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PPT – ECE 4853: Optical Fiber Communication Waveguide/Fiber Modes (Slides and figures courtesy of Saleh PowerPoint presentation | free to download - id: 7c8c95-NDFjO

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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

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

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 EM 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

becomes

or

or (simpler)

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

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.

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

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 ?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.

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 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.

End view, cylindrical modes

Fiber Optics Communication Technology-Mynbaev

Scheiner

Cylindrical mode characteristics

- Each mode has a specific
- Propagation constant ß (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 V2.405, 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

Fundamentals of Photonics - Saleh and Teich

Approximations

Step index fiber The number of modes will be

defined (approximately) by

- Low V, M?4V2/p22
- higher V, 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 lt 2.405

The only mode that can exist is the HE11 mode.

Birefringence if n1x and n1y are different.

Graded-index Fiber

for r between 0 and a.

- for which the number of modes is

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

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

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!