=Paper=
{{Paper
|id=Vol-1490/paper5
|storemode=property
|title=Calculation of mode set in weakly guiding fiber
|pdfUrl=https://ceur-ws.org/Vol-1490/paper5.pdf
|volume=Vol-1490
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==Calculation of mode set in weakly guiding fiber==
https://ceur-ws.org/Vol-1490/paper5.pdf
Computer Optics and Nanophotonics
Calculation of a mode set in weakly guiding fibers
Aleksandrova A.V.
Samara State Aerospace University
Abstract. The aim of the paper is to calculate the mode set in weakly guiding
fibers using a calculation of the eigenmodes of the optical fiber with a step
index of refraction. The superposition of modes with various properties of self-
reproduction for a given set of physical characteristics was determined. The
propagation of light signals in the non-ideal optical waveguides has been
studied by computer simulation using the commercial software BeamProp.
Keywords: optical waveguides, weakly guiding fibers, calculation of the
eigenmodes
Citation: Aleksandrova AV. Calculation of a mode set in weakly guiding
fibers. Proceedings of Information Technology and Nanotechnology (ITNT-
2015), CEUR Workshop Proceedings, 2015; 1490: 37-44. DOI: 10.18287/1613-
0073-2015-1490-37-44
Introduction
The optical fiber is considered now to be the perfect physical environment for
information transfer as well as the most preferable environment for significant data
flows over considerable distances. Optical fibers have wider applications in computer
networking and telecommunications thanks to a number of the features inherent in
optical waveguides.
Success achieved in the production of optical fibers allows information to be
transferred at high speeds over hundreds of kilometres without regeneration of a
signal. High-noise immunity, safety of the transmitted data and electromagnetic
compatibility of communication channels are serious arguments in favour of fiber-
optical systems.
There are two types of optical fibers: single-mode and multi-mode. Fibers with
various refractive index profiles (step-index profile or gradient-index profile) are
used, depending on the field of application. For step-index optical fiber, a refractive
index profile characterized by a uniform refractive index within the core and a sharp
decrease in the refractive index at the core-cladding interface is used so that the
cladding is of a lower refractive index. Gradient-index fiber is an fiber whose core has
a refractive index that decreases with increasing radial distance from the optical
axis of the fiber. In this investigation, we have looked at fibers with a step profile
index of refraction because of their wide extension.
The term "mode division multiplexing" (MDM) is used for multimodal optical
fibers when describing methods for data transmission channel multiplexing, with each
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Information Technology and Nanotechnology (ITNT-2015)
Computer Optics and Nanophotonics Aleksandrova A.V. Calculation of mode set…
spatial fiber mode being treated as a separate channel that carries its own signal [1,2].
The essence of mode division multiplexing is as follows: as a linear superposition of
fiber modes, laser beams can be used to generate signals that will effectively transmit
data in a physical carrier - a multimodal fiber; the data transmitted can be contained
both in the modal composition and in the energy portion associated with each laser
mode [3-13]. In addition, the division of the vortex basis connected with orbital
angular momentum is especially perspective [7-14].
Recent years have witnessed a number of research activities in the field of singular
optics [15, 16, 25-27]. In terms of quantum theory vortex modes they are
characterized as spin-orbital states that the current speed of transfer on one fiber
allows to increase repeatedly without additional polarizing multiplexing.
1. Mode set in weakly guiding stepped-index fibers
Use of the diffraction optical elements is the most popular method for devices of
generation and selection of vortex modes [8, 10, 17-21]. For most popular commercial
fibers, the core-cladding index contrast, Δn = n1–n2, is less than 1%. For such fibers,
termed weakly guiding fibers, assuming n1≅n2, in place of the hybrid modes of the
propagating electromagnetic field we can consider their linearly polarized
superpositions. Considering that for the LP-mode the transverse field is essentially
linearly polarized, a complete set of modes takes place when only one electric and one
magnetic component are predominant.
The aim of this study is to simulate an ideal optical fiber with a superposition of
linearly polarized modes determined by the given physical characteristics of a
possible set of modes and their superpositions, with various properties of self-
reproduction.
Moreover, the intention is to study the propagation of light signals in non-ideal
optical waveguides with a Beam PROP simulation tool (RSoft Design, USA), which
implements the well-known Beam Propagation Method (BPM).
