Download Introduction to modes and their designation in circular and elliptical

Introduction to modes and their designation in circular and elliptical fibers
Deepak Kumar and P. K. Choudhury
Faculty of Engineering, Multimedia University, Cyberjaya 63100, Malaysia
共Received 29 June 2006; accepted 2 February 2007兲
Some basic ideas are presented to clarify the concept of modes in optical fibers. We describe the
formation of modes and the indexing schemes in circular and elliptical core fibers. © 2007 American
Association of Physics Teachers.
关DOI: 10.1119/1.2711825兴
I. INTRODUCTION
Optical fibers 共or waveguides兲 are important components
of integrated optical communication systems and information technology.1–10 The aim of this paper is to describe the
modes in optical fibers, which are bounded mediums in the
transverse direction. These modes consist of certain patterns
of electromagnetic waves formed within the fiber due to the
structurally imposed 共transverse兲 boundaries on the propagating fields. Each mode is a pattern of electric and magnetic
field distributions that is repeated along the fiber at equal
intervals. In modern communication systems we use as few
modes as possible 共preferably a single mode兲 so that the
interaction among several modes is minimized. Because optical fibers are used as basic mediums for transmission of
optical signals, it is useful to make a detailed study of mode
identification and designation in optical fiber communication
systems.
II. MODES IN WAVEGUIDES
The propagation of light along a waveguide can be described in terms of a set of guided electromagnetic waves
called the guided modes of the waveguide. These guided
modes are known as the bound or trapped modes of the
waveguide; each guided mode is a pattern of E- and H-field
distributions that is repeated along the guide at equal intervals. Only a discrete number of modes can propagate along
the guide. The wavelength of the mode and the size, shape,
and nature of the waveguide determine which modes can
propagate. For monochromatic light fields of angular frequency ␻, a mode traveling in the positive z-direction 共along
the fiber axis兲 at time t is described by ei共␻t−kzz兲, where kz is
the z-component of the wave vector k = 2␲ / ␭ 共␭ is the freespace wavelength兲. For guided modes kz has only certain
discrete values determined by the requirement that the electric field must satisfy Maxwell’s equations and the E- and Hfield boundary conditions at the core-cladding interface.
III. TRANSVERSE ELECTRIC AND TRANSVERSE
MAGNETIC MODES
Many propagating modes can coexist in hollow metallic
waveguide. The proper mode for propagation is selected by
suitably launching the signal, and other less important modes
must be eliminated or isolated. A mode filtering scheme may
also be implemented depending on the requirements of the
system. It is, therefore, necessary to identify the existing
modes. At the metallic boundary of a microwave guide transverse electric 共TE, Ez = 0兲 and transverse magnetic 共TM, Hz
= 0兲 modes exist along the guide. No fields exist outside the
guide.
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There are also dielectric waveguides or optical fibers in
which the central core region of higher permittivity or refractive index is surrounded by a cladding of lower refractive
index. In these dielectric waveguides the electric and magnetic field components can exist along the boundary of the
guide, that is, the boundary between the cylindrical core and
its cladding.
In addition to the modal 共or wave兲 approach, another
method for characterizing the propagation characteristics of
light in an optical fiber is geometrical optics or ray tracing,
which provides a fairly good approximation to the light guiding properties of optical fibers when the ratio of the fiber
radius to the wavelength is large 共the small wavelength
limit兲. The advantage of this approach is that, compared with
the exact electromagnetic wave analysis, it gives a more
physical interpretation of the characteristics of the propagation of light in the guide.
The concept of a light ray is very different from that of a
mode. A guided mode traveling in the z-direction along the
fiber axis can be decomposed into a family of superimposed
plane waves that collectively form a standing-wave pattern
in the direction transverse to the fiber axis. The phases of the
plane wave are such that the envelope of the collective set of
waves remains stationary. Because we can associate a light
ray with any plane wave that is perpendicular to the phase
front of the wave, the family of plane waves corresponding
to a particular mode forms a set of rays. Such a set of rays is
called a ray congruence. Each ray belonging to this particular
set travels in the fiber at the same angle relative to the fiber
axis. Because only a certain number of discrete guided
modes exist in a fiber, the possible angles of the ray congruences corresponding to these modes are limited to the same
number. Although a simple ray picture appears to allow rays
at any angle greater than the critical angle to propagate in a
fiber, discrete propagation angles result when the phase condition for standing waves is introduced into the ray picture.
