found in table 2.4 and the labeling is explained in table 2.2. The modes are sorted

by their propagation constants, with the fundamental mode being mode number one. It can

be observed that the mode-groups form a structure where each higher mode-group has two

additional members.

Equation (2.40) can also be used to calculate the group-velocity for each mode, by cal-

culating the propagation constant for several different frequencies and taking the derivative,


Mode Number m Spat. / Pol. Mode Mode Number m Spat. / Pol. Mode

1 2 LP01x 22 10 LP41ay2 2 LP01y 23 10 LP41bx3 4 LP11ax 24 10 LP41by4 4 LP11ay 25 10 LP22ax5 4 LP11bx 26 10 LP22ay6 4 LP11by 27 10 LP22bx7 6 LP21ax 28 10 LP22by8 6 LP21ay 29 10 LP03x9 6 LP21bx 30 10 LP03y10 6 LP21by 31 12 LP51ax11 6 LP02x 32 12 LP51ay12 6 LP02y 33 12 LP51bx13 8 LP31ax 34 12 LP51by14 8 LP31ay 35 12 LP32ax15 8 LP31bx 36 12 LP32ay16 8 LP31by 37 12 LP32bx17 8 LP12ax 38 12 LP32by18 8 LP12ay 39 12 LP13ax19 8 LP12bx 40 12 LP13ay20 8 LP12by 41 12 LP13bx21 10 LP41ax 42 12 LP13by

Table 2.2: Labeling of the spatial and polarization modes. They are sorted by their propagation constant when

calculated with eq. (2.40).


according to eq. (2.35). Since the index-difference ∆ is kept constant at each frequency,

this is only valid when neglecting profile dispersion [38, pp. 471].

The field distribution of each mode in an infinite parabolic-index fiber can be approx-

imated by Laguerre-Gaussian modes as [38, pp. 468][52]. The x-polarized electric field

component of a spatial mode with order l and p can then be written as:

Ex,l,p(r, ϕ) = C′lp(r

ξ)lL

(l)p−1

(r2

ξ2

)exp (

−r2

2ξ2)

⎧⎨⎩cos (lϕ)sin (lϕ) , (2.42)with ξ being the spot size parameter:

ξ =

√a

kn1√2∆

(2.43)

and the normalization [52]

C ′lp =

√elπ

√(p− 1)!

(l + (p− 1))!, (2.44)

with

el =

⎧⎨⎩1 for l = 02 for l ̸= 0 . (2.45)L(l)p−1 is a generalized Laguerre polynomial of orders p − 1 and l. The two orthogonal

orientations in eq. (2.42), distinguished by cos(lϕ) and sin(lϕ) are denoted with a and b,

respectively, in order to use the same LP-mode notation as for the step-index fiber. The

y-polarized field can be calculated similarly.

The number of guided modes in a parabolic-index fiber can be approximated as [38, pp.

456]:

M =V 2

4(2.46)

when defining V as in (2.25). By comparing eq. (2.46) and eq. (2.34), it can be seen that

a step-index fiber and a parabolic-index fiber guide the same number of modes if they have

the following V-parameter ratio:

VparabolicVstep

=√2 (2.47)


0 10 20 30 4010−2

10−1

100

101

102

Mode Number

Diff

eren

tial M

ode

Del

ay (n

s/km

)Step IndexInfinite Parabolic Index

Figure 2.6: Differential Mode Delay (DMD), as defined in eq. (2.48) for a step-index and an infinite parabolic-

index fiber that support roughly the same number of modes. The mode labeling is according to table: 2.2, the

fiber parameters for the step-index fiber as in the previous section, the parameters for the infinite parabolic-

index fiber according to table 2.4. Since the first two modes of both fibers have both zero DMD, they are not

shown on the logarithmic scale. Values are calculated for the scalar wave equation.

Advantage of the Graded-Index Profile

Figure 2.6 shows the Differential Mode-Delay per length for a step-index fiber (V = 8.8)

and an infinite parabolic-index fiber (V = 12.4) that support a similar number of modes.

The DMD is defined as the group-delay difference between a certain mode and the funda-

mental mode of the fiber as:

τ̃lp = |τlp − τ01| (2.48)

The group-delay for the step-index fiber is calculated by numerical evaluation of eq.

(2.37) and for the infinite parabolic-index fiber by numerical differentiation of eq. (2.40).

The first two modes (LP01x and LP01y) of both fibers have both zero DMD, hence they are

not shown on the logarithmic scale.

It can be observed that the maximum DMD is about two orders of magnitude smaller

for the infinite parabolic-index fiber compared to the step-index fiber. The advantage of

low DMD can be recognized when considering e.g. a classical MMF-based transmission

system, where all modes are excited with one signal. The DMD is then the main limiting

factor for the data rates [21, pp. 81], since the different group-velocities of the modes lead

to a temporal broadening of the received signal.