To research a set of modes, we considered the weakly guiding cylindrical optical
fiber with a step profile of index of refraction. The core radius is a, the cladding
radius is b and the respective refractive indices of the core and cladding are n1 and n2
(Fig.1). The electromagnetic field extending in such a waveguide is conveniently
described using the Bessel functions [22-24].
Fig. 1. – The structure of a typical single-mode fiber
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Information Technology and Nanotechnology (ITNT-2015)
Computer Optics and Nanophotonics Aleksandrova A.V. Calculation of mode set…
Approximation LP-modes are applicable for weakly guiding fibers:
J p (u pq r / a)
, 0r a
cos( p) J p (u pq ) (1)
LPpq (r, ) .
sin( p) K p (wpq r / a)
, ar b
K p (wpq )
In Eq. (1), the first-kind Bessel functions J p x describe the field in the fiber
core, whereas the modified Bessel functions K p x are used for the cladding.
uJ m u wKm w (2)
0,
J m1 u Km1 w
2
where the parameters u2 w2 V 2 , V a n12 n22 form the cut-off number and λ
is the wavelength of laser light in air. The cut-оff number V, which includes the main
parameters of fibers and laser radiation, is the number of modes propagating in the
fiber. The numerical simulation parameters are as follows: the core radius is a=5μm,
the cladding radius is b=62.5 μm, and the respective refractive indices of the core and
cladding are n1=1.45 and n2=1.44. For example, when λ=0.633 m and V≈8,4398,
fiber with the above parameters in addition to the fundamental mode LP01 will
behave as LP02, LP03, LP11, LP12, LP21, LP41. Figure 2 shows cross-section
distributions for some modes for a stepped-index fiber with the cut-off number
V=8.4398.
Fig. 2. – Superposition of the (p,q) modes: (1,1)+(–1,1) with different complex coefficients: (a)
transverse amplitude distribution, (b) transverse intensity distribution, and (c) phase in the
plane z=0, and (d) phase distribution at distance z=200 μm
We consider the propagation of a linear superposition of LP-modes in an ideal
stepped index optical fiber:
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Information Technology and Nanotechnology (ITNT-2015)
Computer Optics and Nanophotonics Aleksandrova A.V. Calculation of mode set…
U0 (r, ) Cpq pq (r, ) (3)
p,q
where C pq are the complex coefficients and pq (r, ) are the modes at z=0, whose
angular component is represented in a different way without a loss of generality:
pq (r, , z) exp i pq z Tp Rpq r
J p (u pq r / a)
J (u ) , 0 r a
exp i pq z exp ip
p pq (4)
.
p pq r / a) , a r b
K ( w
K p (wpq )
Fig. 3. – The (p,q) modes: (0,3), (1,2), (2,1), (4,1): (a) transverse amplitude distribution
(negative), (b) transverse intensity distribution (negative) in the plane z=0; transverse phase
distribution (white: zero phase, black: 2π) in the planes (c) z=0 and (d) z=100 μm
Models of optical fibers with the different parameters, structures and shapes have
been established in the software BeamProp in order to research, analyse and compare
non-ideal optical waveguides (Fig. 4).
Fig. 4. – Various models of waveguides
The main characteristic of optical fiber is the set of modes extending within it
(Fig. 5).
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Information Technology and Nanotechnology (ITNT-2015)
Computer Optics and Nanophotonics Aleksandrova A.V. Calculation of mode set…
Fig. 5. – Some of the modes supported by a simple 3D fiber structure
Figure 6 shows the distribution of radiation S-shaped optical fiber and power of
the propagating radiation.
Fig. 6. – The completed s-bend circuit in the CAD window; the simulation results found using
the arc waveguides
Figure 7 shows the distribution of radiation of X-shaped coupler and the power of
the propagating radiation zero and first modes.
By modelling the propagation of radiation in optical fibers, it is possible to
determine their output values and make sure the elements have the given parameters,
and to predict their behaviour depending on external influences.
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Fig. 7. – X-shaped coupler
Conclusions
Using the BeamProp program it is possible to investigate various samples of non-
ideal optical fibers with a step profile of index of refraction, according to following
characteristics:
─ profile and index of refraction;
─ difference of indices of refraction;
─ waveguide length;
─ a type of function on which the index of refraction changes;
─ etc.
Studying the resistance of vortex modes to fiber bends under various
characteristics of a core and cover of optical fiber is of great interest for the
researchers.
Acknowledgements
This work was financially supported by the Russian Ministry of Education and
Science.
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