Step-index silica fiber

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 fibers

–Introduction



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

n g

0.6 0.8 1 1.2 1.4 1.6 1.8 2

l @mmD

-600

-500

-400

-300

-200

-100

0

D@

spêH

mn

mk

LD

Loss

[d

B/k

m]


–Materials 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



Fabrication of the preform





Drawing fiber

4000o F

10

-20

m/sec

Several km of fiber are typical on a single reel.





Types of modes

• Exact solution

– Meridonal rays (ν=0): TE & TM

– Skew rays (ν≠0): HE & EH

• Weakly guiding approximation

– LP modes

– Can be expressed as sum of TE,

TM, HE, EH that become

degenerate for small δn


–Types of modes



Wave equation in cylindricalfor Ez and Hz

Monochromatic vector WE

zEyExEE zyxˆˆˆ 2222 ∇+∇+∇=∇

r Cartesian vector

Laplacian

0

2

0

2

=

⋅

∇−∇=+∇ EEkE

rrr

ε

εε

Scalar simplification

( ) ( ) ( ) ( )zzrEzrErzrEzrE zrˆ,,ˆ,,ˆ,,,, φφφφφ φ ++=

rWrite E in cylindrical coordinates:

E radial

E azimuthal

E radial Ez

Ez does not couple to Er and Eφ fields:


–Derivation of the wave equation

011

0

2

02

2

2

2

2

2

0

2

=+∂

∂+

∂

∂+

∂

∂

∂

∂

=+∇

zzzz

zz

Ekz

EE

rr

Er

rr

EkE

εφ

ε Scalar WE for Ez

∇2 in cylindrical coord.

Ez

Note that Hz obeys the same W.E.



Solution of the WESeparation of variables


–Solution of the wave equation

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

Plug into wave equation. Note now ordinary differential eq.

011 2

02

2

2

2

22

2

=Φ+Φ+Φ

+

+Φ ZRk

dz

ZdR

d

dRZ

rdr

dR

rdr

RdZ ε

φ

Multiply by r2/RΦZ

0111 22

02

22

2

2

2

22 =++

Φ

Φ+

+ rk

dz

Zd

Zr

d

d

dr

dRr

dr

Rdr

Rε

φ

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

( ) 011 222

02

2

2

22 =−+

Φ

Φ+

+ rk

d

d

dr

dRr

dr

Rdr

Rβε

φ

Depends on r Depends on rDepends on φ

( ) 2

2

2222

02

22 11

νφ

βε =Φ

Φ−=−+

+

d

drk

dr

dRr

dr

Rdr

R



Solution of the WERadial and azimuthal functions

( ) νφφνφ

jeA

d

d ±±=Φ∴Φ−=

Φ 2

2

2

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

02

222

0

2

2

22 =

−−+

+ R

rkr

dr

dRr

dr

Rdr

νβε

Solution of radial equation.

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



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

[ ] ( ) 0222 =−+′+′′ fzfzfz ν

( ) ( )42

2

large cos πνπ

πν −− → zzJzz

( ) z

zzezK

− → πν 21

large

[ ] ( ) 0222 =−−+′+′′ fzfzfz ν

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 Jν:

Core: nco > N Core: ncl < N



Form of fields

( )( )( )

( )( )( )

>

<=

>

<=

−

−

−

−

areerKD

areerkJBzrH

areerKC

areerkJAzrE

zjj

r

zjj

r

z

zjj

r

zjj

r

z

βνφν

βνφν

βνφν

βνφν

αφ

αφ

,,

,,

Forms of modes:

22

0

22

22

0

22

clclr

cocor

nNkk

Nnkkk

−=−=

−=−=

βα

β

where, as before,



Through Maxwell’s curl equations, we can find all the transverse fields in

terms of these two longitudinal fields:

( )( )( )( )φ

εωε

β

φ

β

βφ

φ

ωµ

β

φ

β

βφ

β

εωε

β

ωµ

∂

∂

∂

∂

−

−

∂

∂

∂

∂

−

−

∂

∂

∂

∂

−

−

∂

∂

∂

∂

−

−

−=

+=

+=

−=

ziz

i

zz

i

zz

i

zz

i

E

rr

H

k

j

r

H

rr

E

ik

j

r

EH

rk

j

r

r

HE

rk

j

H

H

E

E

0

22

22

0

22

22

0

0

i=core or cladding

We thus have 5 unknowns A,B,C,D and β and four continuity conditions

on Ez, Eφ, Hz, and Hφ leaving one free variable which is the total mode

amplitude.