2.1.6 Numerical Calculation of Fiber Modes

By using a vector finite difference mode solver [53], it is possible to calculate the eigen-

modes of fibers with arbitrary refractive-index profiles. The mode solver calculates the true

vector modes, being solutions of the vector-wave equation [54, pp. 157]. The vector-wave

equation contains an additional term that is proportional to ∇n2 [54, pp. 157] that is usu-ally neglected, leading to the scalar wave equation (2.13). The solutions to the vector-wave

equation are denominated TE, TM , HE and EH modes, while HE and EH modes ap-

pear in a two-fold degeneracy as even and odd modes [54, pp. 154]. With the help of the

superposition principle for modes with equal propagation constants, it is possible to trans-

late the vector modes into LP-modes under the weakly guiding approximation. Figure 2.7

shows the electric fields of two exemplary modes, a HE21 and a TE01 mode, both being

non-linearly polarized solutions of the vector wave equation. The LP11 mode is constructed

by adding up the electric fields of the two vector modes. The number of modes that can be

found by the numerical calculation is equal to the number of solutions for the scalar equa-

tion. A general calculation rule for the superposition of vector modes to form LP modes is

given in [54, pp. 157].

Vector Mode LP Spatial / Polarization Mode Circ. Order l

HE1p (even) LP0px

l=

0HE1p (odd) LP0py

HE2p (odd) + TE0p LP1paxl=

1HE2p (odd) - TE0p LP1pbyHE2p (even) + TM0p LP1pbxHE2p (even) - TM0p LP1pay

HE(l+1)p (odd) + EH(l−1)p (odd) LPlpax

l≥

2HE(l+1)p (odd) - EH(l−1)p (odd) LPlpbyHE(l+1)p (even) + EH(l−1)p (even) LPlpbxHE(l+1)p (even) - EH(l−1)p (even) LPlpay

Table 2.3: Superposition of true vector modes to form linearly polarized fiber modes [54, pp. 155].


HE21 TE01LP11

Figure 2.7: Two Vector modes that have been calculated with the Vector Finite Difference Mode Solver [53]

and their superposition to form a LP-mode. The circle represents the core-cladding interface.


2.1.7 Accuracy of the Infinite Parabolic-Index Assumption

The infinite parabolic-index assumption can be a good help since it offers analytical solu-

tions for the fiber modes. However, it is of interest to assess the error that it implies when

compared to real fiber designs, as they are shown in figure 2.9.

The DMD has a large influence on the nonlinear signal propagation in multi-mode fibers,

as it will be shown later in this work. Several studies have been performed to investigate

the accuracy of the infinite parabolic-index assumption regarding the DMD. In [55, pp. 38],

the DMDs for a truncated parabolic-index fiber have been compared with the analytically

calculated values for an infinite parabolic-index fiber. A good agreement was observed for

those modes that propagate in the center region of the fiber, while those that propagate in

the core-cladding interface showed some deviations. This result indicates that numerical so-

lutions for an infinite parabolic-index profile should show a good agreement with all modes

of the analytical solutions for this profile. The numerical solutions in [55] were based on a

scalar wave equation.

The numerical mode solver that was used in this work, however, calculates the solutions

to the vector wave equation. Figure 2.8 shows the DMD of the infinite parabolic-index fiber

when calculated with the numerical mode solver. The vector modes are labeled according

to table D.1. The analytical solutions consider the scalar wave equation (eq. (2.40), and

eq. (2.35)), fiber parameters are shown in table 2.4. For the infinite parabolic index fiber,

it can be observed that the total DMD spread is about four-times larger when numerically

calculated, compared to the analytical values that were obtained from the scalar wave equa-

tion. This result has further been verified with a different mode solver [56] and might be

an indication that the negligence of the ∇n2 term in the vector wave equation might have alarger impact than considered so far. This effect was briefly discussed in [57], however, as

it is not the core focus of this thesis, it will be left as an open point that should be subject to

future investigations.

The parabolic-index profile is not the optimum profile at λ = 1.55µm in terms of min-

imal DMD, especially when considering profile dispersion [58, 59, 60]. But even withouth

profile dispersion, it was shown[61] that for λ = 1.55µm, a profile exponent of g = 1.95

leads to a minimal DMD, even without considering profile dispersion, in a fiber that supports

a total of 20 modes (including polarization). The red graph in figure 2.8 shows the DMD

for an infinite power-law fiber with exponent g = 1.95 when calculated with the numerical

vector mode solver. The maximum values match comparably good with the analytically

obtained values for the infinite parabolic-index. As the DMD is used later in this thesis as

an input parameter for the assessment of nonlinear effects, its value will be calculated from

the analytical solutions for the infinite parabolic-index fiber, as these values are comparable


10 20 30 400

0.2

0.4

0.6

0.8

1

Mode Number

Diff

eren

tial M

ode

Del

ay (n

s/km

)

Numerical Inf. PI Exp = 2Analytical Inf. PI Exp = 2Numerical Inf. PI Exp = 1.95

Figure 2.8: Comparison of numerically and analytically calculated values of the differential-mode delay. The

mode labeling for the analytical modes can be found in table 2.2, for modes that are calculated with the numer-

ical mode solver in table D.1. The numerical values for the infinite parabolic-index profile are about four times

larger than the analytical values, while the numerical values for the exponent g = 1.95 fit comparably good

with the analytical values for the parabolic-index fiber. Note that naturally only integer values are plotted with

connecting lines for better visibility.

with values that have been published even for fibers with a core-cladding interface [61].

In [62], it was shown that the chromatic dispersion for power-law fibers becomes equal

for all modes, when neglecting profile dispersion. This was experimentally verified [61],

even if profile dispersion is present. This work thus assumes equal chromatic dispersion in

all modes. The chromatic dispersion parameter is calculated by taking the second derivative

of the propagation constant with respect to the angular frequency ω [51, pp. 38].

