Fiber lecture

EE247 - Optical Fiber Communications

1/13/2003, # 1C

opyright O. Solgaard, 2003

Optical Fibers

Geometry of a step-index cylindrical waveguide (optical fiber) with core radius a. The cladding radius (125 µm for standard single mode fiber) is large enough that we can consider it infinite

a

CladdingCore

Optical fibers have revolutionized the communications industry due to their low loss and inexpensive fabricationOptical fibers are fabricated by pulling preformed rods with carefully designed index profiles into long fibersThis process is well controlled and result in highly uniform core and cladding diametersThe uniformity of the fibers is so good that for most practical purposes we consider the fiber a perfectly cylindrical waveguideWe will assume that the cladding is infinite, so the fiber can be characterized by three parameters:

The core radius, a, and the core and cladding refractive indices, ncore and ncladding

Together with the wave vector of the optical field, these parameters completely determine the propagation of light on the fiber.


1/13/2003, # 2

Copyright O

. Solgaard, 2003

The vector Laplacian is very complex in cylindrical coordinates because the radial and azimuthal fields cannot be decoupledThe z-components of the electric and magnetic fields are however not changed during propagation, so the eigenmodes of the cylindrical waveguide can be found from the scalar wave equation for Ez and Hz, and the other components of the fields are found from theseThe total fields will are required to meet the boundary conditions to find the complete field solutionsThe procedure is tedious, but conceptually very similar to the one we used for the simple slab waveguide

A

B

Z

Coupling of radial and azimuthal fields on a cylindrical coordinate system. The radial field component at cross section A, becomes azimuthal at cross section B. The z-component, on the other hand, is unchanged by propagation.

Wave equations in cylindrical coordinates

To find the optical fields propagating on the fiber we must solve the wave equation

02

22 =

∂∂

−∇tEEr

rµε which for time harmonic

fields becomes02

02 =+∇ EnkE

rr


1/13/2003, # 3C


Guided Modes on Step-Index FibersFor step-index fibers we find the longitudinal fields, and use these to express the radial and azimuthal components.The longitudinal field of the guided modes can be expressed:

( ) ( )( ) ( ) ..,,

..,,

cceerBJzrH

cceerAJzrEarzjj

z

zjjz

+⋅=

+⋅=<−⋅

−⋅

βφνν

βφνν

κφ

κφ

( ) ( )( ) ( ) ..,,

..,,

cceerDKzrH

cceerCKzrEarzjj

z

zjjz

+⋅=

+⋅=>−⋅

−⋅

βφνν

βφνν

γφ

γφ

where ν is the angular mode number, Jn(κr), are Bessel functions of the first kind and order n, Kn(γr) are modified Bessel functions of the second kind of order ν, κ is the transverse wave vector, and γ is the decay parameter in the cladding. These parameters are defined as

222 βκ −= corek

222cladk−= βγ

aV

aV

<<⇒−

= κκγ 02

22

222cladcore nnaV −

⋅=

λπ


1/13/2003, # 4

Copyright O

. Solgaard, 2003

Fiber ModesThere are four different kinds of guided modes on optical fibers:

( ) ( )( ) ( ) ..,,

..,,

cceerBJzrH

cceerAJzrE

ar

zjjz

zjjz

+⋅=

+⋅=

<

−⋅

−⋅

βφνν

βφνν

κφ

κφ

( ) ( )( ) ( ) ..,,

..,,

cceerDKzrH

cceerCKzrE

ar

zjjz

zjjz

+⋅=

+⋅=

>

−⋅

−⋅

βφνν

βφνν

γφ

γφ

A=0 -> TE modes (non-degenerate)B=0 -> TM modes (non-degenerate)A>B -> HE modes (Ez dominates

over Hz) (twofold-degenerate)A<B -> EH modes (Hz dominates

over Ez) (twofold-degenerate)The TE and TM modes are non-degenerate, while HE and EH modes each have two degenerate solutions due to the arbitrary azimuthal axis of originJν are Bessel functions of the first kind

They are oscillatory and can be thought of as equivalent to harmonic functions in cylindrical coordinatesThey are defined everywhere and are orthogonalThey describe the field distributions in the core of cylindrical waveguides