We first briefly consider the modal fields formed in dielectric slab waveguides 共see Fig. 1兲 in which the guiding region
of refractive index n1 is bounded by two plane surfaces so
that it is separated from the cladding regions of lower refractive index n2. This geometry represents the simplest form of
an optical waveguide and can serve as a model for optical
fibers. A cross-sectional view of the slab waveguide looks
the same as the cross-sectional view of an optical fiber cut
along its axis. Figure 1 shows the field patterns of several
low-order TE modes. The order of a mode is equal to the
number of field zeros across the guide. The order of the
mode is also related to the angle that the ray congruence
corresponding to this mode makes with the plane of the
waveguide 共or the axis of the fiber兲; the steeper the angle, the
higher the order of the mode. The plots show that the electric
© 2007 American Association of Physics Teachers
546
Fig. 1. Electric field distributions for some lower-order guided in a symmetrical slab waveguide.
fields of the guided modes are not completely confined to the
central dielectric slab 共they do not go to zero at the guidecladding interface兲, but partially extend into the cladding 共the
evanescent field兲. The fields vary harmonically in the guiding
region of refractive index n1 and decay exponentially outside
of this region. For low-order modes the fields are concentrated near the center of the slab 共or the axis of an optical
fiber兲 with little penetration into the cladding region. For
higher order modes the fields are distributed more toward the
edges of the guide and penetrate further into the cladding
region.
In TE modes the axial or longitudinal component of the
electric field is zero, and the electric field lies in the transverse plane. In TM modes there is no component of the
magnetic field in the direction of propagation; the magnetic
field lies entirely in the transverse plane. In contrast, channel
waveguides 共such as the circular or elliptical fiber兲 can sustain only hybrid modes. The electric and the magnetic fields
in these hybrid modes are not confined to the transverse
plane, but there are also components of these fields in the
axial direction.
To attain a detailed understanding of the optical power
propagation mechanism in a fiber, it is necessary to solve
Maxwell’s equations subject to cylindrical boundary conditions at the core-cladding interface of the fiber. The two
types of rays that can propagate in a fiber are meridional
共Fig. 2兲 and skew rays 共Fig. 3兲. Meridional rays are confined
to the meridian planes of the fiber, which are planes that
contain the axis of symmetry of the fiber 共the core axis兲.
Because a given meridional ray lies in a single plane, its path
is easy to track as it travels along the fiber. Meridional rays
can be divided into two general classes—bound rays that
propagate along the fiber axis according to the laws of geo-
Fig. 2. Ray optics representation of a meridional ray traveling in an optical
fiber core.
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Fig. 3. Cross-sectional view of skew reflections of a ray in an optical fiber
core.
metrical optics, and unbound rays that are refracted out of
the fiber. Skew rays are not confined to a single plane, but
instead follow a helical path along the fiber. These rays are
more difficult to track as they travel along the fiber, because
they do not lie in a single plane. Before giving an account of
hybrid modes according to the wave theory, we mention that,
according to the simpler ray theory, TE and TM modes are
represented by meridional rays and hybrid modes are represented by skew rays.
IV. RADIATION AND LEAKY MODES
The solution of Maxwell’s equations subject to the cylindrical boundary conditions at the fiber core-cladding interface shows that, in addition to supporting a finite number of
guided modes, an optical fiber has a continuum of radiation
modes that are not trapped in the core and guided by the fiber
but are still solutions of the same boundary-value problem.
Each fiber has an acceptance angle given by sin−1关共n21
− n22兲1/2 / n兴, where n is the refractive index of the medium
from which the light is launched into the fiber. For a particular light ray launched into the fiber with an angle higher than
the acceptance angle of the fiber, the ray will not be guided
by the core region. Thus, the radiation field results from the
optical power being refracted out of the core. Because of the
finite radius of the cladding, some of this radiation is trapped
in the cladding and causes cladding modes to appear. As the
core and the cladding modes propagate along the fiber, mode
coupling occurs between the cladding modes and the higherorder core modes. Such a coupling occurs because the electric fields of the guided core modes are not entirely confined
to the core but extend partially into the cladding 共as shown in
Fig. 1兲. A transfer of power back and forth between the core
and the cladding modes thus occurs, resulting into a loss of
power from the core modes and causing the generation of
radiation modes.
