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Transcript of 2nd year project_spatial mode division MUX
1
Second Year Project Report
Mode Division Multiplexed Transmission for Long Haul Fiber
Communication Systems
April 2014
By:
Hamdi Ghodbane
Saddam Gafsi
Supervisors:
Dr. Amine Ben Salem
Pr. Mourad Zghal
2
Acknowledgements
We wish to express our sincere gratitude and appreciation to everyone who assisted
and supported us during this project. We would like to make special mention of the following
people:
Our supervisor, Professor Mourad ZGHAL, first for giving us the opportunity to work
on such an exciting subject in the field of optics, the field we were passionate about and
aspired to work in, second for his encouragement and appreciation for our efforts, and finally
for introducing us to great people that we now enjoy their friendship and look forward to
further work with.
Our second supervisor doctor Amine Ben Salem for his continuous support, his
guidings, although some times strict have always been in our best interest.
Finally, we would like to thank our friend and colleague Abderrahmen trichilli for his
valuable help and encouragement during this project.
3
Table of contents
General introduction :................................................................................................................................................. 9
1. Chapter 1 : The capacity crunch and the need for new multiplexing techniques ............................ 11
1.1 Introduction : ................................................................................................................................................... 11
1.2 Motivation : ...................................................................................................................................................... 11
1.3 Existing multiplexing techniques : ............................................................................................................. 12
1.3.1 Wavelength division multiplexing :........................................................................................... 12
1.3.2 Time division multiplexing : ........................................................................................................ 12
1.3.3 Polarization division multiplexing : ...........................................................................................13
1.3.4 Code division multiplexing : ........................................................................................................13
1.3.5 Subcarrier multiplexing : ..............................................................................................................13
1.4 Space Division Multiplexing : ......................................................................................................................14
1.4.1 Overview :...............................................................................................................................................14
1.4.2 Presentation of optical fibers :.....................................................................................................14
1.4.3 Concept of SDM : ........................................................................................................................... 15
1.4.4 Engineering challenges and current state : .............................................................................. 17
1.5 Conclusion : ......................................................................................................................................................19
2. Chapter 2 : Modes determination and optical fiber modeling ............................................................... 20
2.1 Introduction : ................................................................................................................................................... 20
2.2 Optical Fiber Modes Determination : ......................................................................................................... 20
2.2.1 Maxwellโs equations : ............................................................................................................................. 20
2.2.3 Bessel Equation : .....................................................................................................................................23
2.2.4 Mode determination: .............................................................................................................................. 27
2.2.5 Solutions for m=0 and m=1 : ................................................................................................................ 28
2.2.6 Solutions for m>1 : ..................................................................................................................................30
2.2.7 Field amplitude distribution of the modes: .......................................................................................31
2.2.8 Analogy with microwave waveguides: ..............................................................................................33
2.3 Modeling the optical fiber : ..........................................................................................................................34
2.4 Conclusion : ......................................................................................................................................................36
3. Chapter 3 : Study of key system performance parameters in a MDM transmission system .........37
3.1 Introduction : ...................................................................................................................................................37
3.2 Definition and perspective : .........................................................................................................................37
3.3 Mode coupling and its origins : ..................................................................................................................38
3.4 Coupled power theory : .................................................................................................................................38
3.5 Simulation and parametric study: ...............................................................................................................41
3.5.1 Justification of the perturbation model : ...........................................................................................41
4
3.5.2 Refractive index perturbation : ............................................................................................................ 45
3.5.3 Input powers : .......................................................................................................................................... 48
3.5.4 Core radius : ............................................................................................................................................. 50
3.5.5 Wavelength : ............................................................................................................................................ 50
3.5.6 Modes : ....................................................................................................................................................... 51
3.5.7 Coupling of modes of the same group: .............................................................................................. 51
3.5.8 Effect of the number of modes : .......................................................................................................... 52
3.6 Conclusion : ...................................................................................................................................................... 58
Conclusion : ................................................................................................................................................................. 58
4. Bibliography : .................................................................................................................................................... 60
5
List of figures
Figure 1 Forecast of the worldwide IP traffic in petabytes per month, according to the Cisco Visual Networking Index ...................................................................................................................................................... 11 Figure 2 Multimode and single mode fibers ........................................................................................................14 Figure 3 Multicore fibers .......................................................................................................................................... 15 Figure 4 SDM over a few mode fiber ....................................................................................................................16 Figure 5 SDM over a multicore fiber .....................................................................................................................16 Figure 6 Cylindrical coordinate system ................................................................................................................ 24 Figure 7 solutions of the eigenvalue equation for m=0 in the (u,w) plane .................................................. 29 Figure 8 solutions of the eigenvalue equation for m=0 and m=1 in the (u,w) plane ................................30 Figure 9 Field intensity distribution for m=0 ......................................................................................................32 Figure 10 Field intensity distribution for m=1 Figure 11 Field intensity distribution for m=2 ................................................................................................................................32 Figure 12 correspondence between modes in metallic guides and optical fibers ......................................33 Figure 13 Dividing a sphere into smaller finite elements (meshing) progressively .................................34 Figure 14 Different types of finite elements that can be used in meshing ..................................................34 Figure 15 Meshed structure .....................................................................................................................................35 Figure 16 calculated solution ..................................................................................................................................35 Figure 17. LP11 mode : transverse electric field and electric field norm .......................................................41 Figure 18. LP01 mode : transverse electric field and electric field norm .......................................................41 Figure 19 3D plot of the sin(2x)*cos(2y) function .............................................................................................. 42 Figure 20 Evolution of the perturbed refractive index as a function of the distance z ............................ 42 Figure 21 Evolution of the refractive index of the fiber as a function of the distance z for a higher perturbation value .....................................................................................................................................................43 Figure 22 Exact plot of the fiberโs cross section (5.5 um core radius) ...........................................................43 Figure 23 zoomed-in snapshot of the exact plot of the fiberโs cross section (5.5 um core radius) ........44 Figure 24 Power coupling for ฮดn2(x, y) = 1.5 *10^-2 sin(rc x)cos(rc y) .........................................................44 Figure 25 Crosstalk between the two propagating modes............................................................................... 45 Figure 26 Coupling coefficient as a function of K .............................................................................................46 Figure 27 Point of equilibrium as a function of K ..............................................................................................46 Figure 28 Power coupling for ฮดn2(x, y) = 10^-2 sin(rc x)cos(rc y) ................................................................. 47 Figure 29 Power coupling for ฮดn2(x, y) = 2*10^-2 sin(rc x)cos(rc y) ............................................................. 47 Figure 30 Point of equilbrium as a function of P01 ........................................................................................... 48 Figure 31 Power coupling for P01=4mW and P11=1mW .................................................................................49 Figure 32 Power coupling for P01=1.2mW and P11=1mW .............................................................................49 Figure 33 Coupling coefficient as a function of the core radius .................................................................... 50 Figure 34 Coupling coefficients of the different modes with the fundamental mode .............................. 51 Figure 35 Power coupling between the LP02 and the LP21 modes ............................................................... 52 Figure 36 Power coupling in a four mode fiber .................................................................................................. 54 Figure 37 Power coupling in a six mode fiber (first configuration) .............................................................. 55 Figure 38 Power coupling in a six mode fiber (second configuration) ....................................................... 56 Figure 39 Power coupling in a six mode fiber (third configuration) ............................................................ 56 Figure 40 Power coupling in a six mode fiber (fourth configuration) ......................................................... 57 Figure 41 Power coupling in a six mode fiber (fifth configuration) ............................................................. 57
6
List of tables Table1.1: State of the art SDM demonstrations 18
Table 2.1: Determination of possible solutions 28
Table 3.1: Modes effective indices 54
Table 3.2: Coupling coefficients for the different modes 54
7
List of acronyms: SMF single mode fiber
MMF multimode fiber
FMF few mode fiber
GI graded index fiber
SI step index fiber
WDM wavelength division multiplexing
DWDM dense wavelength division multiplexing
CWDM coarse wavelength division multiplexing
TDM time division multiplexing
FDM frequency division multiplexing
PDM polarization division multiplexing
CDM code division multiplexing
CDMA code division multiple access
SCM subcarrier multiplexing
CATV cable television
SDM spatial division multiplexing
MCF multicore fiber
Tx transmitter
Rx receiver
DSP digital signal processing
FEM finite element method
DMGD differential mode group delay
MIMO multi-input/multi-output
EDFA erbium doped fiber amplifier
8
List of symbols : โ : Nabla operator
๐ฝ : Propagation constant
ํ0 : Vacuum dielectric permittivity
๐0 โถ Vacuum magnetic permiability
๐ : wavelength
๐ : Anguler frequency
๏ฟฝโ๏ฟฝ : Electric field
๏ฟฝโโ๏ฟฝ : Magnetic field
๏ฟฝโ๏ฟฝ : Magnetic induction
๏ฟฝโ๏ฟฝ : Polarization vector
๐ : Light celerity in vacuum
๐ฝ๐: 1st order Bessel function
๐พ๐ : Modified Bessel function
๐๐ : Bessel function of the 2nd kind
๐ ๐(๐ง) : Real part of the complex number z
๐ผ๐(๐ง) : Imaginary part of the complex number z
๐ : Normalized frequency
๐0 : Wavenumber
โ๐๐ : Coupling coefficient between modes p and q
๐ : Transverse component of the wavenumber ๐0
๐: Core radius
๐๐๐ : Refractive index inside the claddinng
๐๐๐: Refractive index inside the core
9
General introduction :
Over the past forty years, a series of technological breakthroughs have allowed the
capacity-per-fiber to increase around 10x every four years. Transmission technology has
therefore thus far been able to keep up with the relentless, exponential growth of capacity
demand. The cost of transmitting exponentially more data was also manageable, in large
part because more data was transmitted over the same fiber by upgrading equipment at the
fiber ends. But in the coming decade or so, an increasing number of fibers in real networks
will reach their capacity limit. Keeping up with demand will therefore mean lighting new
fibers and installing new cables, potentially also at an exponentially increasing rate. Further,
this fiber capacity limit is not specific to a particular modulation format or transponder
standard, it is fundamental and can be derived from a straightforward extension of the
fundamental Shannon capacity limit to a nonlinear fiber channel under quite broad
assumptions.