Characteristic equationand relations between amplitudes


–Characteristic equation

Steps:

1. Find the fields Eφ, and Hφ from the forms of Ez and Hz using the

expressions on the last page.

2. Set them equal across the boundary to generate four equations that

are linear in the four unknowns A,B,C,D and transcendental in β.

3. Require that the four equations be satisfied by setting the determinant

of the 4 × 4 matrix equation equal to zero.

4. The resulting equation involves only β and is the characteristic

equation for the modes. (See section 4.5 for the details)

( )( )

( )( )

( )( )

( )( )

′+

′

′+

′=

+

aK

aKk

akJk

akJk

aK

aK

akJk

akJ

ka rr

rcl

rr

rco

rr

r

rr

r

rr αα

α

αα

α

α

βν

ν

ν

ν

ν

ν

ν

ν

ν22

2

22

211

For each ν = 0,1,2… there will be solutions with m = 0,1,2…radial zeros.

This is analogous to the slab WG but with a new mode number ν.

Using the same B.C.s one can relate the field amplitudes:

( ) ( )aKCakJA rr ανν =Continuity of Ez:

( ) ( )aKDakJB rr ανν =Continuity of Hz:

( )( )

( )( )

′+

′=

+

aK

aK

akJk

akJB

ka

jA

rr

r

rr

r

rr αα

α

αωµ

βν

ν

ν

ν

ν22

0

11Continuity of Eφ:

( )( )

( )( )

+=

′+

′

22

2211

rrrr

rcl

rr

rco

kB

aK

aKn

akJk

akJnajA

ααα

α

βν

ω

ν

ν

ν

νContinuity of Hφ:

The last two are redundant, which is to be expected since one variable

should remained undetermined. We need both forms, as we’ll see.



TE and TM modes


–TE and TM modes

( )( )

( )( )

( )( )

( )( )

022

=

′+

′

′+

′

aK

aKk

akJk

akJk

aK

aK

akJk

akJ

rr

rcl

rr

rco

rr

r

rr

r

αα

α

αα

α

ν

ν

ν

ν

ν

ν

ν

ν

For meridional rays that pass through the axis,

ν = 0. The characteristic equation becomes:

If this term = 0, then A = 0 by the continuity of Eφ:

( )( )

( )( )

011

22

0

=

′+

′=

+

aK

aK

akJk

akJB

ka

jA

rr

r

rr

r

rr αα

α

αωµ

βν

ν

ν

ν

ν

( ) ( ) 0== akJAaKC rr νν αand by the continuity of Ez: so Ez = 0 for all

space and thus these are TE modes.

( )( )

( )( )

( )( )

( )( )

00

1

0

1

0

0

0

0 =

−−=

′+

′

aK

aK

akJk

akJ

aK

aK

akJk

akJ

rr

r

rr

r

rr

r

rr

r

αα

α

αα

α

The characteristic equation for the TE modes is thus:

Yielding the eigenvalues TE

m0β

An analogous argument for the second term=0 yields B=D=0 so Hz = 0 for

all space and thus TM modes. The characteristic equation for is: TM

m0β

( )( )

( )( )

00

1

0

1

2

=

−

−

aK

aK

akJk

akJ

n

n

rr

r

rr

r

cl

co

αα

α Like the slab, this looks like

TE but with a term like the

ratio of the indices squared.



TE01 modeEφ, Hr and Hz

If A = 0 and ν = 0, Er = 0 (see Eq. 4.21 in book) and Eφ varies like J1.

Electric field. Color plot = Ez,

arrows = Ex and Ey

Note that Ez = 0 (TE).

Magnetic field. Color plot =

Hz, arrows = Hx and Hy

Note that Hz ≠ 0 (not TM).

Note that power is flowing out

of the screen.


–TE and TM modes



TM01 modeEr, Ez and Hφ

If B = 0 and ν = 0, Hr = 0 (see Eq. 4.21 in book) and Hφ varies like J1.


arrows = Ex and Ey

Note that Ez ≠ 0 (not TE).



Note that Hz = 0 (TM)


of the screen.

In the weak guiding limit nco ≈ ncl these two modes will be ~ degenerate.