Kν are Modified Bessel functions of the 2nd kind They decay with increasing radial coordinateThey are the equivalent of exponentially decaying functions in cylindrical coordinatesThey describe the cladding field of guided modes


1/13/2003, # 5C


Eigenvalue EquationThe longitudinal wavevector is found from the characteristic equation of the step-index fiber

The Jn is oscillatory and gives several solutions indexed by the integer labels ν and m

ν the angular mode number (the number of 2π phase shifts of the field through one rotation around the fiber axis) m the radial mode number (corresponding to the number of radial nulls in the field distribution)

The longitudinal wavevector (β) for the modes is found by numerically or graphically solving the characteristic equationThe computation of the complete β−ω diagram, which determines the dispersion of the modes, is therefore quite time consumingOnce β is determined for a specific mode, we can calculate the complete field distribution of that mode

( )( )

( )( )

( )( )

( )( )

⋅⋅⋅′⋅

+⋅⋅

⋅′⋅

⋅⋅

⋅′+

⋅⋅⋅′

=

+

aKaKnk

aJaJnk

aKaK

aJaJ

acladcore

γγγ

κκκ

γγγ

κκκ

κγνβ

ν

ν

ν

ν

ν

ν

ν

ν22

022

02

222

22 11


1/13/2003, # 6

Copyright O

. Solgaard, 2003

Characteristic Equation(s) for EH, HE, TE, and TM Modes

The cylindrical waveguide is more complex than the slab waveguide (TE and TM) solutions, because the electric and magnetic fields in general have r, φ, and z componentsWe can group the modes into EH and HE modes

In HE modes, the longitudinal electric field dominateIn EH modes, the longitudinal magnetic field dominates

The characteristic equation is quadratic in J’n(κr)/Jn(κr), so we find two roots corresponding to the two types of modesSolving the characteristic equation for J’n(κr)/Jn(κr), we find

EH modes:( )( )

( )( )

−+

⋅⋅⋅′+

=⋅⋅

⋅+ RaaaK

aKnnn

aaJaJ

core

cladcore222

221

2 κν

γγγ

κκκ

ν

ν

ν

ν

HE modes:( )( )

( )( )

−+

⋅⋅⋅′

+−=

⋅⋅⋅− R

aaaKaK

nnn

aaJaJ

core

cladcore222

221

2 κν

γγγ

κκκ

ν

ν

ν

ν

where( )

( )

21

2

2222220

2222

2

22 112

+

⋅+

⋅⋅

⋅′

−=

κγνβ

γγγ

ν

ν

aankaaKaK

nnnR

corecore

cladcore


1/13/2003, # 7C


TE Modes

( )( )

( )( )aaKaK

aaJaJ

⋅⋅⋅

−=⋅⋅

⋅γγ

γκκ

κ

0

1

0

1

Consider the special case of ν=0In this case there is no azimuthal dependence, and all field components are radially symmetricThe characteristic equation for EH modes simplifies to

These mode does not have any longitudinal E field, but they do have longitudinal H fieldsThese are the TE modes of the cylindrical waveguideIf the longitudinal wave vector has the solutions βm, m=1,2,3…, then we designate the transversal electric field solutions as TE0m, where the first subscript indicates that only TE modes with ν=0 exist.

TE01

( ) ( )xKxK 10 −=′where we have used


1/13/2003, # 8

Copyright O

. Solgaard, 2003

TM Modes

Similarly, we find for the HE modes, using the relations

( ) ( )xKxK 10 −=′

( ) ( )xJxJ 11 −=−

( )( )

( )( )aaKaK

nn

aaJaJ

core

clad

⋅⋅⋅

−=⋅⋅

⋅γγ

γκκ

κ

0

12

2

0

1

These modes have no longitudinal magnetic fieldThey are called TM modes, designated TM0mThe characteristic equations can be solved graphically much like we solved the characteristic equation for the slab waveguide

Using the relations

TM01


1/13/2003, # 9C


2 4 6 8 10 12 14

-2

-1.5

-1

-0.5

0.5

1

1.5

2

Graphical Solutions to TE & TM

Plots of the left and right hand sides of the characteristic equations for TE and TM modes of cylindrical waveguides. The V-parameter is 10, and the ratio of the square of the indexes is 2 (which is unrealistically large, but illustrative of the differences between the TE and TM modes).