The upcoming potential โcapacity crunchโ, then, is an era of unfavorable cost
scaling. For some carriers who have access to a limited number of dark fibers, very
expensive installation of new cables will be the only alternative as the capacity of existing
fibers is filled. โFiber-richโ carriers who attempted to future-proof their fiber plant by
including large numbers of premium fibers in each cable (thus putting off the need for
subsequent new cables) will be forced to overbuild, i.e. deploy multiple systems over parallel
fibers, to keep up with demand. However, multiple systems over parallel fibers suggest that
transmission costs and power consumption will scale linearly with growing capacity. The
fear is that, without further innovation to lower the cost-per-bit, the capacity crunch will
apply pressure to constrain growth, and we will finally reach the end of the seemingly
boundless connectivity that drives our economy and enriches our experiences.
In order to achieve higher spectral efficiency, mode division multiplexing in few-
mode fibers is a new research area. The idea faces lots of technical issues including
intermodal delay and mode coupling which limit the achievable length of the system. This
report is designated to make an analysis of mode coupling in step-index few-mode fibers.
We analyze numerically all the parameters of fiber, which could impact mode coupling in
few-mode fibers and identify the conditions which can minimize power coupling and the
crosstalk between modes.
10
Chapter one provides an introduction to the area of interest in which MDM falls. In
particular, it is the purpose of this chapter to present multiplexing techniques currently
used in optical transmission systems.
In chapter two we introduce fundamental notions indispensable for understanding
the concept of mode division multiplexing such as modes in optical fibers and essential fiber
characteristics. We then demonstrate how to model an optical fiber using the finite element
method.
Finally, in chapter three, we try to develop a deeper understanding of mode coupling
effects in fibers which represent the main obstacle for MDM transmission and evaluate key
system performance parameters numerically using a finite element analysis solver
and simulation software, and MATLAB to implement the coupled power theory equations
and to obtain numerical results.
11
1. Chapter 1 : The capacity crunch and the need for new multiplexing techniques
1.1 Introduction :
In this chapter we will first present the motivation behind this project. Then, we will give
an overview of the existing technologies that helped satisfy the capacity demand until today
which leads us to the introduction of the concept of spatial division multiplexing and
especially mode division multiplexing.
1.2 Motivation :
Some of the best numbers we have on bandwidth usage come from Cisco's Visual
Networking Index, which shows that worldwide IP (Internet protocol) traffic hit 20.2 exabytes
per month in 2010, and 242 exabytes per year. (1 exabyte = 1000 000 000 gigabytes !).
According to Cisco, global IP traffic increased eightfold over the five years leading up to
2010 and will quadruple by 2015, hitting 966 exabytes (nearly one zettabyte) for the full year.
That will be the equivalent of all movies ever made crossing IP networks every four minutes.
Figure 1 Forecast of the worldwide IP traffic in petabytes per month, according to the Cisco Visual Networking Index
12
Recent fundamental studies have shown that the bandwidth provided by our current fiber
optic technologies are very close (within a factor of 2 or 3) to the "Shannon limits" of
(nonlinear) optical fiber transmission. Scientists conclude that space is the additional degree
of freedom required to achieve higher transmission, whether it's multiple single mode fibers,
multimode fibers, or various coupled mode configurations.
1.3 Existing multiplexing techniques :
Optical communications technology has made enormous and steady progress for
several decades, providing the key resource in our increasingly information-driven society
and economy. Much of this progress has been in finding innovative ways to increase the
data carrying capacity of a single optical fiber. In this search, researchers have explored (and
close to maximally exploited) every available degree of freedom, and even commercial
systems now utilize multiplexing in time, wavelength, polarization, and phase to speed
more information through the fiber infrastructure.
1.3.1 Wavelength division multiplexing :
A powerful technique in optical communications is wavelength division
multiplexing (WDM). WDM creates several channels over the same fiber, either SMF or
MMF, using a different wavelength for each channel. At the receiving side of a WDM
system, optical filters are required in order to demultiplex the transmitted signals. The
format of the transmitted signals can be arbitrary since the demultiplexing is based on
wavelength differentiation. There are two WDM variants, namely dense WDM (DWDM)
and coarse WDM (CWDM). CWDM, sometimes referred to as wideband WDM, uses a much
wider spacing in the wavelengths of the optical sources and therefore it has increased
tolerance with respect to wave-length drifting and consequently to temperature
fluctuations. CWDM is a lower cost technique than DWDM due to the more relaxed
requirements in the system design and related components. Therefore CWDM seems more
suitable for application in MMF systems.
1.3.2 Time division multiplexing :
In digital communications, it is possible to divide the transmission time in slots and
transmit each digital channel periodically. This technique is called time division
multiplexing (TDM). Similarly to WDM and FDM, TDM can apply directly in the optical or
electrical domain. TDM can apply over the intensity of the transmitted optical carrier and it
requires a digital signal format. Optical TDM aims at achieving a very high capacity per
13
transmission wavelength in long-haul SMF transmission systems. Electrical TDM can be a
cost-effective approach in LANs and optical access systems.
1.3.3 Polarization division multiplexing :
In SMFs, the optical field propagates in one mode with two orthogonal polarizations.
Therefore, polarization division multiplexing (PDM) can be achieved and two channels can
be transmitted over an SMF. The two polarizations should be separated at the receiving end
to demultiplex the two channels, which can transport signals of any format. PDM requires
that polarization is maintained along propagation and it is an example of spatial
multiplexing. It is usually employed in transmission experiments where record capacities
are pursued. In principle, PDM can also apply in MMF transmission to create two
independent channels, as long as polarization maintenance can be achieved.
1.3.4 Code division multiplexing :
In all multiplexing techniques, a minimum level of orthogonality is needed in a
certain domain among the received signals in order to demultiplex the channels. The
previously mentioned techniques achieve the necessary orthogonality in the wavelength,
frequency, time and polarization (space) domains. It is possible to create several
communication channels by using a unique code at each channel to transmit a digital data
stream. The necessary orthogonality can then be achieved with the use of mutually
orthogonal codes. This technique is called code division multiplexing (CDM) or code
division multiple access (CDMA), depending on the application and whether it uses
synchronous or asynchronous transmission. CDMA has been originally introduced in radio
communications but optical CDMA has been investigated as well. In CDM/CDMA, the
communication channels can use the same wavelength, frequency, time or polarization (in
general, spatial mode).
1.3.5 Subcarrier multiplexing :
Similarly to WDM, in radio communications, frequency division multiplexing (FDM)
is applied. In a sense, WDM is an optical form of FDM. It is possible to use a radio FDM
signal to modulate the laser intensity of an optical link. At the end of such a link, the
electrical received signal can be processed with an FDM demultiplexer. Therefore several
radio channels can be multiplexed over the same fiber, either SMF or MMF. This technique
is known as subcarrier multiplexing (SCM) and it is mainly used in radio-over-fiber systems,
such as the cable television (CATV) distribution systems. In SCM, the transmission channels
14
are transparent to the transmission format and their bandwidth is limited by the subcarrier
spacing. SCM transmission has been considered over MMF, and combined with DWDM has
yielded a very high aggregate bit rate of 204 Gbit/s over 3 km of 50/125 ยนm silica-based GI-
MMF [9].