When comparing fibers with an infinite parabolic-index profile and such with a core-

cladding interface, as they are shown in figure 2.9, it seems obvious that the infinite parabolic-

index assumption can lead to further deviations from the true modal behavior when consid-

ering those modes that spread into the core-cladding boundary.

The field distributions of the propagating modes are of special importance for the non-

linear signal propagation, as it will be shown later in this thesis. Therefore, this section

intends to analyze the validity of the analytically calculated field distributions for infinite

parabolic-index fibers from eq. (2.42) with numerical values for different power-law index

profiles. In order to compare the field-distributions that are calculated with the analytical

solutions and with the numerical mode solver, an overlap integral is defined as the figure of


n(r)

ra a2a2 a

n(r)

raa

n(r)

raa

(a) (b)

(c)

≜ 𝑁𝐴

≜ 𝑁𝐴2

Figure 2.9: Different refractive index profiles : (a) Infinite power-law profile, the core diameter is specified in

order to make use of eq. 2.40, (b) truncated power-law profile, (c) trench assisted power-law index profile. For

an exponent of α = 2, the power-law profile becomes a parabolic-index profile.

merit:

Overlap(q) =

∫∫A ψ

(q)anaψ

(q)∗num∫∫

A ψ(q)anaψ

(q)∗ana

(2.49)

ψ(q)ana is the analytically derived mode field of mode q and is therefore a scalar field. ψ

(q)num

is the field distribution of the corresponding LP-mode when superposing the appropriate

vector fields, according to the calculation rule, given in table 2.3. Figure 2.10 shows the

overlap values between the analytical solutions for the infinite parabolic-index profile and

the numerical solution for a parabolic profile with very large extend (total radial simulation

domain is twice the core-radius parameter) and a truncated parabolic-index profile. Only

12 values are plotted, since only 12 different field distributions exist in the 42-mode fiber2,

as it can be seen in table 2.2. It can be noticed that the analytical field distributions are in

a very good agreement with the numerical ones for the infinite parabolic-index fiber. The

higher order modes of the truncated parabolic-index fiber differ slightly from the analytical

solution of the infinite parabolic-index fiber. The largest deviation of about 8% can be

observed for the LP13 mode. This appears reasonable as this is the mode with the largest

radial order and thus with the largest radial extend.

2All other mode fields are either only a different polarization or a different orientation


1 3 7 11 13 17 21 25 29 31 35 390.9

0.92

0.94

0.96

0.98

1

Mode Number

Mod

e O

verla

p

Infinite PI Truncated PI

Figure 2.10: Overlap between the analytically calculated mode-fields and the numerically calculated fields of

the same infinite parabolic-index and a truncated parabolic-index fiber that were used in the previous section.

The plot shows only 12 data points per graph, corresponding to the 12 different LP-modes; the labeling is

according to table 2.2. Fiber parameters can be found in table 2.4. Note that only integer values are plotted with

solid lines for better visibility.

Figure 2.11 shows the overlap between the analytical solution of the infinite parabolic-

index fiber and the infinite power-law fiber with exponent g = 1.95, that is used to minimize

the DMD. The infinite fiber with exponent 1.95 has overall a very good agreement with the

Laguerre Gaussian modes for the infinite parabolic-index fiber.

Furthermore, figure 2.11 shows the mode-overlap for a truncated and a trench-assisted

power-law fiber with exponent 1.95. The trench helps to minimize bending losses as well

as lowering the DMD [61]. The lower order modes of both, the truncated and the trench

assisted fiber show a good agreement with the analytical results of the infinite parabolic-

index fiber. The LP13 mode of the truncated power-law fiber shows a deviation of about

10%, while this is similar to the deviation of the truncated parabolic-index profile. The

trench assisted fiber shows overall a good agreement with the analytical solution for the

parabolic-index fiber, with LP51 mode showing a deviation of about 6%.


1 3 7 11 13 17 21 25 29 31 35 390.9

0.92

0.94

0.96

0.98

1

Mode Number

Mod

e O

verla

p

Infinite PowLaw, exp. 1.95 Truncated PowLaw, exp. 1.95 Trench Assis. PowLaw, exp. 1.95

Figure 2.11: Mode Overlap between the Laguerre Gaussian Mode Field approximation for the infinite parabolic-

index fiber and numerically calculated mode fields of different power-law fibers with exponent 1.95. The infinite

power-law fiber has almost perfect overlap, while the truncated fiber has a deviation of up to 10 % for the LP13mode. The trench assisted power-law fiber has mode-overlaps that are closer to one with a deviation of about 6

% for the LP51 mode. Mode Labeling according to table 2.2; fiber parameters can be found in table 2.4. Note

that only integer values are plotted with solid lines for better visibility.

Concluding this section, it can be observed that the analytical solutions for the field-

distributions in infinite parabolic-index fibers constitute a good approximation for the actual

field distribution, even when considering truncated or trench assisted power-law fibers with

exponents slightly different from 2.


