Title: EE 230: Optical Fiber Communication Lecture 3 Waveguide/Fiber Modes
1 EE 230 Optical Fiber Communication Lecture
3Waveguide/Fiber Modes
From the movie Warriors of the Net
2Optical Waveguide mode patterns
- Optical Waveguide mode patterns seen in the end
faces of small diameter fibers
Optics-Hecht Zajac Photo by Narinder Kapany
3Multimode Propagation
In general many modes are excited in the
guide resulting in complicated field and
intensity patterns that evolve in a complex way
as the light propagates down the guide
Fundamentals of Photonics - Saleh and Teich
4Planar Mirror Waveguide
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
5Mode Components Number and Fields
Fundamentals of Photonics - Saleh and Teich
6Mode 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
7Planar Dielectric guide
Geometry of Planar Dielectric Guide
The bm all lie between that expected for a plane
wave in the core and for a plane wave in
the cladding
Characteristic Equation and Self-Consistency Cond
ition
Number of modes vs frequency
For a sufficiently low frequency only 1 mode can
propagate
Propagation Constants
Fundamentals of Photonics - Saleh and Teich
8Planar Dielectric Guide
Field components have transverse variation across
the guide, with more nodes for higher
order modes. The changed boundary conditions for
the dielectric interface result in some
evanescent penetration into the cladding
The ray model can be used for dielectric
guides if the additional phase shift due to the
evanescent wave is accounted for.
Fundamentals of Photonics - Saleh and Teich
9Two 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 that can propagate qc
Fundamentals of Photonics - Saleh and Teich
10Modes in cylindrical optical fiber
- Determined by solving Maxwells equations in
cylindrical coordinates
11Key parameters
- q2 is equal to ?2eµ-ß2 k2 ß2. It is
sometimes called u2. - ß is the z component of the wave propagation
constant k, which is also equal to 2p/?. The
equations can be solved only for certain values
of ß, so only certain modes may exist. A mode
may be guided if ß lies between nCLk and nCOk. - V ka(NA) where a is the radius of the fiber
core. This normalized frequency determines how
many different guided modes a fiber can support.
12Solutions to Wave Equations
- The solutions are separable in r, f, and z. The
f and z functions are exponentials of the form
ei?. The z function oscillates in space, while
the f function must have the same value at (f2p)
that it does at f. - The r function is a combination of Bessel
functions of the first and second kinds. The
separate solutions for the core and cladding
regions must match at the boundary.
13Resulting types of modes
- Either the electric field component (E) or the
magnetic field component (H) can be completely
aligned in the transverse direction TE and TM
modes. - The two fields can both have components in the
transverse direction HE and EH modes. - For weakly guiding fibers (small delta), the
types of modes listed above become degenerate,
and can be combined into linearly polarized LP
modes. - Each mode has a subscript of two numbers, where
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 is 0, the mode is meridional.
Otherwise, it is skew.
14Mode characteristics
- Each mode has a specific
- Propagation constant ß
- Spatial field distribution
- Polarization
15w-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
16Step Index Cylindrical Guide
Fundamentals of Photonics - Saleh and Teich
17High Order Fiber modes
Fiber Optics Communication Technology-Mynbaev
Scheiner
18High Order Fiber Modes 2
Fiber Optics Communication Technology-Mynbaev
Scheiner
19The Cutoff
- For each mode, there is some value of V below
which it will not be guided because the cladding
part of the solution does not go to zero with
increasing r. - Below V2.405, only one mode (HE11) can be
guided fiber is single-mode. - Based on the definition of V, the number of modes
is reduced by decreasing the core radius and by
decreasing ?.
20Number 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
21Number of ModesStep Index Fiber
- At low V, M?4V2/p22
- At higher V, M?V2/2
22Graded-index Fiber
- For r between 0 and a.
- Number of modes is
23Comparison 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 mirror 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 Mirror Guide 1-d Dielectric Guide 2-d
Mirror Guide 2-d Dielectric Guide 2-d
Cylindrical Dielectric Guide
24Power propagating through core
- For each mode, the shape of the Bessel functions
determines how much of the optical power
propagates along the core, with the rest going
down the cladding. - The effective index of the fiber is the weighted
average of the core and cladding indices, based
on how much power propagates in each area. - 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.
25Total energy in cladding
- The total average power propagating in the
cladding is approximately equal to
26Power 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
dereased 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 lable
27Macrobending 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!