1.4 Space Division Multiplexing :
1.4.1 Overview :
The term SDM is nowadays taken to refer to multiplexing techniques that establish
multiple spatially distinguishable data pathways through the same fiber, although in earlier
days the same terminology was previously applied to describe the case of multiple parallel
fiber systems: the benchmark that needs to be beaten on a cost-per-bit perspective if any of
the SDM approaches currently under investigation are ever to be commercially deployed.
There are two main approaches in spatial division multiplexing, each one of these
approaches uses a different type of optical fiber, but before we dive any deeper in the
concept of SDM it is essential to have a basic knowledge about these fiber technologies. In
the following section we will describe them briefly.
1.4.2 Presentation of optical fibers :
An optical fiber is a dielectric cylindrical waveguide. Light propagates in the core of
the optical fiber. The core is surrounded by the cladding, which has a smaller refractive
index. Therefore the mechanism of light propagation in optical fibers is total internal
reflection. The diameters of the core and the cladding, the profile of the refractive index, as
Figure 2 Multimode and single mode fibers
15
well as the material of the fiber define the type of the optical fiber and give its particular
characteristics.
An optical fiber is multimode when light propagates in more than one spatial guided
mode. A spatial guided mode, or simply a mode, can be viewed as a solution to the
electromagnetic wave propagation problem of monochromatic light in an optical fiber . It is
common not to refer to the two orthogonal polarizations of the electromagnetic field as two
different modes, but rather as the two polarizations of a mode. Alternatively, a distinct ray-
trace of light propagation in the optical fiber corresponds to a certain mode. A single mode
fiber supports only one mode in its specified wavelength operation range. Besides the
optical power that propagates along the fiber, some of the power is not bound and it is
radiated. This is usually described by the radiation modes.
A multicore fiber can be described as multiple optical fibers in one. In fact multicore
fibers have several cores in a single cladding and each core is considered as a single channel.
The Cladding diameter of this fiber is larger than standard single mode fiber.
1.4.3 Concept of SDM :
As we mentioned before, the notion of increasing fiber capacity with Space Division
Multiplexing (SDM) is almost as old as fiber communications itself, with the fabrication of
fibers containing multiple cores, the first and most obvious approach to SDM, reported as
far back as 1979. Yet only recently has serious attention been given to building a complete
networking platform as needed to make use of this multicore fiber (MCF) approach. The
Figure 3 Multicore fibers
16
alternative approach of using modes within a multimode fiber (MMF) as a means to define
separate spatially distinct channels is called โMode Division Multiplexingโ or MDM.
The current frenzied progress in SDM is occurring now because of a convergence of
enabling technological capabilities and a rapidly emerging need. On the one hand, SDM
draws on the accumulated progress of fiber research. This includes subtle improvements in
traditional fibers, and the fantastically precise fabrication methods developed to produce
hollow-core and other complex microstructure fibers. Sophisticated mode control and
analysis methods along with tapered devices can be borrowed from high-power fiber laser
research, which itself has needed to develop means to better exploit the spatial domain in
the drive to achieve ever higher power levels. These enabling technologies have made SDM
a viable strategy just as a severe need for innovation emerges.
The anticipated promise of SDM is not only that it will provide the next leap in capacity-
per-fiber, but that this will concurrently enable large reductions in cost-per-bit and
Figure 4 SDM over a few mode fiber
Figure 5 SDM over a multicore fiber
17
improved energy efficiency. This is a formidable challenge. SDM is very different from
wavelength division multiplexing (WDM) which inherently allows the sharing of key
components: e.g., an EDFA and dispersion compensation module can easily be shared by
many WDM channels with minimal added complexity. The benefits of SDM are more
speculative, and assume that many system components can be eventually integrated and
engineered to support this new platform.
1.4.4 Engineering challenges and current state :
For mode division multiplexed (MDM) transmission in MMF where the
distinguishable pathways have significant spatial overlap and, as a consequence, signals are
prone to couple randomly between the modes during propagation. In general the modes will
exhibit differential mode group delays (DMGD) and also differential modal loss or gain. The
energy of a given data symbol launched into a particular mode spreads out into adjacent
symbol time slots as a result of mode-coupling, rapidly compromising successful reception
of the information it carries. Crosstalk occurs when light is coupled from one mode to
another and remains there upon detection. Inter-symbol interference occurs when the
crosstalk is coupled back to the original mode after propagation in a mode with different
group velocity. As in wireless systems, equalization utilizing multiple-input multiple-output
(MIMO) techniques is required at the receivers to mitigate these linear impairments.
Conventional MMFs with core/cladding diameters of 50/125 and 62.5/125um support
more than 100 modes and have large DMGDs, and thus are not suitable for long-haul
transmission because the DSP complexity for MIMO equalization would be too high. Recent
advancements have led to fibers supporting a small number of modes, the so-called โfew-
mode fibersโ (FMFs), with low DMGD. The most significant research demonstrations have
so far concentrated on the simplest FMF, which supports three modes, the LP01 and
degenerate LP11 modes, for a total of 6 polarization and spatial modes (referred to as 3MF).
The DMGD in step-index core designs (as used in the first demonstrations of MDM in 3MF)
is a few ns/km, meaning that the number of taps required for MIMO processing was
impractical for transmission distances much greater than 10km. Consequently work has
been undertaken to develop core designs offering substantially reduced values of DMGD.
Using a graded-index (GI-) core design, DMGD values as low as 50 ps/km have been
achieved for 3MF. Moreover, it has been shown that DMGD cancellation is possible by
combining fibers fabricated to have opposite signs of DMGD. In this way transmission lines
with net values of DMGD as low as ~5ps/km (and with low levels of inherent mode-
18
coupling between mode-groups) have been realized, enabling transmission over >1000km
length scales when incorporated with an appropriate amplification approach. Whilst these
results are in themselves technically impressive, the question arises as to how scalable the
basic approach will ultimately prove. To this end, experiments have been undertaken on
both 6-mode and 5-mode FMF with encouraging initial results obtained. However, just as
with the MCF approach it is clear that scaling MDM much beyond this is likely to prove
very challenging, not least in terms of developing scalable, accurate, low-loss mode launch
schemes and ensuring that the required DSP remains tractable.
Whilst zero crosstalk would be ideal, there is a developing school of thought that
contends that mode-coupling is inevitable, that full 2Mx2M MIMO is thus necessary, and
that strong coupling should be actively exploited. If mode-coupling is weak then a data
symbol carried by multiple modes with different group indices will spread in time linearly
with fiber length. In contrast, if the coupling is strong, then the temporal spread follows a
random-walk process, and will scale with the square-root of fiber length. Strong coupling
can therefore potentially reduce the number of MIMO taps required and consequently the
DSP complexity. Indeed this is analogous to spinning of current single-mode fiber during
fabrication to reduce PMD. Similarly, the impact of differential modal gain and loss can in
principle be mitigated by strong mode-coupling over a suitable length scale relative to the
amplifier spacing.
Many successful SDM transmission demonstration have been reported in the past
few years. The table below features some of the most recent of these demonstrations :
Table1.1: State of the art SDM demonstrations
Year Fiber type Number of cores/spatial
modes
Distance (km) Net total capacity
(Tb sโ1)
2013 FMF 3 500 26.63
2013 FMF 6 177 24.58
2013 FMF 3 0.31 57.6
2012 MCF 12 52 1012.32
2012 MCF 7 55 28.88
19
These experiments show the feasibility of scaling capacity using SDM in FMF in combination
with MIMO signal processing.
1.5 Conclusion :
In this chapter we illustrated how the capacity of existing standard single-mode fibers is
approaching its fundamental limit regardless of significant realization of transmission
technologies which allow for high spectral efficiencies. Than we pointed out how space
division multiplexing (SDM) is currently the most promising technique of dealing with this
capacity crunch. Then we explained the concept of SDM and the engineering challenges to
overcome in order to prove it a feasible solution to the problem of saturation of the capacity
of optical transmission systems
20
2. Chapter 2 : Modes determination and optical fiber modeling
2.1 Introduction :
In this chapter we will study the propagation of light in optical fibers in order to determine
the optical modes. Then we will describe the technique we are going to use to simulate the
light propagation in optical fibers.
2.2 Optical Fiber Modes Determination :
In this chapter, we aim to describe mathematically the propagation of electromagnetic
waves in optical fibers. To do so, we will start by exposing Maxwellโs equations then we will
derive and solve the propagation equation.
2.2.1 Maxwellโs equations :
Since 1862, light was considered an electromagnetic phenomenon that can be
described accurately by the following Maxwellโs equations:
Maxwell-Gauss equation for the electric flux:
โโโ . ๏ฟฝโ๏ฟฝ = 0
Maxwell-Thomson equation for the magnetic flux:
โโโ . ๏ฟฝโโ๏ฟฝ =0
Maxwell-Faraday equation of magnetic induction:
โโโ . ๏ฟฝโ๏ฟฝ = โ๐0๐๏ฟฝโโ๏ฟฝ
๐๐ก
Maxwell-Ampรจre equation for electric currents:
โโโ ร ๏ฟฝโโ๏ฟฝ =๐
๐๐ก(ํ0๏ฟฝโ๏ฟฝ + ๏ฟฝโ๏ฟฝ )
Where:
๏ฟฝโ๏ฟฝ and ๏ฟฝโโ๏ฟฝ are the electric and the magnetic fields.