Infin

ite

Para

bolic

-Ind

ex

Profile Exponent g 2

(Core)-Radius a 15µm

NA 0.205

n1 @ λ = 1.55µm 1.4574

Number of guided modes (λ = 1.55µm) 42Tr

unca

ted

Para

bolic

-Ind

ex

Profile Exponent g 2

Core-Radius a 15µm

NA 0.205

n1 @ λ = 1.55µm 1.4574

Number of guided modes (λ = 1.55µm) 42

Infin

ite

Pow

er-L

aw-I

ndex Profile Exponent g 1.95

Core-Radius a 15µm

NA 0.205

n1 @ λ = 1.55µm 1.4574


Trun

cate

d

Pow

er-L

aw-I

ndex Profile Exponent g 1.95

Core-Radius a 15µm

NA 0.205

n1 @ λ = 1.55µm 1.4574


Tren

ch-A

ssis

ted

Pow

er-L

aw-I

ndex

Profile Exponent g 1.95

Core-Radius a 15µm

NA 0.205

n1 @ λ = 1.55µm 1.4574

Trench Radius a2 20µm

Trench Depth NA2 0.08


Table 2.4: Fiber parameters for the fibers that are compared in section 2.1.7. The different fiber designs are

shown in figure 2.9.


2.1.8 Few-Mode Fibers

A very popular fiber type for current mode-division multiplexed transmission experiments

are few-mode fibers. Few-mode fibers will be analyzed in section 4 of this thesis. Even

though the term few-mode is not clearly defined, it usually refers to fibers that support 3 to

6 spatial modes (6-12 modes including polarization) [63]. Real optical fibers are often not

exactly circular symmetric, as it was considered in the previous sections, in fact the core

shape may be slightly elliptical [64]. The index of refraction for an elliptical power-law

index profile is not only a function of the radial but also the azimuthal coordinate and can

be written as [65]:

n(r, ϕ)2(r) =

⎧⎨⎩n(r, ϕ)2 = n21

(1− 2∆( ra(ϕ))

g)

for r ≤ a(ϕ)

n22 for r > a(ϕ)(2.50)

with

a(ϕ) =a0√

1− ϵ2 cos(ϕ)2(2.51)

and √1− ϵ2 = b

a(2.52)

a0 is then the core-radius at ϕ = 0 and ba is ellipticity. A cross-section of such a fiber is

shown in figure 2.12.

a

b

core

cladding

Figure 2.12: Cross-section of a graded-index fiber with an elliptical core. Note that the scale in this schematic

image is highly exaggerated for demonstrational purpose.

As a consequence of this slight ellipticity, the eigenmodes of such a fiber become linearly

polarized. Figure 2.13 shows the 6 eigenmodes of a slightly elliptical fiber (LP01x, LP01y,


Core-Radius a0 = 6µm

Numerical Aperture NA = 0.205

Ellipticity ab = 0.98

Profile Exponent g = 2

Table 2.5: Fiber parameters for the elliptical graded-index few-mode fiber

LP11ax, LP11ay, LP11bx and LP11by), calculated with the numerical mode solver, for the

fiber parameters given in table 2.5. While the LP11a/bx/y modes in the perfectly cylindrical

fiber have equal propagation constants [54] they are different in elliptical fibers, where

the LP11ax/y modes have a different propagation constant than the LP11bx/y modes [66].

This might lead to a propagation scenario, where the two orthogonal polarization of each

fiber mode exhibit strong coupling, while the spatial modes remain uncoupled throughout a

transmission link.


(a) (b)

(c) (d)

(e) (f)

Figure 2.13: Electrical fields of the 6 fiber modes of the few-mode fiber (LP01x, LP01y ,

LP11ax,LP11ay ,LP11bxLP11by), according to the fiber parameters in table 2.5. The circle indicates the core-

cladding boundary.

2.2. NONLINEAR PROPAGATION IN MULTI-MODE FIBERS 33

2.2 Nonlinear Propagation in Multi-Mode Fibers

The propagation properties of optical fibers are not only depending on the wavelength of an

electromagnetic wave that propagates through it, but also on the intensity of the wave, as

it will be discussed in the following sections. The dependance of a medium on the electric

field E(t) of a wave, propagating through this medium, can be described by the induced

electrical polarization [67]:

P (t) = ϵ0

(χ(1)E(t) + χ(2)E2(t) + χ(3)E3(t) + · · ·

)= P (1)(t) + P (2)(t) + P (3)(t) + · · · , (2.53)

when considering only a scalar field. χ(i) is a material property, called the ith order

susceptibility. Optical fibers for long-haul transmission are made of silicon dioxide that has

mainly a contribution from the linear susceptibility χ(1) and the third order susceptibility

χ(3) [64]. When applying an electric field E(t) to the material, the total polarization can be

written as [68]:

P (t) = ϵ0(χ(1)E(t) + χ(3)E3(t)) (2.54)

It is customary to define a nonlinear material index [64, pp. 33] which is denoted by ñ2:

ñ2 =3

8nRe(χ(3)

)(2.55)

The nonlinear index is thus connected to the linear index of refraction n. In this work,

the value of the nonlinear index is considered to be ñ2 = 2.6 · 10−20m2/W [64, pp. 36].When including the nonlinear response of the signal into the propagation equations, it

is beneficial to make use of the slowly varying envelope approximation [64, pp. 32]. The

x-polarized electric field of mode p can then be defined as3 [64, pp. 34]:

E(p)x (z, t) = Re(A(x)p (z, t)ψp(x, y)exp(jω0t− jβ

(p)0 z)) (2.56)

ψp(x, y) represents the field distribution of mode p, as defined in the previous sections.

