Modal Analysis of Step-Index Fibers

8/11/2019 Modal Analysis of Step-Index Fibers

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ECE 4006/5166 Guided Wave Optics

Robert R. McLeod, University of Colorado 65

Step-index silica fiber

Material and fabrication

Types and naming of modes

Derivation and solution of the WE

Solution of the WE

TE/TM modes

Hybrid modes LP modes

Modal analysis of step-index fibersIntroduction


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Optical properties

SiO2 & SiO2/GeO2

0.6 0.8 1 1.2 1.4 1.6 1.8 2

l @mmD

1.47

1.48

1.49

1.5

1.51

ng

0.6 0.8 1 1.2 1.4 1.6 1.8 2

l @mmD

-600

-500

-400

-300

-200

-100

0

D

@

spH

mn

mk

LD

Loss[dB/km]

Modal analysis of step-index fibersMaterials and fabrication

Absorption:

Large at small wavelength due to Rayleigh scattering off of inhomogeneities

in the glass

Large at long wavelength due to molecular vibrational resonances

(absorption) farther out in the IR.

Is remarkably low loss around 1.5 microns

Dispersion:

Pure and doped silica have nearly identical dispersion

Zero dispersion around 1.3 microns

0.6 0.8 1 1.2 1.4 1.6 1.8 2

l@mmD

1.44

1.45

1.46

1.47

1.48

n


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Fabrication of the preform



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Drawing fiber

4000o F

10-20m/sec

Several km of fiber are typical on a single reel.



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Types of modes

Exact solution

Meridonal rays (=0): TE & TM

Skew rays (0): HE & EH

Weakly guiding approximation

LP modes

Can be expressed as sum of TE,

TM, HE, EH that become

degenerate for smalln

Modal analysis of step-index fibersTypes of modes


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Wave equation in cylindrical

for Ezand Hz

Monochromatic vector WE

zEyExEE zyx 2222

++=

r Cartesian vector

Laplacian

0

2

0

2

=

=+ EEkE

rrr

Scalar simplification

( ) ( ) ( ) ( )zzrEzrErzrEzrE zr ,,,,,,,, ++=r

Write E in cylindrical coordinates:

E radial

E azimuthal

E radial Ez

Ez does not couple to Er and E fields:

Modal analysis of step-index fibersDerivation of the wave equation

011

0

2

02

2

2

2

2

2

0

2

=+

+

+

=+

zzzz

zz

Ekz

EE

rr

Er

rr

EkE

Scalar WE for Ez

2 in cylindrical coord.

Ez

Note that Hzobeys the same W.E.


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Solution of the WESeparation of variables

Modal analysis of step-index fibersSolution of the wave equation

( ) ( ) ( ) ( )zZrRzrEz =,, Separation of variables

Plug into wave equation. Note now ordinary differential eq.

011 202

2

2

2

22

2

=+++

+ ZRk

dzZdR

ddRZ

rdrdR

rdrRdZ

Multiply by r2/RZ

0111 22

02

22

2

2

2

22 =++

+

+ rk

dz

Zd

Zr

d

d

dr

dRr

dr

Rdr

R

Assume sinusoidal dependence in z: Z(z) = exp[-j z]

( ) 011 222022

2

22 =+

+

+ rk

d

d

dr

dRr

dr

Rdr

R

Depends on r Depends on rDepends on

( ) 22

2222

02

22 11

=

=+

+

d

drk

dr

dRr

dr

Rdr

R


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Solution of the WERadial and azimuthal functions

( )

jeA

d

d ==

2

2

2

Solution of azimuthal equation. Since must be periodic,=integer.

02

222

0

2

2

22 =

+

+ R

rkr

dr

dRr

dr

Rdr

Solution of radial equation.

Note similarity to d2/dr2 + (k2 kz2-(/r)2) = 0. Solutions:

Modal analysis of step-index fibersSolution of the wave equation

0 2 4 6 8 10

z

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4

z

0.5

1

1.5

2

2.5

3

3.5

4( )0

222

=+

+ fzfzfz

( ) ( )42

2

largecos

zzJ

zz( ) z

zz ezK

21

large

( )0

222

=+

+ fzfzfz

zero

1 2.4048 3.8317 5.1356 6.3802 7.5883 8.7715

2 5.5201 7.0156 8.4172 9.7610 11.0647 12.3386

3 8.6537 10.1735 11.6198 13.0152 14.3725 15.7002

4 11.7915 13.3237 14.7960 16.2235 17.6160 18.9801

5 14.9309 16.4706 17.9598 19.4094 20.8269 22.2178

Zeros of J:

Core: nco > N Core: ncl < N


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Form of fields

( ) ( )

( )

( ) ( )

( )

>

0 has two possible angularphases and all have two possible polarizations

Modal analysis of step-index fibersLP modes


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Summary The solution of the wave-equation in cylindrical

coordinates yields J in the core (like cos) and K in thecladding (like e-x).

Matching these functions across the boundary with theEM boundary conditions yields a complexcharacteristic equation for which has different

solutions m for each angular mode number. If there is no angular variation (=0), the solutions are

TE0m and TM0m.

Otherwise the solutions involve all 6 field componentsand are labeled HEm and HMm.

The HE11 mode is ~linearly polarized and never cutsoff.

The number of modes in terms of Vis roughly equal tothe number in a slab waveguide of the sameparameters, squared.

TE/TM modes occur for V> 2.405.

Increases in the decrease the maximum m by ~1/2. In the limit of weak guiding, sets of these modes have

degenerate effective indices and can thus be linearly

combined. The resulting LP modes are linearly polarized, have

small axial fields and have two degeneratepolarizations and (for modes with angular variation)two angular phases.

Modal analysis of step-index fibersSummary

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Modal Analysis of Step-Index Fibers