(2.1)
(2.2)
(2.3)
(2.4)
21
The vector ๏ฟฝโ๏ฟฝ is the induced electric polarization, which describes the response of the
medium to the electric excitation.
ฮต0 and ยต0 are respectively the vacuum dielectric permittivity and the vacuum magnetic
permeability and are related to the light celerity in vacuum as follows : ํ0๐0 =1
๐2
2.2.2 Wave equation :
The propagation equation for an electromagnetic pulse in an optical fiber is a second
order partial differential equation that is derived directly from Maxwellโs equations.
First, we start by taking the curl of Maxwell-Faraday equation:
โโโ ร (โโโ ร ๏ฟฝโ๏ฟฝ ) = โ๐0๐
๐๐ก(โโโ ร ๏ฟฝโโ๏ฟฝ ) = โ๐0
๐ยฒ
๐๐กยฒ(ํ0๏ฟฝโ๏ฟฝ + ๏ฟฝโ๏ฟฝ )
Using the usual identity: โโโ ร (โโโ ร ๏ฟฝโ๏ฟฝ ) = โโโ (โ.โโโ ๏ฟฝโ๏ฟฝ ) โ โ2๏ฟฝโ๏ฟฝ we get the following wave equation:
โ2๏ฟฝโ๏ฟฝ = ๐0ํ0๐ยฒ๏ฟฝโ๏ฟฝ
๐๐กยฒ+ํ0๐0ํ0
๐ยฒ๏ฟฝโ๏ฟฝ
๐๐กยฒ
Which can be written as:
โ2๏ฟฝโ๏ฟฝ โ1
๐2๐2๏ฟฝโ๏ฟฝ
๐๐ก2=
1
ํ0๐ยฒ
๐ยฒ๏ฟฝโ๏ฟฝ
๐๐กยฒ
Generally, to identify the induced polarization rigorously, it is necessary to do some
complex quantum-mechanical analysis that will lead to the following equation:
๏ฟฝโ๏ฟฝ (๐ , ๐ก) = ํ0โซ ๐(1)(๐ก โ ๐กโฒ). ๏ฟฝโ๏ฟฝ (๐ , ๐กโฒ)๐๐กโฒ +โ
โโ
+ ํ0โซ โซ ๐2(๐ก โ ๐ก1 , ๐ก โ ๐ก2): ๏ฟฝโ๏ฟฝ +โ
โโ
+โ
โโ
(๐ , ๐ก1)๏ฟฝโ๏ฟฝ (๐ , ๐ก2)๐๐ก1๐๐ก2
+ โซ โซ โซ ๐3(๐ก โ ๐ก1 , ๐ก โ ๐ก2, ๐ก โ ๐ก3) โฎ ๏ฟฝโ๏ฟฝ (๐ , ๐ก1)๏ฟฝโ๏ฟฝ (๐ , ๐ก2)๏ฟฝโ๏ฟฝ (๐ , ๐ก3)๐๐ก1๐๐ก2๐๐ก3 +โ
โโ
+โ
โโ
+โ
โโ
+โฏ
(2.5)
(2.6)
22
where ๐๐ is the so-called jth order susceptibility represented by a tensor of rank j+1.
Obviously, this formula can lead to an enormous complexity. Hence, it is necessary to simplify
it as much as possible without neglecting any important physical effects.
First of all, in optical telecommunications, terms of order higher than three have no
considerable impact on our study.
Furthermore, thanks to the symmetry of silica glass molecules SiO2, ๐(2) vanishes.
This leaves us to consider only the 1st order term of susceptibility which is responsible
for the linear effects and the 3rd order susceptibility accounting for non- linear effects.
Thus, the polarization vector can be separated into a linear term and a non-linear term
as written below:
๏ฟฝโ๏ฟฝ = ๏ฟฝโ๏ฟฝ ๐ฟ + ๏ฟฝโ๏ฟฝ ๐๐ฟ
Where the linear polarization ๏ฟฝโ๏ฟฝ ๐ฟ is given by the relation :
๏ฟฝโ๏ฟฝ ๐ฟ = ํ0โซ ๐1(๐ก โ ๐กโฒ). ๏ฟฝโ๏ฟฝ (๐ , ๐กโฒ)๐๐กโฒ+โ
โโ
And the non-linear polarization ๏ฟฝโ๏ฟฝ ๐๐ฟ is given by:
๏ฟฝโ๏ฟฝ ๐๐ฟ = ํ0โซ โซ โซ ๐3(๐ก โ ๐ก1, ๐ก โ ๐ก2, ๐ก โ ๐ก3): ๏ฟฝโ๏ฟฝ (๐ , ๐ก1)๏ฟฝโ๏ฟฝ (๐ , ๐ก2)๐๐ก1๐๐ก2 +โ
โโ
+โ
โโ
+โ
โโ
Because of the high complexity, it is necessary to make more simplifying
approximations. Therefore, the nonlinear polarization is treated as a small perturbation to the
total induced polarization i.e : ๏ฟฝโ๏ฟฝ ๐๐ฟ = 0โ . It is also useful to work in the frequency domain:
โ2แบผ + ํ0(๐)๐2
๐2แบผ = 0
where แบผ(r,ฯ) is the fourier transform of E(r,t) defined as:
๐ธ(๐, ๐ก) =1
2๐โซ แบผ(๐, ๐) exp(โ๐๐๐ก) ๐๐+โ
โโ
At this level, we define the frequency-dependent dielectric constant ํ(๐) as follows:
ํ(๐) = 1 + ๐(1)(๐)
(2.8)
(2.9)
(2.10)
1
(2.11)
(2.12)
23
where ๐(1)(๐) stands for the Fourier transform of ฯ(1).
Note that generally, ๐1(๐) is complex, so the frequency dependent dielectric constant
is also a complex number. This number can be used to determine the refractive index ๐ and
the absorption given by a coefficient ๐ผ using this formula :
ํ = (๐ +๐๐ผ๐
2๐)2
Therefore, we can deduce explicitly the expressions of ๐ and ๐ผ:
๐(๐) = 1 +1
2๐ ๐[๐~(1) (๐)]
๐ผ(๐) =๐
๐๐๐๐ผ๐[๐~(1)(๐)]
The imaginary part of ฮต is small in comparison to the real part. Thus, we can replace ฮต
by ๐2(๐).
Finally, we can write the wave equation in the frequency domain as:
โ2แบผ + ๐2๐2
๐2แบผ = 0
We can now define the wave number as follows:
๐0 =๐
๐=2๐
๐
Hence, the wave equation becomes:
โ2แบผ + ๐2๐02แบผ = 0
Which has the form of a well-known equation called the Helmholtz equation.
2.2.3 Bessel Equation :
Since the geometry of the fiber is cylindrical, it is more convenient to continue our
study in the cylindrical frame of reference (see figure 1.1) characterized by the following
spatial coordinates(๐, ๐, ๐ง).
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
24
Moreover, let us presume that the electrical field inside the fiber can be decomposed
as follows:
๐ธ = ๐ธ0๐๐๐
Here :
๐ = ๐(๐, ๐) : the transversal part of the electric field.
๐ = ๐(๐ง) = exp (โ๐๐ฝ๐ง) : the amplitude distribution in the plane normal to the
z-axis which includes a constant ๐ฝ known as the propagation constant
๐ = exp (๐๐๐ก) : the time varying component of the electric field which denotes
a monochromatic wave oscillating at the angular frequency ๐.
Substituting this expression of the electric field in the wave equation in the cylindrical
coordinates leads to the equation:
1
๐
๐
๐๐(๐๐
๐๐๐ธ0๐๐๐) +
1
๐2๐2
๐๐2๐ธ0๐๐๐ +
๐2
๐๐ง2๐ธ0๐๐๐ =
๐2
๐2๐2
๐๐ก2๐ธ0๐๐๐
Obviously, the factor ๐ธ0 appears in all terms of the equation, thus, it is cancelled out.
The first term of the wave equation can be equivalently rewritten as:
1
๐
๐
๐๐(๐๐
๐๐๐๐๐) = ๐๐
1
๐
๐
๐๐(๐๐
๐๐๐) = ๐๐
1
๐(๐
๐๐๐ + ๐
๐2
๐๐2๐) = ๐๐ (
1
๐
๐
๐๐๐ +
๐2
๐๐2๐)
The second term becomes:
1
๐2๐2
๐๐2๐๐๐ = ๐๐
1
๐2๐2
๐๐2๐
(2.19)
(2.20)
Figure 6 Cylindrical coordinate system
25
and the third:
๐2
๐๐ง2๐๐๐ = ๐๐
๐2
๐๐ง2๐ = โ๐ฝ2๐๐๐
On the right hand side:
๐2
๐2๐2
๐๐ก2๐๐๐ = ๐๐
๐2
๐2๐2
๐๐ก2๐ = โ๐๐๐๐2
๐2
๐2= โ๐2๐0
2๐๐๐
We notice now, that the factor ZT is common to all terms, so this factor is also
cancelled out.