β(p)0 is the phase constant of mode p at ω0. A

(x)p (z, t) is the pulse envelope that carries the

signal information [64, pp. 34] and is centered at the angular frequency ω0.

In order to keep the parameter space as small as possible, the following derivations will

only consider the x-polarization of the electric field. Thus, the modes and signals will only

be labeled by their LP-mode and orientation.3Please note that in the following, the index p refers to an integer counting variable and is not to be confused

with the radial mode order p in the LP-notation.


2.2.1 Generalized Nonlinear Schrödinger Equation for Multi-Mode Fibers

In general, one is interested in the evolution of the pulse envelope Ap(z, t), when propagat-

ing along the fiber [64]. A propagation equation for the pulse propagation in single-mode

fibers that includes linear and nonlinear effects is commonly known [64]. Since this equa-

tion is of Schrödinger type, it is often referred to as the Nonlinear Schrödinger Equation

(NLSE) [64]. A NLSE that describes the propagation of the slowly varying envelopes in

multi-mode fibers has been introduced by [69, 70, 24]. The propagation of the slowly vary-

ing pulse envelope Ap(z, t) that propagates in mode p of a fiber that supports a total of n

modes can in general be described by [69, 70]:

∂Ap(z, t)

∂z= −αAp(z, t) + j

(β(p)0 − β0

)Ap(z, t)−

(β(p)1 − β1

) ∂Ap(z, t)∂t

− j β(p)2

2

∂2Ap(z, t)

∂t2+ j

n∑q=1

n∑r=1

n∑s=1

γpqrsAq(z, t)Ar(z, t)A∗s(z, t) (2.57)

The first four terms on the right side represent linear transmission effects. α is the

attenuation coefficient as defined in eq. (2.8). β(p)m is defined as [64, pp.35]:

β(p)m =

(dmβ(p)

dωm

)ω=ω0

. (2.58)

being the mth order coefficient of the Taylor series expansion of the propagation con-

stants when developed around ω0 as:

β(p)(ω) = β(p)0 + β

(p)1 (ω − ω0) +

β(p)2

2(ω − ω0)2 + . . . (2.59)

Here, ω0 corresponds to a wavelength of λ = 1.55µm. When comparing the Taylor coef-

ficients with the discussion of the previous section, it can be observed that β(p)0 corresponds

to the phase constant, β(p)1 to the group-delay per unit length and β(p)2 to the chromatic dis-

persion parameter of the pth mode. The phase constant of each mode is normalized by the

phase constant of the fundamental mode [70] (β(1)0 = β(1)0 ). The group-delay parameter is

normalized by an arbitrarily set value β1, here chosen as the group-delay parameter of the

fundamental mode (β1 = β(1)1 ). This normalization corresponds to the retarded time frame

in single-mode fibers [64, pp. 40].

The last term of the multi-mode NLSE represents the Kerr-effect based nonlinear effects.

γpqrs is the nonlinear parameter, defined as [64, pp.35]:

γpqrs =ñ2ω0

cA(pqrs)eff

(2.60)


c is the speed of light in vacuum, ñ2 defined in eq. (2.55) and A(pqrs)eff is called the

effective area. It is defined with the following overlap [64, pp. 371]:

A(pqrs)eff =

[∫∫|ψp|2 dxdy

∫∫|ψq|2 dxdy

∫∫|ψr|2 dxdy

∫∫|ψs|2 dxdy

]1/2∫∫ψ∗pψqψ

∗rψsdxdy

(2.61)

where ψ(p,q,r,s) are the field distributions of the {p, q, r, s}th fiber modes, respectively,calculated with eq. (2.42). In the following, it is assumed that the nonlinear parameter γpqrsis constant over frequency [64, pp. 35].

2.2.2 Single-Mode NLSE

This section covers the pulse propagation in fibers where only one fiber mode exists or

when only one fiber mode of a multi-mode fiber is excited. For simplicity, only nonlinear

propagation effects are considered at this point (β2 = 0, α = 0). Furthermore, only one

polarization is considered.

When a pulse Ap(z, t) propagates along the fiber, eq. (2.57) can be written as:

∂Ap(z, t)

∂z= j

ñ2ω0

cA(pppp)eff

Ap(z, t)Ap(z, t)A∗p(z, t) = jγpppp |Ap(z, t)|

2Ap(z, t) (2.62)

A(pppp)eff is called the intramodal effective area and is equal to the commonly used overlap

integral for the effective area in single-mode fibers. It can be found with eq. (2.61) for

p = q = r = s and is defined as [64, pp.35]:

A(pppp)eff =

[∫∫|ψp|2 dxdy

]2∫∫

|ψp|4 dxdy(2.63)

Figure 2.14 shows the intramodal nonlinear parameters of the infinite parabolic-index

multi-mode fiber from the previous sections. The mode-fields are calculated with eq. (2.42)

for the fiber parameters, given in table 2.4. With ñ2 = 2.6 · 10−20m2/W , γpppp solelydepends on the field distribution of mode p. Thus, only 12 values are plotted for the 12

different LP-modes. The mode counting is according to table 2.2.

It can be observed that the fundamental mode has the largest nonlinear parameter with

γ1111 ≈ 0.91/(Wkm), corresponding to an effective area of about 125µm2. The other fibermodes have a smaller nonlinear parameter than the fundamental mode, while they are still

in the order of at least 35% of the fundamental mode.