( )( )aaJaJ⋅⋅

⋅κκ

κ

0

1

( )( )aaKaK

nn

core

clad

⋅⋅⋅

−γγ

γ

0

12

2

( )( )aaKaK⋅⋅

⋅−

γγγ

0

1

κa


1/13/2003, # 10

Copyright O

. Solgaard, 2003

Cut-off (TE &TM)

We see that the left hand side of the equations go to infinity at each root of the equation J0(x)=0If we designate these roots x0m, then we have the following cut-off condition

220

022

00

2 cladcore

m

mcladcorem

nnxa

xnnakxV

−>

⇒>−⇒>

πλ

The first three roots of J0(x)=0 are

654.8520.5405.2

03

02

01

===

xxx

( )( )

( )( )aaKaK

aaJaJ

⋅⋅⋅

−=⋅⋅

⋅γγ

γκκ

κ

0

1

0

1 ( )( )

( )( )aaKaK

nn

aaJaJ

core

clad

⋅⋅⋅

−=⋅⋅

⋅γγ

γκκ

κ

0

12

2

0

1

Characteristic Equation - TE Characteristic Equation - TM

Higher order roots can be found from the asymptotic formula π

−≅

41

1 mx m


1/13/2003, # 11C


Hybrid ModesWhen n≠0, the modes are neither TE nor TM, and all their field components are non-zeroGraphical representations of the characteristic equation are used to find the longitudinal wave vectors or the modesThe graphical solution to the EH modes are not conceptually different from TE and TM modes (only more complex), and the same is true for HE modes of azimuthal eigenvalues ν>1With ν=1, however, the HE modes becomes interesting and deserve closer investigation

2 4 6 8 10 12 14

-4

-2

2

4Graphical solutions of the characteristic equation for HE1m modes

Parameters: ncore=1.4514nclad=1.4469V=10

Notice that the HE11 mode exist for all values of V

( )( )aaJaJ⋅⋅

⋅κκ

κ

1

1κa

( )( )

−+

⋅⋅⋅′

+− R

aaaKaK

nnn

core

cladcore22

1

12

22 12 κγγ

γ


1/13/2003, # 12

Copyright O

. Solgaard, 2003

Cut-Off (HE1m and EH1m)For large V-numbers, we have many possible modesUnlike the TE and TM modes, the fundamental HE1m mode (HE11) is never cut off! The other HE and EH modes all reach cut-off as the V-parameter value is decreasedFor ν=1, the left hand side of the characteristic equation go to infinity at each root of the equation J1(x)=0, which we designate x1mThe cut-off condition for modes HE1(m+1) and EH1m is then:

221

12 cladcore

mm

nnxaxV

−>⇒>

πλ

The first three roots of J1(x)=0 are

173.10016.7832.3

13

12

11

===

xxx

Higher order roots can be found from the asymptotic formula π

+≅

41

1 mx m


1/13/2003, # 13C


Cut-Off (HEνm & Ehνm – ν>1)For ν >1, the cut-off conditions for EHnm modes are

222 cladcore

m

nnxa

−>

πλν

where xnm are the roots of the equation Jν(x)=0

For the HE nm modes the cut-off conditions are

222 cladcore

m

nnza

−>

πλν

where z nm are the roots of the equation

( ) ( ) 1 1)1( 12

2>

+−= − νν νν zJnnzzJclad

core


1/13/2003, # 14

Copyright O

. Solgaard, 2003

Total Number of Guided Modes

The left hand side of the characteristic equation of the HE and EH modes has the denominator Jn(κa). Each zero of this denominator corresponds to modes with an extra radial null. For large values of the argument, the Bessel function can be approximated as

( )

−−⋅

⋅≈⋅

42cos2 πνπκ

πκκν a

aaJ

This function has a null for each increase of π of the argument. The total number of radial nulls is then

( ) ( )2

22

5.02

424

421

max

πνπν

πνππ

πνπκπ

+≈++≈⇒

−−=

−−⋅≈

mmV

Vam

Each combination of a radial null and an azimuthal eigenvalue corresponds to four modes (two polarizations, and two angular orientations). The total number of modes is then 2

24

πVN ≈

We then have that mmax=V/π and nmax=2V/π, so that 0.5(mmax·nmax)=V2/π2


1/13/2003, # 15C


Modes on Step-index Optical Fibers

ncladd

ncore

β/k

Normalized longitudinal wavevector (propagation constant) for the lower order modes of a step-index fiber as a function of normalized frequency (after D. Gloge, Applied Optics, vol. 10, pp. 2252-2258, 1971.)


