Download Lecture 3: Fiber Modes

EE 230: Optical Fiber Communication Lecture 3
Waveguide/Fiber Modes
From the movie
Warriors of the Net
Optical Waveguide mode patterns
Optical Waveguide
mode patterns seen
in the end faces of
small diameter fibers
Optics-Hecht & Zajac Photo by Narinder Kapany
Multimode 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
Planar 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 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 Components Number and
Fields
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 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
Condition
Number of
modes vs
frequency
Propagation Constants
For a sufficiently low
frequency only 1 mode
can propagate
Fundamentals of Photonics - Saleh and Teich
Planar 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
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 that can propagate
qc
Fundamentals of Photonics - Saleh and Teich
Modes in cylindrical optical fiber
• Determined by solving Maxwell’s
equations 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

 2
 q Hz  0
2
2
r r
r
r 
Key parameters
• q2 is equal to ω2εμ-β2 = k2 – β2. It is
sometimes called u2.
• β is the z component of the wave propagation
constant k, which is also equal to 2π/λ. The
equations can be solved only for certain
values of β, so only certain modes may exist.
A mode may be guided if β 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.
Solutions to Wave Equations
• The solutions are separable in r, φ, and z.
The φ and z functions are exponentials of the
form eiθ. The z function oscillates in space,
while the φ function must have the same
value at (φ+2π) that it does at φ.
• 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.
Resulting 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.
Mode characteristics
Each mode has a specific
• Propagation constant β
• Spatial field distribution
• Polarization
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
Step Index Cylindrical Guide
æ2p ö
2
2
V = k0a(n1 - n2 )» çç ÷
an 2D
÷
çè l ÷
ø 1
Fundamentals of Photonics - Saleh and Teich
High Order Fiber modes
Fiber Optics Communication Technology-Mynbaev & Scheiner
High Order Fiber Modes 2
Fiber Optics Communication Technology-Mynbaev & Scheiner
The 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 V=2.405, 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 ∆.
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
Number of Modes—Step Index Fiber
• At low V, M4V2/π2+2
• At higher V, MV2/2
Graded-index Fiber

r
nr   n1 1  2 
a
For r between 0 and a.
Number of modes is
M

 2
akn1  
2
Comparison of the number of modes
1-d Mirror Guide
M= 2
d
l0
1-d Dielectric Guide
M» 2
d
NA
l0
2-d Mirror Guide
p æ2d ö
÷
M » çç ÷
÷
÷
4 çè l 0 ø
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.
2
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
2-d Dielectric Guide
ö
p æ2d
÷
M » ççç NA÷
÷
÷
4 èl 0
ø
2-d Cylindrical Dielectric Guide
æd
ö
4
÷
M » 2 V 2 = 16 çç NA÷
÷
÷
çèl 0
p
ø
2
V=2p
a
NA
λ0
Power 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.
Total energy in cladding
The total average power propagating 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 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
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!