When Ap only contains one wavelength channel (no WDM), the effect that is described

by eq. (2.62) is called self-phase modulation (SPM) as the phase of the signal is modulated

by its own power, represented by the term |Ap(z, t)|2.


1 3 7 11 13 17 21 25 29 31 35 390

0.2

0.4

0.6

0.8

1

Mode Index p

γ ppp

p (W

−1/k

m)

Figure 2.14: Intermodal nonlinear parameters for the 12 different mode fields of the infinite parabolic-index

fiber with parameters, according to table 2.4. The LP-modes are calculated with eq. (2.42) and the mode

labeling is according to table 2.2. It is visible that higher order modes tend to have a lower nonlinear parameter,

leading to generally lower nonlinear impact. Note that the mode number p an integeter index and not to be

confused with p in the LP-notation.

2.2.3 Two-Mode NLSE

The next case models a (theoretical) two-mode fiber or a multi-mode fiber where only two

modes are excited. When neglecting linear propagation effects and considering a pulse 4 Apin the pth and Aq in the qth fiber mode, eq. (2.57) can be written as:

∂Ap∂z

= j(γppppApApA∗p + γpppqApApA

∗q

+ γppqpApAqA∗p + γppqqApAqA

∗q + γpqppAqApA

∗p

+ γpqpqAqApA∗q + γpqqpAqAqA

∗p + γpqqqAqAqA

∗q) (2.64)

As only two modes propagate in the fiber, the triple summation in eq. (2.57) produces

eight different addends for the nonlinear part of eq. (2.64). For further simplifications, it is

necessary to analyze each of the eight nonlinear parameters. The two propagating modes

are chosen to be the LP01x (p) and the LP11ax mode (q), while all other modes are not

excited. Figure 2.15 shows that four out of the eight nonlinear parameters are zero. The

other four values are exactly those that contain only the mode p or those where two indexes

are p and two are q. Furthermore, it can be observed that the overlap integrals that contain

two different modes have equal value: γ1133 = γ1331 = γ1313.

Consequently, eq. (2.64) can be simplified as:

4The z and time dependance are not explicitly written down for clarity


Index pqrs

0

0.2

0.4

0.6

0.8

1

� pqr

s (W

-1/k

m)

Figure 2.15: Nonlinear parameter for all mode-combinations between the LP01x and the LP11ax mode of

the infinite parabolic-fiber, calculated with the field distributions according to eq. (2.42) for the fiber with

parameters according to table 2.4. The index is according to the labeling in table 2.2.

∂A1∂z

= j(γ1111|A1|2A1 + 2γ1133|A3|2A1 + γ1331A3A3A∗1)

= j(γ1111|A1|2 + 2γ1133|A3|2 + γ1331|A3|2e2j(ϕ3−ϕ1))A1

≈ j(γ1111|A1|2 + 2γ1133|A3|2)A1 (2.65)

The last term in the second row of eq. (2.65) depends on the phase difference between

the signals in p and q as: 2j(ϕq − ϕp). This term usually vanishes, since the phase relationchanges very often, leading to a negligible impact as a result of averaging effects. This

averaging will be discussed in detail in section 2.2.4.

The effect that is described by eq. (2.65) can be considered an intermodal cross-phase

modulation, meaning that the power of the pulse Aq changes the phase of the pulse Ap.

This effect is closely related to the intramodal cross-phase modulation (XPM), being a well

known effect from single-mode fiber transmission systems [64, pp. 226]. Figure 2.16 il-

lustrates intra- and intermodal XPM: Figure 2.16 (a) schematically shows a scenario where

two spectral components at angular frequencies ωi and ωj propagate inside one fiber mode,

while the power of the spectral component at ωj modulates the phase of the spectral com-

ponent at ωi. Intramodal XPM can be modeled by eq. (2.65) by modifying it so that all

intermodal effective areas are replaced by the intramodal effective area [64, pp. 229].

Figure 2.16 (b) shows the intermodal XPM as it is described by eq. (2.65): The pulse

Aq, centered at the angular carrier frequency ωj in mode q changes the phase of the pulse

Ap that is centered at angular carrier frequency ωi in mode p through intermodal XPM.


Angular Frequency

Spec

trum

XPM

ωi ωj

(a)

Spec

trum

IntermodalXPMSp

ectr

um

Angular Frequencyωi

ωj

(b)

Figure 2.16: Schematic description of the spectral arrangements for intramodal and intermodal cross phase

modulation. The power of a spectral component at ωk influences the propagation of a spectral component at

ωj . The two spectral components can either propagate in (a) the same fiber mode (Intramodal XPM) or (b) two

fiber modes (Intermodal XPM).

2.2.4 Four-Wave Mixing

Besides self- and cross-phase modulation, Four-Wave Mixing (FWM) is the third nonlinear

effect that is caused by the Kerr-nonlinearity. FWM shows up, when three spectral com-

ponents interact through the Kerr-nonlinearity and create a fourth spectral component. A

FWM process requires two basic conditions to be fulfilled. The first one is called the fre-

quency condition that basically represents the conservation of energy and thereby defines

the frequency spacings between the spectral components that interact through FWM. When

three spectral components with the angular frequencies ωi, ωj and ωk interact through the

Kerr-nonlinearity, they create a new spectral component at angular frequency ωl [64, pp.