1/13/2003, # 16

Copyright O

. Solgaard, 2003

Field Distribution of Fiber Modes

Schematic representation of the transversal electric field of the two sets of lowest order exact modes on a step-index optical fiber.

HE11 HE21

TE01 TM01


1/13/2003, # 17C


Linearly Polarized ModesWeakly guiding optical fibers (i.e. fibers with a small refractive-index difference between the core and cladding) have modes that are, to a good approximation, transverse electro-magnetic waves. In this approximation, the amplitude of the electric field of the fiber modes can be described in the following way:

Core (r < a):

Cladding (r > a):

where Jl is the Bessel function and Kl the modified Bessel function of the second kind. The parameters h and q are found from the equations:

( )( )

( ) ( )φlrqKaqKahJEE l

l

lx cos0 ⋅

⋅⋅

=

( ) ( )φlrhJEE lx cos0 ⋅=

( )( )

( )( )

( ) ( )222222

11

2 claddingcore

l

l

l

l

nnaVqha

aqKaqKq

ahJahJh

−

==+

⋅⋅

±=⋅⋅ ±±

λπ


1/13/2003, # 18

Copyright O

. Solgaard, 2003

Field Distribution in LP01

At 670 nm wavelength V = 4.236. The characteristic equation has 4 solutions

2 4 6 8 10

0.2

0.4

0.6

0.8

1

LP01: l = 0, h = 0.488154, q = 0.953337,

Core (r < a):

Cladding (r > a):

-7.5 -5 -2.5 0 2.5 5 7.5

-7.5

-5

-2.5

0

2.5

5

7.5

( )rhJEEx ⋅= 00( )( )

( )rqKaqKahJEEx ⋅

⋅⋅

= 00

00

Electrical field amplitude of LP01 as a function of radius [µm]

Contour plot of LP01 mode at 670 nm


1/13/2003, # 19C



There are four degenerate LP11modes. In addition to the one shown (two orthogonal polarizations), there are two orthogonally polarized modes with a sinφ angular dependence.

2 4 6 8 10 12 14

0.2

0.4

0.6

0.8

1

-7.5 -5 -2.5 0 2.5 5 7.5

-7.5

-5

-2.5

0

2.5

5

7.5

( ) φcos10 rhJEEx ⋅=

( )( )

( ) φcos11

10 rqK

aqKahJEEx ⋅

⋅⋅=

LP11: l = 1, h = = 0.76807, q = 0.746468,

Core (r < a):

Cladding (r > a):




1/13/2003, # 20

Copyright O

. Solgaard, 2003


LP21: l = 2, h = 1.00806325, q = 0.361877

Core (r < a):

Cladding (r > a):

2 4 6 8 10 12 14

0.2

0.4

0.6

0.8

1

( ) ( )φ2cos20 rhJEEx ⋅=

( )( )

( ) ( )φ2cos22

20 rqK

aqKahJEEx ⋅

⋅⋅

=


-7.5 -5 -2.5 0 2.5 5 7.5

-7.5

-5

-2.5

0

2.5

5

7.5



1/13/2003, # 21C


Field Distribution in LP02LP02: l = 0, h = 1.0482116, q = 0.219998,

Core (r < a):

Cladding (r > a):

2 4 6 8 10 12 14

-0.4

-0.2

0.2

0.4

0.6

0.8

1

-7.5 -5 -2.5 0 2.5 5 7.5

-7.5

-5

-2.5

0

2.5

5

7.5

( )rhJEEx ⋅= 00

( )( )