In addition to the bound and radiation modes, a third category called leaky modes is also present in optical fibers.
These leaky modes are only partially confined to the core
region, and are attenuated by continuously radiating their
power out of the core as they propagate along the fiber. To
study the behavior of such modes we use the differential
Deepak Kumar and P. K. Choudhury
547
Table I. State of the E- and H-field components for different types of modes
in optical fibers.
Nomenclature
TEM 共transverse
electromagnetic兲
Longitudinal
components
Transverse
components
Ez = 0
E T, H T
Hz = 0
Ez = 0
Hz ⫽ 0
Hz = 0
Ez ⫽ 0
Hz ⫽ 0
Ez ⫽ 0
TE
TM
HE or EH
E T, H T
E T, H T
E T, H T
Fig. 4. Polarization states of hybrid modes in optical fibers.
equations and boundary conditions similar to those used to
calculate the phenomenon of tunneling in quantum mechanics.
V. HYBRID MODES
There are many types of propagating modes. To analyze
optical fibers with circular cross sections, we use a cylindrical polar coordinate system 共r , ␾ , z兲 and assume the direction
of propagation is along the z-axis.
Hybrid modes are those where all six field components are
nonzero. These modes can be classified on the basis of amplitude coefficient ratio of the axial components of the electric and magnetic fields,10,11 the polarization states,12,13 and
the field configurations14 of the propagation modes. If the
amplitude of the axial component of the electric field is
strong compared to that of the axial component of magnetic
field, the modes are called EH modes; otherwise the modes
are HE.4 The classifications based on the amplitude coefficient ratio and the polarization states have disadvantages for
certain classes of refractive index profiles.15,16 The amplitude
coefficient ratios of axial components also cannot be utilized
for the designation of hybrid modes in more complicated
three-layer cylindrical dielectric waveguides.17 In terms of a
ray analysis, the rays corresponding to the HE and the EH
modes in optical fibers are circularly polarized skew rays in
opposite polarization states to each other12,13 共see Fig. 4兲.
Thus, the descriptive names HE and EH can be used without
confusion about their polarization states. However, the polarization states and the associated field configurations of hybrid modes change with increasing frequency in certain
classes of fibers. The designation of a mode should be determined by the characteristic curve obtained from the dispersion relation and not change with frequency. To be more
explicit, the characteristic curve for a system of cylindrical
optical fiber through which electromagnetic waves are propagating can be obtained by plotting the dispersion relation 共or
the eigenvalue equation兲 for the system against the propagation vector. The curve will determine the variation of the
modal characteristics of the fiber with the propagation vector.
The amplitude coefficient ratio P of the axial components
of electric and magnetic fields is given by P = −共␻␮0 / kz兲
⫻共Hz / Ez兲, where kz is the z-component of the propagation
vector of the guided mode and ␮0 is the free-space permeability. Snitzer8 suggested that the modes for which P ⬎ 0
should be designated as EH, and those for which P ⬍ 0 as
HE. Khun11 used the sign of P to classify hybrid modes in
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homogeneous fibers with finite cladding. Similarly, Tanaka et
al.18 proposed a mode designation scheme based on the
multilayer approximation of the refractive index profile, and
the approximate factorization of eigenvalue equations.
Hybrid modes have, in general, no circular symmetry but
in some cases some circularly symmetric TE and TM modes
can exist. Other less important modes, such as radiating or
evanescent modes, also exist. Each propagating mode requires two indices for its identification. For example, there
are HElm and EHlm modes, where l, m take integral values.
