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ECE 4853: Optical Fiber Communication Waveguide/Fiber Modes (Slides and figures courtesy of Saleh

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Title: ECE 4853: Optical Fiber Communication Waveguide/Fiber Modes (Slides and figures courtesy of Saleh


1
ECE 4853 Optical Fiber Communication
Waveguide/Fiber Modes (Slides and figures
courtesy of Saleh Teich) (Modified, amended and
adapted by R. Winton)
From the movie Warriors of the Net
2
Waves bounded by geometry optical waveguide mode
patterns
  • Optical Waveguide mode patterns seen in the end
    faces of small diameter fibers

Optics-Hecht Zajac Photo by Narinder Kapany
3
EM wave bound by two metallic planes Wave path
analysis
The planar mirror waveguide can be solved by
starting with Maxwells Equations and the boundary
condition that the parallel component of the E
field vanish at the mirror or by considering that
plane waves already satisfy Maxwells
equations and they can be combined at an angle so
that the resulting wave duplicates itself
Fundamentals of Photonics - Saleh and Teich
4
Mode number and wave context (metallic
reflections)
Fundamentals of Photonics - Saleh and Teich
5
Mode velocity and polarization degeneracy
Group Velocity derived by considering the
mode from the view of rays and geometrical optics
TE and TM mode polarizations
Fundamentals of Photonics - Saleh and Teich
6
Planar slab dielectric wave guide
Geometry of planar dielectric guide
Characteristic equation and self-consistency
condition for identifying allowed values of qm
(Characteristic equation consequence of either
geometrical or EM wave propagation analysis)
Fundamentals of Photonics - Saleh and Teich
7
Planar slab dielectric wave guide modes
The bm must be between that expected for a plane
wave in the core and that expected for a plane
wave in the cladding
Number of modes vs frequency
Propagation Constants
Note For a sufficiently low frequency only 1
mode can propagate
8
Planar dielectric layer bound modes and
evanescent penetration into cladding
The field components have a transverse variation
across the guide. There are more nodes for
higher-order modes. The changed boundary
conditions for the dielectric interface result
in an evanescent penetration into the cladding.
Fundamentals of Photonics - Saleh and Teich
9
Dielectric layer bounded waves
The ray model is mathematically accurate for
dielectric guides if the additional phase shift
due to the evanescent wave is acknowledged.
10
Waveguides obey Maxwell equations, which for
simple, isotropic dielectric material with no
free charges are
(Faraday law, Gauss law)
(Ampere law, Gauss law)
11
And the relationships between field types (for
simple, isotropic dielectric material with no
free charges) are
  • And if we put all of these equations together
    (vector analysis) we end up with the wave
    equation

which is the same for the magnetic field
12
The rectangular cross-section has the simplest
mathematics. The wave equation in rectangular
coordinates is
Which, using
becomes
or
or (simpler)
13
The mathematics that fit the rectangular geometry
(shown) and this equation are in the form of sin(
) and cos( ) functions. For example the Ez 0
(TE mode forms) will be
And there are two mode numbers, one for each
geometrical dimension
m mode number for x-direction number of ½l
within boundaries x 0, a n mode number for
y-direction number of ½l within boundaries y
0, b
14
Typical end-view representations of some of these
modes
15
Two Dimensional Rectangular Planar Guide
In two dimensions the transverse field depends on
both kx and ky and the number of modes goes as
the square of d/l The number of modes is limited
by the maximum angle qc that can propagate
Fundamentals of Photonics - Saleh and Teich
16
Modes in cylindrical optical fiber are determined
by the wave equation(s) in cylindrical
coordinates

17
Solutions to cylindrical wave equation
  • are separable in r, f, and z. The f and z
    functions are exponentials of the form eix. The
    z function is a propagation oscillation. The
    function in f is an azmuthal function that must
    have the same value at (f 2p) that it does at
    f.

With the azmuthal coordinate separated, the
residual wave equation in the r coordinate is of
the form
This is called Bessels equation and will have
solutions that are (a) Bessel functions of the
first kind (for the core) and (b) of second kind
(for the cladding). The solutions for the core
and cladding regions must match at the boundary.
18
Solutions to the cylindrical wave equation for
core/cladding optical fiber profile
For r lt a (core), Bessel function first kind,
Jn(ur), where u2 k2 b2 and b lt ( k k1)
required.
For r gt a (cladding), Bessel function second
kind Kn(wr), where w2 b2 k2and b lt ( k
k2) required
Both kinds of Bessel functions are shown below,
plots taken from http//en.wikipedia.org/wiki/Bess
el_function
19
Bessel functions (shown) are not unlike sin(mx)
cos(mx) functions associated with the rectilinear
geometries, except their mahematical profile is
in the r coordinate. Jn(x) is not a closed
function but one generated by an infinite series.
D. Gloge, Weakly guided fibers, Applied Optics,
Oct 1971, pp 2252 - 2257
20
Step index cylindrical waveguide Bessel
function boundary matching
21
Step index cylindrical waveguide Graphical
solutions to boundary matching
Roots defined by
22
Defining parameters for cylindrical functions
  • For the Bessel equation q2 ?2eµ ß2 k2
    ß2.
  • q2 is defined as u2 for r lt a.
  • q2 is defined as -w2 for r gt a.
  • ß bZ is the z component of the propagation
    constant k 2p/?. The boundary conditions for
    the Bessel equations can be solved only for
    certain values of ß, so only certain modes exist.
  • A mode is guided if (n2k k2) lt ß lt (k2 n2k)
    where n1, n2 refractive indices of core and
    cladding, respectively.