( )rqKaqKahJEEx ⋅

⋅⋅

= 00

00



Counting all polarizations and both helical polarities of the LP11 mode, we find a total of 12 modes for V = 4.236. Not all of these produce distinguishable intensity patterns. One has to be careful not to confuse patterns produced by combinations of modes with true modes


1/13/2003, # 22

Copyright O

. Solgaard, 2003

Comparison of exact and LP modes

nclad

ncore

β/k

Normalized longitudinal wavevector (propagation constant) for the lower order modes of a step-index fiber as a function of normalized frequency

Linearly Polarized ModesExact Modes

Normalized propagation constant (note definition) for the lower order LP modes of a step-index fiber as a function of normalized frequency

After D. Gloge, Applied Optics, vol. 10, pp. 2252-2258, 1971.

(β/k

-ncl

ad)/(

n cor

e-n c

lad)


1/13/2003, # 23C


The Fundamental Mode of Optical Fibers

The HE11 mode of the cylindrical waveguide is guided for all values of the normalized frequency (V-number)When the V-number is less than 2.405, the HE11 mode is the only guided mode! The cut-off condition is often rewritten in terms of the wavelength

Standard single mode fibers:

Cut off: 1,180nmThe HE11 mode is of special importance, because single-mode operation is preferred in high-capacity communication systemsThe field distribution of the HE11 mode is used in calculations of mode propagation, dispersion, coupling, switching, cross talk, and modulationThe difficulty of these calculations is often substantial because of the complex nature of the HE11 mode.

1.831= 24469.124514.155.1

955.32−

⋅=

mmV

µµπ

405.22

405.222

220

cladcoreccladcore

nnannakV −⋅=>⇒<−=

πλλ


1/13/2003, # 24

Copyright O

. Solgaard, 2003

Gaussian Approximation to HE11

The HE11 mode can be approximated as a Gaussian functionThis approximation is sufficiently accurate for the majority of fiber mode calculationsExceptions include cross calculations, which depend critically on the power in the tails of the mode profile In the Gaussian approximation, the mode field is

where the 1/e-field beam radius (1/e2 intensity), ω, is chosen to give the best match to the HE11 mode shape. For a step-index fiber with radius a, the beam radius is

The Mode Field Diameter is defined as twice the Gaussian beam radius

( )

−=

2exp

ωrErE x

r

65.1 87.2619.165.0 −− ⋅+⋅+= VVaω


1/13/2003, # 25C


Gaussian vs. LP01 mode

Comparison of LP01 mode (blue) and the Gaussian approximation (green) to the HE11 modeThe two mode shapes are well matched, but that the Gaussian falls off quicker at large radii The energy in the tail is important in cross-talk calculations

2 4 6 8 10 12 14

0.2

0.4

0.6

0.8

1Fiber parameters:

V-number: V=2.166Core radius: a=3.955µmWavelength: λ=1.310 µm

r/ω


1/13/2003, # 26

Copyright O

. Solgaard, 2003

Power Confinement

As for the slab waveguide, the total power in a mode is calculated by integrating

( )HESzrr

×= Re21

over the cross sectional area of the waveguide. In the weakly guiding limit, the ratio of the power carried in the core to the total power is

( )( ) ( )

⋅⋅⋅

−

−=

−+ aKaKaK

Va

PPtotal

coreγγ

γκ

νν

ν

11

2

2

2211

and

total

core

total

cladPP

PP

−=1

As in the slab waveguide we studied before, the power confinement factor for a given mode is very low at the cut-off for the mode, and increases with increasing V-number above cut-off.


1/13/2003, # 27C


SummaryAs with slab waveguides, we find the modes of fibers by solving the wave equation in each homogeneous region and apply the boundary conditions

The cylindrical geometry leads to Bessel function solutions, notharmonic and exponentialsCoupling of polarizations make the math more complex

Optical fibers are characterized by their V-number:

In the weakly-guiding regime, the LP modes are good approximationsThe fiber has a single mode (HE11, LP01) if V<2.405The HE11 (LP01) mode can be approximated as a Gaussian

( )

−=

2exp

ωrErE x

r 65.1 87.2619.165.0 −− ⋅+⋅+= VVaω

220 cladcore nnakV −=

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