We shall later interpret the indices l and m. Table I shows the
description of the field components for various kinds of
modes that can exist in dielectric optical fibers.1
VI. CUTOFF CONDITION OF MODES
We now introduce the concept of a cutoff condition. Assume power is launched into an optical fiber and excites the
fiber modes. A given mode, after traveling some distance
along the fiber core, will deliver its power to the cladding
section, and this mode is then cut off. That is, the cutoff
condition is the point at which a mode is no longer bound to
the fiber core, and the power of that mode goes to the cladding. For a given mode the core parameter ␥1 is defined by
␥1 = 共k2n21 − kz2兲1/2 ,
共1兲
where k = 2␲ / ␭; ␭ is the free-space wavelength, n1 is the
refractive index of the core, and kz is the longitudinal component of the propagation vector associated with the mode.
The limiting condition of a mode to remain guided is ␥1 = 0,
that is kz = n1k.
Two other useful parameters, the cladding parameter ␥2
and the normalized frequency parameter V, are
␥2 = 共kz2 − k2n22兲1/2 ,
共2兲
and
V=
2␲a 2
共n1 − n22兲1/2 ,
␭
共3兲
respectively, where a is the core radius of the fiber and n2 is
the refractive index of the cladding material. The normalized
dimensionless frequency parameter determines how many
modes a fiber can support. Except for the lowest-order HE11
mode, each mode can exist only for values of V that exceed
a certain limiting value 共which depends on the mode兲. The
modes are cut off when kz = n2k, which occurs when V
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548
Table II. The cutoff parameters for different types of modes in a simple
dielectric optical fiber.
Mode
Fig. 5. Plot of Bessel function used for calculating the cutoff conditions of
modes.
ⱕ 2.405,1 the value at which the lowest-order Bessel function
becomes zero. The HE11 mode has no cutoff and ceases to
exist only when the core diameter is zero as explained in
greater detail in the following. A fiber remains single mode if
V ⬍ 2.405; this is the principle on which the design of singlemode fibers is based.
As V approaches the cutoff for a particular mode, more of
the power of that mode exists in the cladding. At the cutoff,
the mode becomes radiative with all the optical power of the
mode residing in the cladding region. Thus, a mode remains
guided as long as kz satisfies the condition n2k ⱕ kz ⱕ n1k,
which gives the permissible range for kz for a bound solution. Therefore, the boundary condition between guided
modes and leaky modes is defined by the cutoff condition
kz = n2k. If kz is smaller than n2k, power leaks out of the core
into the cladding region. Leaky modes can carry a significant
amount of optical power in short fibers. Most of these modes
disappear after a few centimeters, but a few have sufficiently
low losses to persist in fiber lengths of a kilometer.
The simplest type of optical fiber has a circular cross section with a step variation 共that is, no grading兲 in the refractive indices of the core 共n1兲 and the cladding 共n2兲, with n1
⬎ n2. Such fibers are called step-index fibers. The cutoff condition of the modes in a step-index circular fiber is
␥1a = xlm = u
共l = 1,2,3, . . . 兲,
共4兲
where xlm is the mth root of the equation
Jl共xlm兲 = 0,
共5兲
and Jl is the Bessel function of order l. As stated, the HE11
mode is never cut off and exists for all frequencies. This
fundamental mode continues to propagate when all the other
modes are cut off. The equation that describes the cutoff for
the HE11 mode is
␥1a = 0.
共6兲
From Eq. 共6兲 we see that the HE11 mode is cut off only when
the core diameter is zero. The cutoff equation for the EHlm
modes is also Eq. 共5兲, but with the additional constraint that
xlm ⫽ 0. Figure 5 shows the cutoff condition for the HE1m and
the EH1m modes. The remaining HElm modes satisfy the cutoff equation
冉 冊
␧1
u
Jl共u兲,
+ 1 Jl−1共u兲 =
␧2
l−1
共7兲
with ␧1 and ␧2 as the permittivity values of the core and the
cladding regions, respectively.
There are two types of hybrid modes for each integer
value of l ⬎ 1. Both the EH1m and HE1共m+1兲 modes have the
same cutoff frequency due to their degeneracy at this fre549
Am. J. Phys., Vol. 75, No. 6, June 2007
HE11
TE01, TM01
HE21
HE12, EH11
HE31
EH21
HE41
TE02, TM02
HE22
Cutoff parameter u
Mode
Cutoff parameter u
0.0
2.405
2.42
3.83
3.86
5.14
5.16
5.52
5.53
EH31
EH51
HE13, EH12
HE32
EH41
HE61
EH22
HE52
6.38
6.41
7.02
7.02
7.59
7.61
8.42
8.43
quency. However, they are not actually degenerate because at
frequencies other than cutoff, they have different values of
the longitudinal components of the propagation vectors.