Therefore, the wave equation becomes:
(1
๐
๐
๐๐๐ +
๐2
๐๐2๐) +
1
๐2๐2
๐๐2๐ โ ๐ฝ2๐ = โ๐2๐0
2๐
We are left with an equation of the amplitude distribution of the electric field in the
plane normal to the propagation direction. To resolve this equation, we have to make a further
factorization: ๐(๐, ๐) = ๐ (๐)ฮฆ(๐)
Substituting this expression in the wave equation and multiplying all terms by ๐2
๐ ฮฆ and
then rearranging all terms leads to the equation below:
โ1
ฮฆ
๐2
๐๐2ฮฆ =
1
๐ (๐2
๐2
๐๐2๐ + ๐
๐
๐๐๐ + ๐2(๐2 โ ๐ฝ2)๐
We see that the right hand side depends only on R whereas the left hand side depends
only on ฮฆ. Therefore, both sides must be equal to a constant which we presume positive and
we will denote it ๐2 .
These assumptions leave us with two independent equations of the form:
1
ฮฆ
๐2
๐๐2ฮฆ+m2 = 0
1
๐ (๐2
๐2
๐๐2๐ + ๐
๐
๐๐๐ + ๐2(๐2 โ ๐ฝ2)๐ = ๐2
ร๐ โ ๐2
๐2
๐๐2๐ + ๐
๐
๐๐๐ + ๐ [(๐2 โ ๐ฝ2)๐2 โ๐2] = 0
By using the abbreviation: ๐ 2 = ๐02 โ ๐ฝ2, the equation above becomes :
(2.21)
(2.22)
(2.23)
(2.24)
(2.25)
(2.26)
26
๐2๐2
๐๐2๐ + ๐
๐
๐๐๐ + ๐ (๐ 2๐2 โ๐2) = 0
Before moving to the mode analysis and resolving the two equations above, it is
necessary to discuss the meaning of the constants ๐, ๐ , ๐๐๐ ๐ฝ.
To do so, all we need is the ray-optical description where the rays make small angles
with the axes.
In this approach, k stands for the wave number as mentioned before.
๐ฝ : the propagation constant, can be seen as the component of the wave number k in
the propagation direction.
Finally, ๐ can be understood as the transverse component of the wave number k.
Note here, that all these assumptions are deduced directly from the equation:
๐ 2 = ๐02 โ ๐ฝ2 which can be written in an equivalent way as: ๐0
2 = ๐ฝ2 + ๐ 2 which
stands for the Pythagorean identity.
The equation :
1
ฮฆ
๐2
๐๐2ฮฆ+m2 = 0
has the general solution :
ฮฆ(๐) = ๐0 cos(๐๐ + ๐0)
Where ๐0 and ๐0 are constants. Surely, ฮฆ(๐) and ๐ฮฆ
๐๐ must be continuous at
๐ = 0 ๐๐๐ ๐ = 2๐ which leads to:
ฮฆ(0) = ๐0 cos(๐0) = ฮฆ(2๐) = ๐0cos (2๐๐ + ๐0)
This relation suggests that m must be an integer.
The radial equation:
(2.27)
(2.29)
27
๐2๐2
๐๐2๐ + ๐
๐
๐๐๐ + ๐ (๐ 2๐2 โ๐2) = 0
is of the form:
๐ฅ2๐ฆโฒโฒ + ๐ฅ๐ฆโฒ + (๐ 2๐ฅ2 โ๐2)๐ฆ = 0
Which stands for a standard Bessel equation. For m integer, the solutions are of the
form:
๐ฆ(๐ฅ) = ๐1๐ฝ๐(๐ ๐ฅ) + ๐2๐๐(๐ ๐ฅ)
Or, in the other form:
๐ฆ(๐ฅ) = ๐3๐ผ๐(๐ ๐ฅ) + ๐4๐พ๐(๐ ๐ฅ)
Where: ๐ฝ๐ and ๐๐ are special functions called Bessel functions.
This type of functions has an oscillatory behavior. Thus, itโs convenient to describe
the propagation inside the core regions. That is why, we are going to consider those functions
whenever the condition: ๐ โค ๐ is satisfied (a stands here for the core radius).
However, ๐ผ๐ and ๐พ๐ are a non-oscillatory Bessel functions that are more convenient
to describe the wave behavior in the cladding region. So, we are going to use these functions
whenever the condition ๐ โฅ ๐ is satisfied.
2.2.4 Mode determination:
Considering that the solution has to be smoothly continuous on the boundaries, which
means that ๐ (๐) and ๐
๐๐๐ are continuous, and the fact that the field amplitude is inversely
proportional to the distance r, we are left with the following condition:
๐ฝ๐(๐ข)
๐ข๐ฝ๐+1(๐ข)=
๐พ๐(๐ค)
๐ค๐พ๐+1(๐ค)
Where ๐ข = ๐ ๐๐๐ and ๐๐๐2 = ๐๐๐2 ๐02 โ ๐ฝ2 is the transverse component of the wave
number inside the core and ๐ค = ๐ ๐๐๐ and ๐ ๐๐2 = ๐ฝ2 โ ๐๐๐
2 ๐02 is the transverse component
inside the cladding region.
Note here that:
(2.30)
(2.31)
(2.32)
(2.33)
28
๐ข2 + ๐ค2 = (๐ ๐๐2 + ๐ ๐๐
2 )๐2 = ๐2(๐๐๐2 ๐0
2 โ ๐ฝ2 + ๐ฝ2 โ ๐๐๐2 ๐0
2) = ๐2๐02(๐๐๐
2 โ ๐๐๐2 )
The quantity: โ๐2๐02(๐๐๐2 โ ๐๐๐
2 ) = ๐๐0โ(๐๐๐2 โ ๐๐๐2 ) is denoted V and called the
normalized frequency.
This constant is of central importance because it resumes all the properties of the
optical fibers. In fact, it includes the geometry of the fiber which is represented by the factor
a, it includes also the chemical properties of the materials which are represented by the
refractive indices of the core and the cladding regions and finally it contains the injected
wavelength which reveals the physical aspect.
The V number can reveal the number of the supported modes when using some well-
known approximation relations.
The solution of the wave equation is:
๐ธ = {๐ธ0๐0๐ฝ๐(๐ ๐) cos(๐๐ + ๐0) exp(๐(๐๐ก โ ๐ฝ๐ง)) ๐๐ ๐ โค ๐
๐ธ0๐๐๐๐พ๐(๐ ๐) cos(๐๐ + ๐0) exp(๐(๐๐ก โ ๐ฝ๐ง)) ๐๐ ๐ โฅ ๐
2.2.5 Solutions for m=0 and m=1 :
Let us now consider the simple case where m=0. This gives the equation:
๐ฝ0(๐ข)
๐ข๐ฝ1(๐ข)=๐พ0(๐ค)
๐ค๐พ1(๐ค)
For a survey of possible solutions, we can consider the following table:
u ๐ฑ๐ ๐ฑ๐ ๐ฑ๐๐๐ฑ๐
๐ฒ๐(๐)
๐๐ฒ๐(๐)
w
0 1 0 โ โ 0
โฎ + + + + +
2.405 0 + 0 0 โ
โฎ - + - - -
3.832 - 0 โ โ 0
(2.34)
(2.35)
29
โฎ - - + + +
5.520 0 - 0 0 โ
โฎ + - - - -
Table 2.1: Determination of possible solutions
Since w cannot be negative, so the regions that lead to a negative sign are the values
of u for which there is no solutions. With a numeric approach, we can compute the exact
values of u and w which we represent in the figure below :
Figure 7 solutions of the eigenvalue equation for m=0 in the (u,w) plane
The denominations 01, 02 and 03 are going to be discussed later.
Since, we have the relation ๐2 = ๐ข2 + ๐ค2 the solutions are the intersection between
the curves of ๐2(๐ข) and ๐ค(๐ข).
Each intersection gives a couple of values (u,w) which means a couple of values for
the propagation constant ๐ฝ: the first one is obtained from u, so it concerns the core region
and the second is derived from w ,thus, it belongs to the cladding region.
A mode is simply defined as the distribution of power in the transverse plane and it
is perfectly characterized by the constant ๐ฝ, thus, we can say for every intersection there is a
supported mode.
30
We also notice from the figure above that the particular value: V=2.405 marks the
transition of a unique solution to more than one solution. So, below this value, the fiber can
support only one mode and it is called single mode fiber.