370]:

ωl = −ωi + ωj + ωk (2.66)

The second condition that determines the strength of the newly created spectral compo-

nent is defined by the phase matching condition. It is usually denoted by ∆β and takes a

certain value, as it will be discussed in the following section. A small or even vanishing

phase matching condition (∆β ≈ 0) leads to the generation of a strong spectral component,however, even for a non-vanashing phase matching condition (∆β ̸= 0), a new spectralcomponent is created at a lower strength [68]. The phase matching condition can be ex-

pressed with the propagation constants of the interacting spectral components.


Intramodal Phase Matching

In this section, FWM processes are discussed, where all spectral components propagate in-

side one fiber mode. These FWM processes are called intramodal FWM. The phase match-

ing condition for an intramodal FWM process where all four spectral components propagate

in mode p can be written as [64, pp. 370]:

∆β(pppp)(ωi, ωj , ωk, ωl) = −β(p)(ωi) + β(p)(ωj) + β(p)(ωk)− β(p)(ωl). (2.67)

A simplification of the intramodal phase matching condition can be done when develop-

ing the propagation constants into Taylor series as defined in eq. (2.58) and (2.59). Equation

(2.59) only considers dispersion terms up to β(p)2 . This is sufficient when operating outside

the zero-dispersion region [64, pp. 40].

For a given angular frequency ωi, it is possible to write the angular frequency ωj and ωkas:

ωj = ωi +∆ω1 (2.68)

ωk = ωi +∆ω2 (2.69)

where ∆ω1 is the spectral separation between ωj and ωi and ∆ω2 is the spectral separa-

tion between ωk and ωi as shown in figure 2.17 (a). With the frequency condition from eq.

2.66, ωl can be written as:

ωl = ωi +∆ω1 +∆ω2 (2.70)

and thus ωl has the spectral distance of the sum of ∆ω1 and ∆ω2 from ωi. When using

this description for the four frequency components and choosing ωi = ω0, it is possible to

write the phase matching condition as5:

∆β(pppp)(ωi, ωj , ωk, ωl) = ∆β(pppp)(∆ω1,∆ω2) = β

(p)2 ∆ω1∆ω2 (2.71)

It can be observed that the frequency definitions connect the interacting frequency com-

poenents so that the phase matching term is only a function of two variables instead of

four. Equation (2.71) shows further that the phase matching condition depends on the dis-

persion parameter β2 and the spacing between the frequency components. It is noteworthy

that both, β(p)0 and β(p)1 are canceled out. Since the dispersion is considered to be non-zero

(β(p)2 ̸= 0), a complete phase matching (∆β(pppp)(∆ω1,∆ω2) = 0) is only possible if either5see Appendix A


Angular Frequency

Spec

trum

ωi ωj ωlωk

Δω1 Δω2Δω1

(a)

1/(2) (GHz)

2/(

2)

(GH

z)

-200 0 200

-200

0

200

-60

-50

-40

-30

-20

Δβ(pppp)(dB)

(b)

Figure 2.17: (a) Arrangement of the spectral components for intramodal Four-Wave Mixing when fulfilling the

frequency condition (2.66). (b) Phase matching coefficient ∆β(pppp) as a function of the two free variables

∆ω1 and ∆ω2 for an intramodal four-wave mixing process.

∆ω1 or ∆ω2 are zero. These special cases are called degenerate FWM for two waves and

are, from a mathematical point, equal to cross-phase modulation, where only two spectral

components interact.

Fig 2.17 (b) shows a contour plot of the phase matching condition (2.71) for an in-

tramodal FWM-process, plotted in dB (10log10(∆β)). The plot thus corresponds to the

strength of the created frequency components at ωl [71]. Blue color corresponds to low val-

ues of ∆β. In accordance with eq. (2.71), the lines where either ∆ω1 or ∆ω2 are zero have

minimum values of ∆β. The two lines cross at the origin of the plot at ∆ω1 = ∆ω2 = 0.

Intermodal Phase Matching

FWM can also appear as an intermodal process. The four spectral components that par-

ticipate in the FWM process can thus propagate in up to four spatial modes. While the

frequency condition for intermodal FWM is the same as for intramodal FWM, the phase

matching condition is considerably altered due to the different propagation constants of

the spatial modes. Several different spectral / modal arrangements will be analyzed in the

following.

Two-Mode FWM

The first spectral / modal arrangement that is discussed here, considers two spectral com-

ponents propagating in one fiber mode and two propagating in a second fiber mode. Three

different spectral / modal arrangements are then possible. The first one assumes that the


spectral components at ωi and ωk propagate in the pth mode and those at ωj and ωl in the

qth mode. The phase matching equation can then be written as:

∆β(pqpq)(ωi, ωj , ωk, ωl) = −β(p)(ωi) + β(q)(ωj) + β(p)(ωk)− β(q)(ωl) (2.72)

where β(p,q) are the propagation constants of the {p, q}th fiber modes, respectively,evaluated at the angular frequency ωi,k ωj,l. Figure 2.18 shows the setup of the spectral

components for this intermodal FWM process, where p and q are exemplary chosen to

be the LP01 and the LP11a mode, respectively. After expanding each mode’s propagation

constant into a Taylor as for the single-mode FWM process in eq. (2.59), and with equ,

(2.68) - (2.70), the phase matching condition can be simplified as 6:

∆β(pqpq)(∆ω1,∆ω2) =(β(p)1 − β

(q)1

)∆ω2 − β(p)2 ∆ω1∆ω2 (2.73)

where β(p,q)1 are the group-velocity parameters of the pth and qth mode, and β(p)2 is the

dispersion parameter of the pth mode, being considered equal for all fiber modes.