Table II shows the first few modes and their respective cutoff
parameter values.1 These parametric values are of much importance in fabricating optical fibers for certain applications.
For example, a single-mode fiber must be designed such that
its cutoff value is less than or equal to 2.405. Depending on
the cutoff parameter, the number of sustainable modes in a
fiber can be estimated, which has great importance in designing few-mode fibers.
For each l there are m possible roots of Eq. 共7兲, namely
m = 1 , 2 , . . . , m. For l = 0 there are two linearly polarized
modes,1 namely the TE0m and TM0m, for which either the
electric field or the magnetic field in the direction of propagation is zero. These modes have circular symmetry.
Neither TE01 nor TM01 is the lowest-order mode, that is,
neither has a nonzero cutoff V-value. The lowest-order mode
has a cutoff frequency corresponding to the first zero of J0共u兲
at u = 2.405, where the parameter u contains the fiber dimension a and the free-space wavelength ␭. The lowest-order
mode is the HE11 mode or the fundamental mode of a cylindrical light guide fiber.10 In other words, the cutoff frequency
of the HE11 mode is zero corresponding to the first zero of
J1共u兲 at u = 0.
In the interval 共0, 2.405兲 of u only the HE11 mode exists.
The electric field of the HE11 mode has two orthogonal polarizations. By carefully decoupling these polarized modes,
we can obtain a truly single-mode single polarization transmission, free from intermodal interferences.
The next higher order modes consist of a group designated
as TE01, TM01, and HE21 in the interval 2.405⬍ u ⬍ 3.832. If
we count the twofold degeneracy of the HE21 mode, there are
six modes in this interval. The next higher order modes have
a cutoff frequency at u = 3.832 and consist of HE12, EH11,
and HE31. Each of them has a twofold degeneracy. Thus, for
3.832⬍ u ⬍ 5.136, the total number of possible modes is 12.
It is necessary to limit the size of the fiber to obtain a singlemode structure to eliminate higher-order mode groups. As
the fiber cross section increases, the number of modes that
can be sustained within the fiber increases.
An optical fiber channel and the number of modes it can
sustain can be viewed in analogy with a passage and the
number of ways a person can pass through the passage, respectively. An extremely narrow passage allows a person to
Deepak Kumar and P. K. Choudhury
549
follow a single path 共single mode兲, whereas a person will
have many different possible ways to pass through a wider
passage 共multimode兲.
VII. WEAKLY GUIDING FIBERS AND LINEARLY
POLARIZED MODES
The exact modal analysis for optical fibers is formidable.
A simpler but highly accurate approximation can be used,
called the weak guidance approximation, which holds for
共n1 − n2兲 / n1 1. Its implementation gives weakly guided
modes as the approximate solutions. These modes are called
linearly polarized 共LP兲 modes.8 These modes are due to the
superposition of HE or EH modes. The linearly polarized
modes are not exact modes of the step-index fiber. Each linearly polarized mode has many degenerate modes. Thus, linearly polarized modes are not true modes, because they are
not exact. If the optical fibers have only a few propagation
modes, the difference between the corresponding true modes
for each linearly polarized mode is of practical importance
for designing fibers and calculating fiber bandwidths.18,19 In
such cases the approximation is not useful.
Apart from degeneracy, there is also an instability of the
lobe orientations of the fields. This instability arises due to
the dependence of the phase velocity of the waves on the
refractive index of the medium, which in turn depends on the
direction of propagation of the waves. In weakly guiding
fibers we can construct modes whose transverse fields are
polarized in one direction. In elliptical fibers the fiber can
support two types of modes, one polarized predominantly in
the x-direction and the other polarized predominantly in the
y-direction.
In fibers used for telecommunication purposes 共n1
− n2兲 / n1 is usually less than 0.02. In contrast, for strong guidance 共n1 − n2兲 / n1, though less than 1, can be close to 1.