Above this value, many modes are supported and the fiber is then called multimode fiber.
We can notice also that at higher wavelengths, V is always below 2.405, so the fiber is
obviously a single mode fiber. This implies that the qualifications single mode or multimode
have meaning only in relation to a specific wavelength.
With similar approach, the case where m=1 gives:
Figure 8 solutions of the eigenvalue equation for m=0 and m=1 in the (u,w) plane
Now there are branches of solutions where there where gaps in the m=0 case.
2.2.6 Solutions for m>1 :
With larger m values, one again finds that allowed and forbidden ranges alternate,
with the transitions occurring where the V number equals zeroes of Bessel functions.
We wrap up what we have discussed :
For V<2.405, there is only one branch of solutions.
For Vโฅ 2.405, there are initially two branches
At certain still higher values of V more branches appears.
31
2.2.7 Field amplitude distribution of the modes:
To complete our analysis, we are going to plot the expression of E and for convenience
the radial and the azimuthal terms will be plotted separately.
The field E is given by:
๐ธ = {๐ธ0๐0๐ฝ๐(๐ ๐) cos(๐๐ + ๐0) exp(๐(๐๐ก โ ๐ฝ๐ง)) ๐๐ ๐ โค ๐
๐ธ0๐๐๐๐พ๐(๐ ๐) cos(๐๐ + ๐0) exp(๐(๐๐ก โ ๐ฝ๐ง)) ๐๐ ๐ โฅ ๐
We now see that the modes form a two-parameter family. The first parameter is m. m
indicates as shown in the formula above the angular dependency of the field distribution of
the mode.
For m=0: the distribution is rotationally invariant ie: on any circular path at a given r
one would find a constant field amplitude and thus intensity.
For m=1: the field amplitude will vary according to a sine function of the azimuthal
angle. It therefore, has two zeroes at opposite positions. In between, there are a positive and
a negative branch, or lobe. Either branch contains the maximum of the intensity while the
algebraic sign indicates the phase of the field. Thus, in one lobe the field oscillates in opposite
phase to the other.
For m=2: a circular path would run through two full periods of the sine function; the
intensity pattern then resembles a four-leaved clover. Again, each one of the leaves in
opposite positions has the same phase while the other has the opposite phase.
When m takes even higher values, the angular dependency of the field amplitude has
2m leaves.
The parameter m fixes also which Bessel function governs the field distribution in the
radial direction: we have found a combination of Jm in the core and Km in the cladding.
Since Jm oscillates at any m, there are infinitely many ways to smoothly connect Jm to
Km even after m has been fixed. This set of possibilities is labeled with p, the second
parameter.
We adopt here the terminology of modes as introduced in 1971 by Gloge: modes are
designated with ๐ฟ๐๐๐ on grounds that they are essentially linearly polarized. Index m
designates the number of pairs of nodes in the azimuthal coordinate, and index p counts the
possibilities in the radial coordinate.
32
This explanation can be visualized through the following figures:
Figure 9 Field intensity distribution for m=0
Figure 10 Field intensity distribution for m=1 Figure 11 Field intensity distribution for m=2
33
2.2.8 Analogy with microwave waveguides:
Microwave waveguides are metal pipes with conducting walls. This enforces a node
of the electrical field on the boundary.
In contrast optical fibers are weakly guiding conduits. Therefore, we could use an
approximation, which is not valid in microwave guides whereas there one finds different
types of modes and uses a different terminology. Many of the modes derived here are linear
combinations of metallic waveguide modes, the following table presents the correspondence:
Figure 12 correspondence between modes in metallic guides and optical fibers
The biggest difference may be that metallic waveguides always have a minimum
frequency even for the fundamental mode; below, no mode is supported at all.This can be
traced directly to the conducting walls.
34
2.3 Modeling the optical fiber :
For our modeling purposes we are going to perform a finite element analysis using a FEM
software.
the finite element method (FEM) is a numerical technique for finding approximate solutions
to boundary value problems for differential equations. It uses variational methods to minimize
an error function and produce a stable solution. Analogous to the idea that connecting many
tiny straight lines can approximate a larger circle, FEM encompasses all the methods for
connecting many simple element equations over many small subdomains, named finite
elements, to approximate a more complex equation over a larger domain.
Figure 13 Dividing a sphere into smaller finite elements (meshing) progressively
Figure 14 Different types of finite elements that can be used in meshing
35
We first create the geometry of our fiber and specify the dimensions, then we choose the
material which is in our case silica glass and specify the refractive index for both the core and
the cladding. After that we mesh the structure (divide it into small elements). The simulation
software then applies the FEM (finite element method) algorithm on the meshed structure
and calculates the solutions.
Figure 15 Meshed structure
Figure 16 calculated solution
36
2.4 Conclusion :
In this chapter we demonstrated mathematically and in detail how to determine the optical
modes in the fiber based on maxwellโs equations and the weakly guided modes
approximation. Then we explained how to model the optical fiber using a finite element
analysis software to solve numerically the differential equations describing the light
propagation in the fiber and thus obtain the values of the field distribution of each mode in
the fiber.
37
3. Chapter 3 : Study of key system performance parameters in a MDM transmission system
3.1 Introduction :
After successfully modeling the fiber as described in chapter 2, in this chapter we will
carry out a parametric study on the propagation of the modes in the fiber to evaluate the
influence of each parameter on mode coupling which is the main obstacle for MDM
transmission that limits the performance and reach of MDM systems.
3.2 Definition and perspective :
The concept of mode coupling is very often used e.g. to describe the propagation of
light in some waveguides or optical cavities under the influence of additional effects, such as
external disturbances or nonlinear interactions. The basic idea of coupled-mode theory is to
decompose all propagating light into the known modes of the undisturbed device, and then
to calculate how these modes are coupled with each other by some additional influence. This
approach is often technically and conceptually much more convenient than, e.g., recalculating
the propagation modes for the actual situation in which light propagates in the device [1].
Some examples of mode coupling are discussed in the following:
โข An optical fiber may have several propagation modes, to be calculated for the fiber being
kept straight. If the fiber is strongly bent, this can introduce coupling e.g. from the
fundamental mode to higher-order propagation modes (even to cladding modes), or coupling
between different polarization states. Bend losses can be understood as coupling to non-
guided (and thus lossy) modes.
โข Nonlinear interactions in a waveguide can also couple the modes (as calculated for low light
intensities) to each other. This picture can serve e.g. to describe processes such as frequency
doubling in a waveguide, where the nonlinear coupling mechanism transfers amplitude (and
optical power) from the pumped mode into a mode with twice the optical frequency.
โข In high-power fiber amplifiers, a mechanism has been identified which can couple power
from the fundamental fiber mode into higher-order modes . This mechanism can involve
38
either a KramersโKronig effect or thermal distortions influencing the refractive index profile.
This leads to a strong loss of beam quality above a certain pump power level.
Technically, the mode coupling approach is often used in the form of coupled
differential equations for the complex excitation amplitudes of all the involved modes. These
equations contain coupling coefficients, which are usually calculated from overlap integrals,
involving the two mode functions and the disturbance causing the coupling. Typically, the
applied procedure is first to calculate the mode amplitudes for the given light input, then to
propagate these amplitudes based on the above-mentioned coupled differential equations (e.g.
using some RungeโKutta algorithm), and finally (if required) to recombine the mode fields to
obtain the resulting field distribution.
An important physical aspect of such coherent mode coupling phenomena is that the
optical power transferred between two modes depends on the amplitudes which are already
in both modes. A consequence of that is that the power transfer from a mode A to another
mode B can be kept very small simply by strongly attenuating mode B. In this way, mode B
is prevented from acquiring sufficient power to extract power from mode A efficiently, so
that mode A experiences only little loss, despite the coupling.
3.3 Mode coupling and its origins :
In an ideal fiber, modes propagate without cross-coupling. In a real fiber,
perturbations, whether intended or unintended, can induce coupling between spatial and/or
polarization modes. Throughout this chapter, we consider only coupling between forward-
propagating modes, since it has a dominant effect on the system properties of interest,
including MD and MDL. In multimode transmission fibers, unintended mode coupling can
arise from several sources. These include manufacturing variations causing non-circularity of
the core, roughness at the core-cladding boundary, variations in the core radius, or variations
in the index profile in graded-index fibers. They also include stresses induced by the jacket,
or by thermal mismatches between glasses of different compositions. Finally, mode coupling
can arise from micro-bending, macro-bending, or twists [2].
3.4 Coupled power theory :
In his book โtheory of dielectric optical waveguideโ , D.Marcuse presented an
exact coupled mode theory that is capable, at least in principle, of handling any kind of mode
conversion and radiation effect that may occur in a multimode or single-mode optical
dielectric waveguide. However, the infinite set of coupled integro-differential equations
39
obtained is very hard to solve. In addition, the problem of a randomly deformed multimode
waveguide is too complex to be treated directly with the coupled wave theory. The complexity
of the coupled wave equations is caused by the fact that they contain too much information.