–TE and TM modes



0 0.5 1 1.5 2

U

- 0.75

- 0.5

- 0.25

0

0.25

0.5

0.75

1

0 2 4 6

U

- 0.75

- 0.5

- 0.25

0

0.25

0.5

0.75

1

Graphical solutionof TE/TM characteristic equation


–TE and TM modes

TE/TM characteristic equation

with normalizations and

polarization factor defined for slab

( )( )

( )( )WKW

WK

UJU

UJP

0

1

0

1 −=γ

2

=

cl

cop

n

nγ for TM, = 1 for TE 222 UVW −=

0

108.2

0015.

5.1

m][ 1

m][ 5

0

=

=

=

=

=

=

mode

co

N

V

n

n

µ

µa

δ

λ

22

72.7

02.

5.1

m][ 1

m][ 5

0

×=

=

=

=

=

=

mode

co

N

V

n

n

µ

µa

δ

λ

V

LHS, TE

LHS, TM

RHS

2.4048 5.5201TM difference exaggerated for clarity

Lig

ht

lin

e

Cu

toff



TE/TM modesObservations


–TE and TM modes

• Unlike the slab, there is a definite cutoff V below which TE and

TM modes do not propagate.

• The first two (one TE and one TM) modes are allowed when

U = V = the first zero of J0 = 2.405

• The second two (one TE and one TM) modes are allowed when

U = V = the second zero of J0 = 5.520

• TM modes should have greater U than TE,

thus NTM < NTE.

22

0

22

0 sin NnakkkaakakU cozcocox −=−==≡ − θ

TE dispersion diagram for fiber with same properties of slab

waveguide example, previous.

a ω / c = a k0

a k

z=

a k

0N

0.25 0.5 0.75 1 1.25 1.5 1.75 2

4

6

8

106

1

=

=

co

cl

n

n

Not allowed

Radiation

Bound



Hybrid modesNature of fields

( ) ( )( )

( ) ( )( )

( ) ( )( )

( ) ( )( )

+−

−

×

+−

−

+=

+−+−

+−+−

−

WKW

WKWK

UJU

UJUJ

n

n

WKW

WKWK

UJU

UJUJ

WUn

N

cl

co

cl

ν

νν

ν

νν

ν

νν

ν

ννν

22

22

11

1111

2

1111

2

22

2


–Hybrid modes

If ν ≠ 0, we must use the complete characteristic equation.

Normalizing:

If ν ≠ 0, then neither of the terms in brackets is zero. From the boundary

conditions,

( )( )

( )( )

′+

′=

+

aK

aK

akJk

akJB

ka

jA

rr

r

rr

r

rr αα

α

αωµ

βν

ν

ν

ν

ν22

0

11

( )( )

( )( )

+=

′+

′

22

2211

rrrr

rcl

rr

rco

kB

aK

aKn

akJk

akJnajA

ααα

α

βν

ω

ν

ν

ν

ν

If either A or B is zero, then both must be zero, and all fields are zero

everywhere.

Thus, the modes have both Ez and Hz and are neither TE nor TM.

If B < A (Hz < Ez), the modes are labeled HE.

If A < B (Ez < Hz), the modes are labeled EH.



0 2 4 6

U

- 0.75

- 0.5

- 0.25

0

0.25

0.5

0.75

1

Graphical solutionSingle and multimode

(left and right)

0 2 4 6

U

2.5

5

7.5

10

12.5

15

17.5

20

0 2 4 6

U

2.5

5

7.5

10

12.5

15

17.5

20

0 2 4 6

U

2.5

5

7.5

10

12.5

15

17.5

20

0 2 4 6

U

2.5

5

7.5

10

12.5

15

17.5

20

0 0.5 1 1.5 2

U

2.5

5

7.5

10

12.5

15

17.5

20

0 0.5 1 1.5 2

U

2.5

5

7.5

10

12.5

15

17.5

20

0 0.5 1 1.5 2

U

2.5

5

7.5

10

12.5

15

17.5

20

0 0.5 1 1.5 2

U

2.5

5

7.5

10

12.5

15

17.5

20

108.2

0015.

5.1

m][ 1

m][ 5

0

=

=

=

=

=

V

n

n

µ

µa

co

δ

λ

72.7

02.