Strong guidance is not important in optical communications,
but is important in illumination engineering. When we want
to reduce the number of guided waves, the weak guidance
condition should be satisfied. Thus, for a single- or fewmode fiber 共n1 − n2兲 / n1 1. This practical requirement in
communication channels permits us to simplify the mathematical analysis by considering what is known as the scalar
wave equation in terms of the field ␺, which can represent
any of the Cartesian components of the E- and H-fields. The
boundary conditions also become simpler so that ␺ and its
radial derivative may be treated as continuous across the
core-cladding boundary.2 The modes are now designated as
LPlm modes. The suffix l represents the lth-order Bessel
function, which corresponds to the cutoff condition for the
mode, and m enumerates the successive zeros of the corresponding Bessel function. If we show the positions of the
antinodes of a particular mode on the cross section of the
fiber, the mode LPlm will have 2l antinodes in a ring of a
given radius, and there will be m such rings on the cross
section. In Fig. 6共a兲 we show the antinodes of LP31, and in
Fig. 6共b兲 the antinodes of LP52.
The lowest-order mode 共fundamental mode兲, which is
never cut off, is represented by LP01, which corresponds to
the HE11 mode of Sec. II. The correspondence between the
mode descriptions is HE11 = LP01 mode, TE01, TM01, HE21
= LP11 mode, etc. We note that for LP0m modes, there is no
radial antinode at the center of the cross section. The features
of LP modes are that each LP0m mode is derived from an
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Am. J. Phys., Vol. 75, No. 6, June 2007
Fig. 6. Illustrations of the 共a兲 antinodes of LP31 mode, and 共b兲 antinodes of
LP52 mode.
HE1m mode; each LP1m mode is derived from TE0m, TM0m,
and HE2m modes; and each LPnm mode 共n ⱖ 2兲 is derived
from an HEn+1,m and an EHn−1,m mode.
The previous description discusses the discrete modes for
the usual optical fibers/waveguides. Choudhury et al.20–24
have reported the bunching tendency of modes for parabolic
cylindrical dielectric and chiral waveguides. The description
of those types of modes is beyond the scope of the present
article. Interested readers may refer to Ref. 25.
VIII. MODES IN ELLIPTICAL FIBERS
So far we have considered only fibers with a circular cross
section. Now, we consider fibers with an elliptical cross section. Because the ellipse is less symmetrical than the circle,
there can be two orientations for the field configuration. The
fields in an elliptical fiber can be described in terms of
Mathieu functions.
A hybrid mode9 is designated by e or o, where e and o
stand for the even mode and the odd mode, respectively. The
axial magnetic fields of even modes are represented by the
even Mathieu functions, and the axial electric fields of odd
modes are represented by the odd Mathieu functions.26,27 The
Deepak Kumar and P. K. Choudhury
550
even and the odd hybrid modes are represented by eHEnm
and oHEnm, respectively. We can describe the EH modes in a
similar manner.
For linearly polarized modes in elliptical fibers28 the hybrid LP11 mode is split into LP11 odd and LP11 even modes,
with well-defined mode intensity patterns. The LP11 odd and
LP11 even modes have significantly different cutoff wavelengths, which allow the existence of a wavelength range
within which only LP01 and LP11 even modes are supported
by the fiber.
The elliptical core fibers that support two stable special
modes, the LP01 and LP11 even/odd modes, are called elliptical core two-mode fibers. One application of these fibers is
to interferometric model/polarimetric sensors, which can be
used to measure strain and/or temperature.29 In fibers with
more complicated cross-sectional shapes, mode designation
becomes more difficult, and the description of these is beyond the scope of this paper.
ACKNOWLEDGMENTS
This paper is dedicated to our revered teacher, the late
Professor Prasad Khastgir, intellectual discussion with whom
has always been the source of inspiration for us as well as for
several generations of students at the Institute of Technology,
Banaras Hindu University 共Varanasi, India兲. We are also
thankful to the late Mr. K. K. Verma, formerly with the D. L.
W. Intermediate College 共Varanasi, India兲 for some timely
discussions.
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