The system of coupled equations contains a detailed description of the phase and amplitude
of all the modes at any point along the waveguide. However, as this report represents a first
approach to studying mode coupling in fibers we will try to use a simpler model.
For many practical purposes, especially MMF systems using spatially and temporally
incoherent light emitting diodes, it would suffice to know the average amount of power
carried by each mode or groups of modes. We are interested to know how the total power
carried by the waveguide is distributed among the modes and how this distribution is
changing as a result of coupling and loss processes.
The coupled power equations hold only for relatively weak coupling. However, this
weak coupling case is most often encountered in practical applications. Weak coupling means
that changes in the power distribution take place over distances that are very long compared
to the wavelength of light. Undesired, random coupling caused by waveguide imperfections
is always weak and is thus accessible to the approximate coupled power theory. This approach
is also suitable for geometrical perturbations, such as bends or tapers, which represent large
deviations from an ideal fiber, provided they vary slowly along the fiberโs length [2,4].
In the simple case of two modes, the power coupling equations [5,ch.5] are :
๐๐1๐๐ง
= โ๐ผ๐1 + โ12(๐2 โ ๐1)
๐๐2๐๐ง
= โ๐ผ๐2 + โ12(๐1 โ ๐2)
With the initial conditions : P1(0)=P01 and P2(0)=P02.
where we have assumed equal attenuation coefficients for the two modes (ฮฑ).
h12: is the intermodal coupling coefficient which measures the probability per unit length of
a transition occurring between modes 1 and 2.
Note that h12 is symmetric which means h12=h21.
(3.1)
(3.2)
40
On the right-hand side, the first term describes loss by power attenuation coefficient
ฮฑ, and the second term describes coupling by non-negative real coefficient h12 [2,ch.11 ;
4,ch.12].
โ12 = โจ|โซ ๐ถ12(๐ง)๐โ๐(๐ฝ1โ๐ฝ2)๐ง๐๐ง
๐ฟ
0
|
2
โฉ
๐ถ12 =๐ํ04โซ โซ ๐ฟ๐2(๐ฅ, ๐ฆ)๐ธ1
โ(๐ฅ, ๐ฆ)๐ธ2(๐ฅ, ๐ฆ)๐๐ฅ๐๐ฆ +โ
โโ
+โ
โโ
Assuming a perturbed index profile of the form :
n2(x, y, z) = n02(x, y) + ฮดn2(x, y)*f(z)
(3.2) โ ๐1 =1
โ12(๐๐2๐๐ง
+ (๐ผ + โ12)๐2)
๐กโ๐๐ ๐๐ ๐ค๐ ๐๐๐๐๐๐ก (3.6) ๐๐ (3.1) ๐ค๐ ๐๐๐ก โถ
๐2๐2๐๐ง2
+ 2(๐ผ + โ12)๐๐2๐๐ง
+ (๐ผ2 + 2๐ผโ12)๐2 = 0
To solve this second order differential equation we have to solve the characteristic
equation first :
๐2 + 2(๐ผ + โ12)๐ + (๐ผ2 + 2๐ผโ12) = 0
The solutions for this second order equation are :
๐1 = โ๐ผ ๐๐๐ ๐2 = โ๐ผ โ 2โ12
which means the general solution for P2 is :
๐2(๐ง) = ๐ถ1๐โ๐ผ๐ง + ๐ถ2๐
(โ๐ผโ2โ12)๐ง
finally , considering the initial conditions we get :
๐2(๐ง) =(๐01 + ๐02)
2๐โ๐ผ๐ง +
(๐02 โ ๐01)
2๐โ(๐ผ+2โ12)๐ง
Similarly :
๐1(๐ง) =(๐01 + ๐02)
2๐โ๐ผ๐ง +
(๐01 โ ๐02)
2๐โ(๐ผ+2โ12)๐ง
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
41
3.5 Simulation and parametric study:
We lunch two modes into the fiber (2 mode fiber, core radius = 5.5um), the
fundamental mode and the LP11 mode :
With the initial powers : P01=2mW and P11=1mW , and we assume a perturbation of
the form :
ฮดn(x, y) = 1.5 10^-2 sin(rc x)cos(rc y);
where rc=(2 pi)/(5.5 10^-6/10) and f(z)=sin(z)
3.5.1 Justification of the perturbation model :
The scattering of light in the fiber is due to small localized changes in the refractive
index of the core and the cladding material. The changes are indeed very localized. We are
looking at dimensions which are less than the wavelength of the light. There are two causes,
both problems within the manufacturing processes. The first is the inevitable slight
fluctuations in the โmixโ of the ingredients. These random changes are impossible to
completely eliminate. The other cause is slight changes in the density as the silica cools and
solidifies.
In the previously adopted model , the peaks ( (x,y,z) spots that correspond to an
extremum of sin(rcx)cos(rcy) and an extremum of sin(z) ) represent the above mentioned
โsmall localized changes in the refractive indexโ.
In order to better understand the index perturbation profile , we drew the following
circular 3D-plot of the sin(2x)cos(2y) function:
Figure 17. LP11 mode : transverse electric field and electric field norm
Figure 18. LP01 mode : transverse electric field and electric field norm
42
The following figure illustrates the evolution of the refractive index of the fiber along
the z axis (which is periodic since f(z)=sin(z)) :
Figure 19 3D plot of the sin(2x)*cos(2y) function
Figure 20 Evolution of the perturbed refractive index as a function of the distance z
43
Of course this is in the case where ฮดn(x, y) = 1.5 10^-2 sin(rc x)cos(rc y) , but if we
consider ฮดn(x, y) = 3 10^-2 sin(rc x)cos(rc y) for example , changes in the refractive index
will be more significant :
The following figure represents the exact plot of the fiberโs cross section (5.5 um core radius) :
Figure 22 Exact plot of the fiberโs cross section (5.5 um core radius)
Figure 21 Evolution of the refractive index of the fiber as a function of the distance z for a higher perturbation value
44
The next figure is a zoomed-in snapshot of the previous figure :
Figure 23 zoomed-in snapshot of the exact plot of the fiberโs cross section (5.5 um core radius)
After calculating the coupling coefficient h12 using the previously mentioned
expressions and the simulation data sets representing the field distribution (using the FEM
simulation software) of the two modes and then implementing them in MATLAB ,we can
easily plot the evolution of the power of both LP01 and LP11 modes :
Figure 24 Power coupling for ฮดn2(x, y) = 1.5 *10^-2 sin(rc x)cos(rc y)
45
The little red circle represents the point of equilibrium (difference between mode
powers is less than 10^-3 mW) and the dotted envelopes represent the power evolution in the
ideal case.
For MDM systems the crosstalk is defined for each mode โiโ by the following formula:
๐๐๐๐ ๐ ๐ก๐๐๐๐ =โ ๐๐โ๐๐
๐๐
Where Pi is the power of mode โiโ and Pj->i is the power transferred from mode โjโ to
mode โiโ.
Using this formula, we plotted the evolution of the crosstalk along the propagation
distance :
Parametric study :
In this section weโll change one parameter (initial powers,index perturbation,modes
โฆ) at a time and observe itโs effect on mode coupling in the fiber, the reference situation being
our first simulation parameters.
3.5.2 Refractive index perturbation :
We can write the index perturbation as a function of a parameter K as follows :
ฮดn2(x, y) = K * 10^-2 sin(rc x)cos(rc y)
Figure 25 Crosstalk between the two propagating modes
46
and then study the evolution of the coupling coefficient and the point of equilibrium
as a function of this parameter :
Figure 26 Coupling coefficient as a function of K
Figure 27 Point of equilibrium as a function of K
47
The following figures illustrate some examples :
Figure 28 Power coupling for ฮดn2(x, y) = 10^-2 sin(rc x)cos(rc y)
Figure 29 Power coupling for ฮดn2(x, y) = 2*10^-2 sin(rc x)cos(rc y)
48
We notice that the more significant the index perturbation, the stronger the coupling
between modes. As we increase the amplitude of the perturbation, the coupling coefficient
increases rapidly and thus the point of equilibrium decreases naturally. The previous figures
demonstrate clearly that the amplitude of the perturbation in the refractive index of the fiber
core is the most influential parameter in mode coupling.
3.5.3 Input powers :
To study the effect of changing the input powers on mode coupling in the fiber, weโll
fix P11 to 1mW and let P01 vary from 1.2 mW to 4 mW. After plotting the results we get this
figure :
Figure 30 Point of equilbrium as a function of P01
49
The following figures illustrate some examples :
Input powers do not affect the coupling coefficients, but increasing the difference
between lunch powers can help by increasing the value of the point of equilibrium. Such
Figure 31 Power coupling for P01=4mW and P11=1mW
Figure 32 Power coupling for P01=1.2mW and P11=1mW
50
solution should be a last resort since increasing the systemโs reach using this method will
increase the energy cost and there is always the limit imposed by the nonlinear effects that
rise as we increase the power injected in the fiber.