5.1

m][ 1

m][ 5

0

=

=

=

=

=

V

n

n

µ

µa

co

δ

λ


–Hybrid modesν

=0

ν=

1ν

=2

ν=

3ν

=4

m=1 m=2

m=1 m=1 m=2,3 m=4,5

m=1 m=2,3

m=1 m=2,3

m=1 m=2



Number of modes


–Hybrid modes

For weak guiding, nco ~ ncl and RHS and a new mode appears with every

zero of Jν. Taking the large argument approximation for J

( ) ( )42

2large

cos πνππν −− → akakJ rakzr r

Note similarity to the slab waveguide. For each n, the number of radial

modes will thus be approximately

π

πνπ42

−−akr# of modes m for each ν =

HE11 mode doesn’t cut off. First TE mode cuts in at U = V = first null of J0

SINGLE MODE CONDITION: V < 2.405

At cut off kra ~ V, so the cut off condition for each mode ν, m is

( )2νπ += mV

ν

mπ

V

Vπ

2

00

Total number of eigenvalues

2

π

V

Times two orientations

and polarizations

2

2

π

V

Which is the square of the slab case.



HE11 modePolarization 1


–Hybrid modes


arrows = Ex and Ey

Note that Ez ≠ 0.

Note that nearly linear

polarized.



Note that Hz ≠ 0.


of the screen.



HE11 modePolarization 2


arrows = Ex and Ey

Note that Ez ≠ 0.

Note that nearly linear

polarized.



Note that Hz ≠ 0.


of the screen.


–Hybrid modes



Hybrid mode shapesElectric fields, only one polarization and orientation

ν=1

ν=2

ν=3

ν=4

m=1 m=2


–Hybrid modes



LP modesDerivation of the characteristic equation

( )( )

( )( )

( )( )

( )( )

′+

′

′+

′=

+

WKW

WK

UJU

UJ

n

n

WKW

WK

UJU

UJ

WUn

N

cl

co

cl ν

ν

ν

ν

ν

ν

ν

νν22

22

2

11

Start with the complete characteristic equation

And let nco be nearly ncl. N must thus also be nearly ncl and the equation

simplifies to

( )( )

( )( )

22

22

2 11

′+

′=

+

WKW

WK

UJU

UJ

WU ν

ν

ν

νν

Using Bessel function identities like

( )( )

( )( )WKW

WK

UJU

UJ

ν

ν

ν

ν 110 ±± ±=

( ) ( ) ( ) ( ) ( )zJz

zJzJz

zJzJ ννννν

νν+−=−=′

+− 11

The LHS can be cancelled, leave a relation much like that of the TE/TM:

+ EH

- HE


–LP modes

The book simplifies this one more step, but I can’t get their formula, so I’ll

stop here. We can see from this expression that ν=0 yields our TE modes.

Since we’ve approximated nco ≈ ncl, TM0m must be degenerate with TE0m.

These modes are also degenerate with HE2m.

For ν≠0, the upper and lower signs yield different equations, associated

with the modes as shown. HEν+1, m is degenerate with EHν-1, m .

We may thus linearly combine modes within any degenerate set to create a

more convenient set of modes, still with the same effective index.



LP modesField distributions


–LP modes

+ =

TE01 HE21

Each degenerate set can be linearly combined to create a new set with

predominately linear polarizatoins.

The axial polarizations (Ez and Hz) are much smaller than the transverse.

This is consistent with the weakly guiding approximation – the angle of

propagation must be very near the axis (NA very small).

For modes with no angular variation, there are two orthogonal polarizations.

For modes with angular variation (e.g. sin(ν φ)), there is also a second

angular phase (e.g. cos(ν φ)), which is functionally orthogonal.



LP modesof a parabolic fiber

Michelson, Chapter 5

m=1 m=10

ν=

0ν

=9

Each mode ν>0 has two possible angular

phases and all have two possible polarizations


–LP modes



Summary• The solution of the wave-equation in cylindrical

coordinates yields J in the core (like cos) and K in the cladding (like e-αx).

• Matching these functions across the boundary with the EM boundary conditions yields a complex characteristic equation for β which has different solutions m for each angular mode number ν.

• If there is no angular variation (ν=0), the solutions are TE0m and TM0m.

• Otherwise the solutions involve all 6 field components and are labeled HEνm and HMνm.

• The HE11 mode is ~linearly polarized and never cuts off.

• The number of modes in terms of V is roughly equal to the number in a slab waveguide of the same parameters, squared.

• TE/TM modes occur for V > 2.405.

• Increases in the ν 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 degenerate polarizations and (for modes with angular variation) two angular phases.


–Summary

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