Download Waveguide&Fiber modes, Saleh&Teich

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
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
E&M 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 Maxwell’s equations
and they can be combined at an angle so that
the resulting wave duplicates itself
Fundamentals of Photonics - Saleh and Teich
Mode number and wave context (metallic reflections)
Fundamentals of Photonics - Saleh and Teich
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
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 E&M
wave propagation analysis)
Fundamentals of Photonics - Saleh and Teich
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
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
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.
Waveguides obey Maxwell equations, which for simple,
isotropic dielectric material with no free charges are:
(Faraday law, Gauss law)
(Ampere law, Gauss law)
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:
The rectangular cross-section has the simplest mathematics. The
wave equation in rectangular coordinates is
Which, using
or
or (simpler)
becomes
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]
Typical end-view representations of some of these modes
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
Modes in cylindrical optical fiber are determined
by the wave equation(s) in cylindrical coordinates:
 2 E z 1 E z 1  2 E z
2



q
Ez  0
2
2
2
r r r 
r
 2 H z 1 H z
1 2H z
2



q
Hz  0
2
2
2
r r
r
r 
Solutions to cylindrical wave equation
are separable in r, φ, and z. The φ and z functions are
exponentials of the form eix. The z function is a propagation
oscillation. The function in φ is an azmuthal function that must
have the same value at (φ + 2π) that it does at φ.
With the azmuthal coordinate separated, the residual wave
equation in the r coordinate is of the form
This is called Bessel’s 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.
Solutions to the cylindrical wave equation
for core/cladding optical fiber profile
For r < a (core), Bessel function = first kind, Jn(ur),
where u2 = k2 –b2 and b < ( k = k1) required.
For r > a (cladding), Bessel function = second kind = Kn(wr),
where w2 = b2 – k2and b < ( k = k2) required
Both kinds of Bessel functions are shown below, plots taken from
http://en.wikipedia.org/wiki/Bessel_function
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
Step index cylindrical waveguide: Bessel
function boundary matching
Step index cylindrical waveguide: Graphical
solutions to boundary matching
Roots defined by:
Defining parameters for cylindrical functions
• For the Bessel equation q2 = ω2εμ – β2
= k2 – β 2 .
q2 is defined as u2 for r < a.
q2 is defined as -w2 for r > a.
• β = bZ is the z component of the propagation constant k = 2π/λ.
The boundary conditions for the Bessel equations can be
solved only for certain values of β, so only certain modes exist.
•
A mode is guided if (n2k = k2) < β < (k2 = n2k) where n1, n2 =
refractive indices of core and cladding, respectively.
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.
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
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 < Ez and it is an HEnm mode.
If transverse E field is larger, Ez < 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.
End view, cylindrical modes
Fiber Optics Communication Technology-Mynbaev & Scheiner
Cylindrical mode characteristics
Each mode has a specific
– Propagation constant β (=bz)
– Spatial field distribution
– Polarization
Step index cylindrical waveguide: mode frequency
Fundamentals of Photonics - Saleh and Teich
Oblique view, cylindrical modes
Superposition gives linearly polarized modes
Composition of two LP11 modes from TE, TM and HE modes
Composition of LP (linearly polarized) modes
Mode degeneracy = modes that can
exist concurrently and independently
LP01 degeneracy:
LP11 degeneracy:
High Order Fiber Modes 2
Fiber Optics Communication Technology-Mynbaev & Scheiner
Below V=2.405, only one mode (= HE11) can be guided;
the fiber is then single-mode.
Number of Modes
Propagation constant of the lowest
mode vs. V number
Graphical Construction
to estimate the
total number of Modes
2
2
V=k 0a(n1 -n2 )=2p
a
NA »
λ0
æ2π ö
çç ÷
÷
÷an1 2Δ
çè λ ø
÷
0
Fundamentals of Photonics - Saleh and Teich
Approximations:
Step index fiber:
The number of modes will be defined
(approximately) by
• Low V,
• higher V,
M4V2/π2+2
MV2/2
Behavior of modes vs normalized propagation
constant b/k and cutoff.
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.
Single-mode fibers: V < 2.405
The only mode that can exist is the HE11 mode.
Birefringence if n1x and n1y are different.
Graded-index Fiber:

r
nr )  n1 1  2D 
a
for r between 0 and a.
for which the number of modes is
M

 2
akn1 )2 D
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.
1-D: reflecting metallic planes
1-D: Dielectric slab planes
For the metallic guide the
number of modes is just the
number of ½ wavelengths that
can fit.
2-D: Rectangular Metallic guide
For dielectric guides it is the
number that can fit but now
limited by the angular cutoff
characterized by the NA of the
guide
2-D: Rectangular dielectric guide
2-D: Cylindrical Dielectric Guide
V=2p
a
NA
λ0
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.
Total energy (power dissipated) in the cladding
The total average power propagated in the cladding
is approximately equal to
Pclad
4

P
3 M
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)
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!