It can be observed that β(p)0 and β(q)0 cancel out in this equation, however, β

(p)1 and β

(q)1

do not cancel out. Equation (2.73) is depicted in figure 2.18 (b). It can be seen that lines of

full phase matching form a similar cross as for the intramodal case that is, however, shifted

by a certain frequency offset to the right. This frequency shift is given by an interaction

frequency (see Apendix A), denoted by fint and obtained as:

fint =∆ωint2π

=β(p)1 − β

(q)1

2πβ(p)2

(2.74)

This means that full phase matching, and consequently strong four-wave mixing, can be

achieved for arbitrary values of ∆ω2, if ∆ω1 = ∆ωint is kept constant. This effect has been

shown experimentally [43].

The second spectral / modal setup assumes that the spectral components at ωi and ωjpropagate in mode p and the spectral components at ωk and ωl in mode q. The phase

matching condition can then be written as:

∆β(ppqq)(ωi, ωj , ωk, ωl) = −β(p)(ωi) + β(p)(ωj) + β(q)(ωk)− β(q)(ωl) (2.75)

This leads with the Taylor series expansion of the two modes’ propagation constants and

eq. (2.68) - (2.70) to:

∆β(pqpq)(∆ω1,∆ω2) =(β(p)1 − β

(q)1

)∆ω1 − β(p)2 ∆ω1∆ω2 (2.76)

6see Appendix A


Angular Frequency

Spec

trum

Spec

trum

ωi

ωj ωl

ωk

Δω1Δω2

Δω1

(a)

1/(2) (GHz)

2/(

2)

(GH

z)

-200 0 200

-200

0

200

-60

-40

-20

Δβ(pqpq)(dB)

(b)

Figure 2.18: (a) Spectral arrangements for an intermodal FWM-process where two frequency components

travel in one fiber mode and two in another one. (b) Phase matching coefficient ∆β(pqpq) for the intermodal

FWM-process from (a).

This equation is similar to eq. (A.7), however the two values ∆ω1 and ∆ω2 with full

phase matching are shifted in the ∆ω2 direction instead of the ∆ω1 direction in the previous

case.

The third spectral / modal setup of this kind assumes ωi and ωl in the pth mode and ωjand ωk in the qth mode. The phase matching equation can than be written as:

∆β(pqqp)(∆ω1,∆ω2) = 2β(q)0 − 2β

(p)0 + 2β

(p)1 (∆ω1 +∆ω2)− 2β

(q)1 (∆ω1 +∆ω2)

+β(p)2

2(∆ω21 +∆ω

22)−

β(q)2

2(∆ω1 +∆ω2)

2 (2.77)

While in the previous two phase matching conditions for the two-mode FWM processes,

the phase constants β(p)0 and β(q)0 canceled out, they sustain in eq. (2.77). According to

[43], the phase constants of all modes are considered to be subject to fast variations along

the fiber. Thus, constant phase matching cannot be achieved in long transmission links.

Consequently, this FWM process cannot create significant spectral components. This effect

has further been experimentally proven in [43], concluding that out of the three possible

two-mode FWM processes only two show measurable impact.

Three- and Four-Mode FWM

In this section, FWM is discussed as it might appear when the four spectral components ωi,

ωj , ωk and ωl propagate in three or four spatial modes p, q, r, s. Two exemplary spectral

/ modal setups are depicted in figure 2.19, while others are certainly possible. To analyze


these FWM setups, it is again necessary to analyze the phase matching condition. It can be

written for the four-mode FWM case as the most general case as:

∆β(pqrs)(ωi, ωj , ωk, ωl) = −β(p)(ωi) + β(q)(ωj) + β(r)(ωk)− β(s)(ωl) (2.78)

After expanding each mode’s propagation constant into a Taylor series, and the fre-

quency definitions as in (2.68) - (2.70), it can be written as:

∆β(pqrs) = −β(p)0 + β(q)0 + β

(r)0 − β

(s)0 +

+ β(q)1 (∆ω1) + β

(r)1 (∆ω2)− β

(s)1 (∆ω1 +∆ω2)+

+β(q)2

2(∆ω1)

2 +β(r)2

2(∆ω2)

2 +β(s)2

2(∆ω1 +∆ω2)

2 (2.79)

This phase matching equation contains the phase constants of all spatial modes βp,q,r,s0that are in general subject to fast variations along the fiber link. This process is consequently

negligible with the same argumentation that was used in the previous section. Equation

(2.79) indicates that in a three-mode FWM setup, only two phase constants could be can-

celed out, leaving two phase constants inside the phase matching equation and consequently

preventing these FWM processes.

2.2.5 Consequence of Intermodal FWM for the NLSE

The conclusions that were drawn about intermodal FWM in the previous sections have a

direct influence on the nonlinear Schrödinger equation, allowing significant simplifications.

When considering the generalized multi-mode NLSE (2.57), the triple sum in the nonlinear

part creates n3 terms, where n is the number of propagating modes in the fiber. It was

shown that for the two-mode NLSE, half of the nonlinear terms vanish due to non-significant

nonlinear parameters. Out of the rem

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