3.5.4 Core radius :
If we maintain the exact same perturbation profile and change the core radius
considering ONLY the coupling between the LP01 and the LP11 modes (as we increase the core
radius, the fiber can support more and more modes, a complete study would demand
necessarily to calculate the coupling between each mode and all the other modes, but in this
case weโre only interested in seeing the effect of increasing the core radius only on the
coupling between the LP01 and the LP11 modes ) we get the following results :
As we increase the core radius of the fiber, the coupling coefficient increases rapidly.
This is one of the reasons why in MDM systems we use few mode fibers and not the standard
multimode fibers.
3.5.5 Wavelength :
Changing the wavelength from 1.55 um to 1.3 um had absolutely no effect on the
coupling coefficient, h=3.6787 10^-4 m-1 in this case.
Figure 33 Coupling coefficient as a function of the core radius
51
3.5.6 Modes :
Instead of just studying the coupling of the LP01 with the LP11 mode only, we can
evaluate the coupling coefficients of the LP01 mode with different modes and then compare
the results.
Those results are summarized in this diagram :
Figure 34 Coupling coefficients of the different modes with the fundamental mode
These results show that not all modes are coupled in the same proportions. In this
case, the LP01-LP11 coupling is stronger than the coupling of LP01 and all the rest of the modes.
The next section elucidates the cause of this observation.
3.5.7 Coupling of modes of the same group:
Most unintended random perturbations of transmission fibers are having longitudinal
power spectra that are lowpass, such lowpass perturbations can strongly couple modes
52
having nearly equal propagation constants (small ฮฮฒ), while they weakly couple modes
having highly unequal propagation constants (large ฮฮฒ). Therefore, in glass MMFs, nearly
degenerate polarization modes or spatial modes within the same group are fully coupled after
distances of order 300 m.[2]
To illustrate this we study the coupling between the LP02 mode (neff=1.442315) and the
LP21 mode (neff=1.442750).
The simulation results confirm indeed the above mentioned observations :
Figure 35 Power coupling between the LP02 and the LP21 modes
Coupling coefficient : 0.0141 m-1.
Point of equilibrium : 300 m .
3.5.8 Effect of the number of modes :
53
In this section weโll be treating more than two modes, therefore more than two
coupled differential equations, it is more convenient to write the system of the coupled
differential equations using a matrix notation :
(
๐๐1๐๐ง๐๐2๐๐งโฎ๐๐๐๐๐ง )
= ๐ด๐ (
๐1๐2โฎ๐๐
)
Where ๐ด๐ = (โ(๐ผ + โ โ1๐
๐๐=2 ) โฏ โ1๐โฎ โฑ โฎโ๐1 โฏ โ(๐ผ + โ โ๐๐
๐โ1๐=1 )
)
The general solution of such a system is :
๐ = ๐ถ1๐1๐๐1๐ง + ๐ถ2๐2๐
๐2๐ง +โฏ+ ๐ถ๐๐๐๐๐๐๐ง
where:
ฮป1, ...,ฮปn are the eigenvalues of the matrix An and V1,...,Vn are itโs eigenvectors.
C1 C2,...,Cn are constants and can be determined using the initial conditions of the
injected powers at the input of the fiber.
1) Four mode fiber :
For this simulation we used a fiber with a 7.5 um core radius which supports four
modes : LP01, LP11, LP21 and LP02.
54
LP01 LP11 LP21 LP02
Effective index 1.444256 1.442102 1.439425 1.438756
Table 3.1: Modes effective indices
We calculated the coupling coefficients between all modes :
Coupl.coeff (m-1) LP01 LP11 LP21 LP02
LP01 โ 8.3249e-04 1.2690e-04 9.2291e-05
LP11 8.3249e-04 โ 5.7740e-04 2.3651e-04
LP21 1.2690e-04 5.7740e-04 โ 0.0065
LP02 9.2291e-05 2.3651e-04 0.0065 โ
Table 3.2: Coupling coefficients for the different modes
We then implemented and solved the above mentioned system of coupled differential
equations. The obtained solutions are illustrated in the following figure :
Figure 36 Power coupling in a four mode fiber
55
2) Six mode fiber :
Following the exact same steps as in the case of the four mode fiber, we made another
simulation, this time with a fiber that has a 12 um core radius supporting six modes.
This time we also tried different lunch power combinations to highlight even more
the coupling between the different modes, especially how it is stronger between modes having
close propagation constants.
The results are illustrated in the following figures :
Figure 37 Power coupling in a six mode fiber (first configuration)
56
Figure 38 Power coupling in a six mode fiber (second configuration)
Figure 39 Power coupling in a six mode fiber (third configuration)
57
Figure 40 Power coupling in a six mode fiber (fourth configuration)
Figure 41 Power coupling in a six mode fiber (fifth configuration)
58
The previous figures illustrate that in MDM systems, the more modes we use, the
more transmission channels we have. Unfortunately, as illustrated by the previous figures,
this has an obvious disadvantage : in a two mode fiber, each mode is only coupled with one
other mode while in six mode fiber each mode is coupled with five other modes. Not to
mention the increase in the coupling coefficients when we increase the core radius.
These figures also illustrate more how modes of the same group tend to couple
stronger than the other modes. One exception is the last figure. This figure illustrates how
mode coupling tends to make an equilibrium between mode powers. If this equilibrium is
established from the beginning (if we lunch all modes with the same initial power) there will
always be mode coupling, but it will preserve this power equilibrium throughout the rest of
the distance of propagation.
3.6 Conclusion :
In this chapter we studied the performance of an MDM system by evaluating the main
obstacle for MDM transmission which is โmode couplingโ, as a function of the different
possible parameters. We then interpreted the obtained numerical results in order to determine
how to optimize a transmission system over FMF for the best MDM efficiency.
Conclusion : In this report we studied the concept of optical mode division multiplexed
transmission systems. The continuously increasing demand for worldwide data traffic
requires new, disruptive technologies in optical transmission systems, and mode division
multiplexing (MDM) is one of the candidates currently drawing a lot of attention from
researchers all over the world.
MDM is a special form of space division multiplexing (SDM) and it has been shown
that compared to other SDM techniques, MDM offers a relatively high potential for synergies,
but at the same time, it also comes along with the highest amount of uncertainty of all SDM
solutions concerning its feasibility. It requires fundamental changes in system design (FMF,
mode multiplexer and demultiplexer, MIMO digital signal processing) and comes along with
new transmission limiting effects compared to systems relying on single mode fiber (SMF).
Today the biggest challenge is linear mode coupling leading to intermodal crosstalk in
59
combination with a high differential mode group delay (DMGD) between different spatial
modes.
In this project we focused primarily on a weakly coupled MDM approach where we
try to minimize mode coupling as much as possible and rely on the maximum required MIMO
DSP to compensate for the high DMGD. This is why our main focus was on optimizing system
parameters to minimize mode coupling as much as possible in order to be able to physically
separate the signals of the different spatial modes at the receiving end.
Unfortunately we had no experimental setup to actually perform any MDM
transmission experiments, so we used the coupled power theory to model mathematically the
coupling phenomena that occurs during the propagation of the different modes in the fiber.
We then used a finite element analysis solver to simulate a FMF and obtained the numerical
results presented in this report.
Finally, after analyzing and interpreting these results we proposed the optimal
configuration for each studied system parameter that result in the lowest levels of mode
coupling.
60
4. Bibliography :
[1] http://www.rp-photonics.com/
[2] J. M. Kahn, Keang-Po Ho and M. Bagher Shemirani"Mode Coupling Effects in Multi-Mode
Fibers" Proc. of Optical Fiber Commun. Conf., Los Angeles, CA, March 4-8, 2013
[3] Fedor Mitschke, โFiber optics : physics and technologyโ, Springer, 2009
[4] A.Yariv and P.Yeh Optical Electronics in Modern Communications, sixth Edition
(Academic Press, 2007).
[5] D. Marcuse, โTheory of Dielectric Optical Waveguideโ second ed, Academic press,1997
[6] D. J. Richardson, J. M. Fini and L E. Nelson, โSpace Division Multiplexing in Optical Fibresโ
[7] P. J. Winzer, โOptical Networking Beyond WDMโ, IEEE Photonics J. 4, 647 (2012)
[8] John Crisp and Barry Elliot, โIntroduction to fiber opticsโ , 3rd edition, Elsevier
[9] Christos P. Tsekrekos, Mode group diversity multiplexing in multimode fiber transmission
systems, thesis
[10] " Cisco Visual Networking Index-Forecast and Methodology, 2007-2012," Cisco Systems,
Inc., May 2013.