23
Combined parameter (normalized frequency
parameter)
An index value V, defined as the normalized
frequency is used to determines how many
different guided modes a (fiber can support.
The normalized frequency is related to the
cylindrical geometry by V
(2pa/l)x(NA) for which a radius of the core.
24
w-b Mode Diagram
Straight lines of dw/db correspond to the group
velocity of the different modes. The group
velocities of the guided modes all lie between
the phase velocities for plane waves in the core
or cladding c/n1 and c/n2
25
Types of cylindrical modes defined by the
cylindrical Bessel functions
  • The E field component is transverse to the z
    direction. Ez 0 and it is a TEnm mode.
  • The H field component is transverse to the z
    direction. Hz 0 and it is a TMnm mode.
  • If neither Ez nor Hz 0 then it is a hybrid
    mode.
  • If transverse H field is larger, Hz lt Ez and it
    is an HEnm mode.
  • If transverse E field is larger, Ez lt Hz and it
    is an EHnm mode.
  • For weakly guided fibers (small D), these type of
    modes become degenerate and combine into linearly
    polarized LPjm modes.
  • Each mode has a subscript of two numbers,n and m.
    The first is the order of the Bessel function
    and the second identifies which of the various
    roots meets the boundary condition. If the first
    subscript n 0, the mode is meridional.
    Otherwise, it is skew.

26
End view, cylindrical modes
Fiber Optics Communication Technology-Mynbaev
Scheiner
27
Cylindrical mode characteristics
  • Each mode has a specific
  • Propagation constant ß (bz)
  • Spatial field distribution
  • Polarization

28
Step index cylindrical waveguide mode frequency
Fundamentals of Photonics - Saleh and Teich
29
Oblique view, cylindrical modes
30
Superposition gives linearly polarized modes
Composition of two LP11 modes from TE, TM and HE
modes
31
Composition of LP (linearly polarized) modes
32
Mode degeneracy modes that can exist
concurrently and independently
LP01 degeneracy
LP11 degeneracy
33
High Order Fiber Modes 2
Fiber Optics Communication Technology-Mynbaev
Scheiner
34
Below V2.405, only one mode ( HE11) can be
guided the fiber is then single-mode.
35
Number of Modes
Propagation constant of the lowest mode vs. V
number
Graphical Construction to estimate the total
number of Modes
Fundamentals of Photonics - Saleh and Teich
36
Approximations
Step index fiber The number of modes will be
defined (approximately) by
  • Low V, M?4V2/p22
  • higher V, M?V2/2

37
Behavior of modes vs normalized propagation
constant b/k and cutoff.
38
Cutoff conditions and evanescent content.
  • For each mode, there is some value of the
    normalized frequency V below which the mode will
    not be contained (and guided) because the Bessel
    function (of the second kind) for the cladding
    does not go to zero with increasing r. The
    evanescent content of the mode is increased as
    the boundary condition is approached.
  • Below V 2.405, only one mode ( HE11) can exist
    in the fiber.
  • It is then called a single-mode fiber.
  • Based on V, the number of modes can be reduced by
    decreasing the core radius and by decreasing the
    relative refractive index ? between core and
    cladding.

39
Single-mode fibers V lt 2.405
The only mode that can exist is the HE11 mode.
Birefringence if n1x and n1y are different.
40
Graded-index Fiber
for r between 0 and a.
  • for which the number of modes is

41
Summary comparison of the number of modes
The V parameter characterizes the number of
wavelengths that can fit across the core guiding
region in a fiber. For the metallic guide the
number of modes is just the number of ½
wavelengths that can fit. For dielectric guides
it is the number that can fit but now limited by
the angular cutoff characterized by the NA of
the guide
1-D reflecting metallic planes
1-D Dielectric slab planes
2-D Rectangular Metallic guide
2-D Rectangular dielectric guide
2-D Cylindrical Dielectric Guide
42
Power propagated along the core
  • For each mode, the radial profile of the Bessel
    function Jn(ua) determines how much of the
    optical power propagates along the core, with the
    rest going down the cladding.
  • The propagation is cited in terms of a weighted
    index. The effective index of the fiber is the
    weighted average of the core and cladding indices
    and is based on how much power propagates in each
    regime.
  • For multimode fiber, each mode has a different
    effective index. This is another way of
    understanding the different speed that optical
    signals have in different modes.

43
Total energy (power dissipated) in the cladding
  • The total average power propagated in the
    cladding is approximately equal to

44
Power Confinement vs V-Number
This shows the fraction of the power that is
propagating in the cladding vs the V number for
different modes. V for constant wavelength, and
material indices of refraction is proportional
to the core diameter a As the core diameter is
decreased, more and more of each mode propagates
in the cladding. Eventually it all propagates
in the cladding and the mode is no
longer guided (Note misleading ordinate label)
45
Macrobending Loss
One thing that the geometrical ray view point
cannot calculate is the amount of bending
loss encountered by low order modes. Loss goes
approximately exponentially with decreasing
radius untill a discontinuity is reached.when
the fiber breaks!
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